diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmeij" "b/data_all_eng_slimpj/shuffled/split2/finalzzmeij" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmeij" @@ -0,0 +1,5 @@ +{"text":"\\section{Equations with the quadratic gradient nonlinearity} \\label{Sec:Analysis_fJMGT_K_III}\nUnlike its Westervelt version, the Kuznetsov versions of these time-fractional equations contain a quadratic gradient nonlinearity, and so their analysis requires the use of higher-order energy estimates. We thus limit our presentation to the analysis of the fJMGT--K III equation, which has the integer-order leading term. \\\\\n\\indent The study of well-posedness for this equation follows by combining the ideas from the previous section concerning the fractional term with the ideas used in the analysis of its integer-order counterpart considered in~\\cite{kaltenbacher2021inviscid, KaltenbacherNikolic}. To formulate the well-posedness result we introduce the solution space\n\\begin{equation}\n\\begin{aligned}\nX_{\\textup{fMGT--K III}}=&\\, \\begin{multlined}[t] \\left\\{\\vphantom{H^{-\\alpha\/2}(0,T; {H_\\diamondsuit^3(\\Omega)})}\\psi \\in L^{\\infty}(0,T; {H_\\diamondsuit^3(\\Omega)}): \\psi_t \\in L^{\\infty}(0,T; {H_\\diamondsuit^3(\\Omega)}),\\right. \\\\ \\left. \\hspace*{-0.3cm} \\psi_{tt} \\in L^\\infty(0,T; {H_\\diamondsuit^2(\\Omega)}) \\cap H^{-\\alpha\/2}(0,T; {H_\\diamondsuit^3(\\Omega)}), \\ \\psi_{ttt} \\in L^2(0,T; H_0^1(\\Omega)) \\right\\} \n\\end{multlined}\n\\end{aligned}\n\\end{equation}\nfor $ \\alpha \\in (0,1)$, and\n\\begin{equation}\n\\begin{aligned}\nX_{\\textup{fMGT--K III}}= H^{3}(0,T; H_0^1(\\Omega)) \\cap W^{2, \\infty}(0,T; {H_\\diamondsuit^2(\\Omega)}) \\cap W^{1, \\infty}(0,T; {H_\\diamondsuit^3(\\Omega)}),\n\\end{aligned}\n\\end{equation}\nfor $\\alpha=1$, {equipped with the norm $\\|\\cdot\\|_{X_{\\textup{fMGT--K III}}}$.}\n\\begin{theorem}[Local well-posedness of the fJMGT--K III equation] \n\t\\label{Thm:fJMGT_K_III} Let $\\alpha \\in (0,1]$, $\\tilde{T}>0$, and $\\varrho>0$. Further, assume that $f \\in H^1(0,\\tilde{T}; H_0^1(\\Omega))$ and that\n\t\\begin{equation}\n\t\\|f\\|^2_{H^1(H^1)} +\\|\\psi_0\\|_{H^3}^2+ \\|\\psi_1\\|_{H^3}^2+\\|\\psi_2\\|_{H^2}^2 \\leq \\varrho^2. \n\t\\end{equation}\n\tThen there exists $T=T(\\varrho) \\leq \\tilde{T}$, such that the initial boundary-value problem\n\t\\begin{equation}\\label{ibvp_fJMGT_K_III}\n\t\\left \\{\n\t\\begin{aligned}\n\t\\tau \\psi_{ttt} + &(1+2\\tilde{k} \\psi_t) \\psi_{tt} - c^2\\Delta\\psi -\\tau c^2\\Delta \\psi_t && \\\\[1mm]\n\t&\\hspace*{2.5cm} - \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\psi_t+\\tilde{\\ell}\\partial_t |\\nabla \\psi|^2= f &&\\text{ in }\\Omega\\times(0,T),\\\\[1mm]\n\t&\\psi=\\,0&&\\text{ on } \\partial \\Omega\\times(0,T),\\\\[1mm]\n\t&(\\psi, \\psi_t, \\psi_{tt})=\\,(\\psi_0, \\psi_1, \\psi_2)&&\\mbox{ in }\\Omega\\times \\{0\\},\n\t\\end{aligned} \\right.\n\t\\end{equation}\n\thas a unique solution $\\psi \\in X_{\\textup{fMGT--K III}}$, which satisfies\t\n\t\\begin{equation} \\label{energy_est2_fJMGT_K_III}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] \\|\\psi\\|^2_{X_{\\textup{fMGT--K III}}}\n\t\\end{multlined}\n\t\\lesssim\\,\\begin{multlined}[t] \t\\|f\\|^2_{H^1(H^1)}+\\|\\psi_0\\|_{H^3}^2+ \\|\\psi_1\\|_{H^3}^2+\\|\\psi_2\\|_{H^2}^2. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tThe proof follows by employing the Banach Fixed-point theorem to the mapping $\\mathcal{T}: w \\mapsto \\psi$, where $\\psi$ solves\n\t\\begin{equation} \\label{fMGT_K_III_linearized}\n\t\\begin{aligned}\n\t\\tau \\psi_{ttt}+(1+2\\tilde{k}w_t)\\psi_{tt} - c^2\\Delta \\psi -\\tau c^2\\Delta \\psi_t- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\psi_t + 2 \\tilde{\\ell}\\nabla w \\cdot\\nabla\\psi_t=f,\n\t\\end{aligned}\n\t\\end{equation}\n\tand\n\t\\begin{equation} \\label{defBR}\n\t\\begin{aligned}\n\tw \\in B_R:=\\{w \\in X_{\\textup{fMGT--K III}}\\, :&\\, \\|w\\|_{X_{\\textup{fMGT--K III}}}\\leq R,\\\\ &w(0)=\\psi_0, \\, w_t(0)=\\psi_1, \\, w_{tt}(0)=\\psi_2 \\}.\n\t\\end{aligned}\n\t\\end{equation}\t\n\t\\noindent (I) The energy estimates for the linear equation can be rigorously derived by a Galerkin procedure with a sufficiently smooth basis; here we present only the derivation of the bound for the semi-discrete solution and omit the superscript $n$ below. We denote \\[p=1+2\\tilde{k}w_t, \\quad \\phi=2 \\tilde{k}w,\\]\n\tthen test the semi-discrete version of \\eqref{fMGT_K_III_linearized} with $\\Delta^2 \\psi_{tt}$ and integrate in space. We can estimate the resulting non-fractional terms and those not involving $f$ in a similar manner to~\\cite[Theorem 6.1]{kaltenbacher2021inviscid}. We include the derivation of these bounds below for completeness. \\\\\n\t\\indent Note that $\\psi_{tt}= \\Delta \\psi=\\Delta \\psi_{tt}=0$ and ${\\textup{D}}_t^{2-\\alpha} \\Delta \\psi=0$ on $\\partial \\Omega$ for smooth Galerkin approximations based on the eigenfunctions of the Dirichlet-Laplacian. Therefore, the following identities hold:\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\prodLtwo{p \\psi_{tt}}{\\Delta^2 \\psi_{tt}} =&\\,\\prodLtwo{p \\Delta \\psi_{tt}+\\psi_{tt} \\Delta p +2\\nabla p\\cdot\\nabla \\psi_{tt}}{\\Delta \\psi_{tt}},\\\\\n\t-c^2\\prodLtwo{\\Delta \\psi}{\\Delta^2 \\psi_{tt}} =&\\, c^2\\frac{\\textup{d}}{\\textup{d}t} \\prodLtwo{\\nabla \\Delta \\psi}{\\nabla \\Delta \\psi_t}-c^2\\nLtwo{\\nabla \\Delta \\psi_t}^2.\n\t\\end{aligned}\n\t\\end{equation}\n\tWe thus have \n\t\\begin{equation} \\label{LinKuzn_id}\n\t\\begin{aligned}\n\t&\\begin{multlined}[t]\\frac12 \\tau\\frac{\\textup{d}}{\\textup{d}t} \\nLtwo{\\Delta \\psi_{tt}}^2 +\\frac12\\tau c^2\\frac{\\textup{d}}{\\textup{d}t} \\nLtwo{\\nabla \\Delta \\psi_t}^2+\\delta \\prodLtwo{\\Dt^{2-\\alpha}\\nabla \\Delta \\psi}{\\nabla \\Delta \\psi_{tt}}\n\t\\end{multlined}\\\\\n\t=&\\,\\begin{multlined}[t] -\\prodLtwo{p \\Delta \\psi_{tt}+ \\psi_{tt} \\Delta p +2\\nabla p\\cdot\\nabla \\psi_{tt}}{\\Delta \\psi_{tt}}-c^2\\frac{\\textup{d}}{\\textup{d}t} \\prodLtwo{\\nabla \\Delta \\psi}{\\nabla \\Delta \\psi_t}\\\\[1mm]+c^2\\nLtwo{\\nabla \\Delta \\psi_t}^2 \n\t- \\prodLtwo{\\nabla \\phi \\cdot \\nabla \\psi_t}{\\Delta^2 \\psi_{tt}}+\\prodLtwo{\\nabla f}{\\nabla \\Delta \\psi_{tt}}.\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tWe next integrate in time and estimate the resulting terms. Note first that\n\t\\[\n\t\\begin{aligned}\n\t&\\int_0^t \\prodLtwo{p \\Delta \\psi_{tt}+\\psi_{tt} \\Delta p +\\nabla p\\cdot\\nabla \\psi_{tt}}{\\Delta \\psi_{tt}}\\, \\textup{d} s \\\\\n\t\\lesssim&\\,\\begin{multlined}[t] \\left \\{(1+R) \\nLtwotLtwo{\\Delta \\psi_{tt}}+\\nLtwotLinf{\\psi_{tt}}\\nLinftLtwo{\\Delta p}\\right. \\\\ \\left. +\\nLinfLfour{\\nabla p}\\nLtwotLfour{\\nabla \\psi_{tt}} \\right \\} \\nLtwotLtwo{\\Delta \\psi_{tt}} \\end{multlined} \\\\\n\t\\lesssim&\\, \\left \\{(1+R) \\nLtwotLtwo{\\Delta \\psi_{tt}}+R\\nLtwotLinf{\\psi_{tt}}+R\\nLtwotLfour{\\nabla \\psi_{tt}} \\right \\} \\nLtwotLtwo{\\Delta \\psi_{tt}},\n\t\\end{aligned}\n\t\\]\n\twhere we have utilized the uniform boundedness of $p=1+2 \\tilde{k}w_t$, which follows from the fact that $w \\in B_R$. Furthermore,\n\t\\[\n\t\\begin{aligned}\n\t-c^2\\int_0^t\\frac{\\textup{d}}{\\textup{d}t} \\prodLtwo{\\nabla \\Delta \\psi}{\\nabla \\Delta \\psi_t}\\, \\textup{d} s =&\\,-c^2\\prodLtwo{\\nabla \\Delta \\psi(t)}{\\nabla \\Delta \\psi_t(t)}+c^2\\prodLtwo{\\nabla \\Delta \\psi_0}{\\nabla \\Delta \\psi_1}\\\\\n\t\\leq&\\,\\begin{multlined}[t] \\frac{1}{2 \\epsilon} T c^4\\nLtwotLtwo{\\nabla \\Delta \\psi_{t}}^2+\\frac{\\epsilon}{2} \\nLtwo{\\nabla \\Delta \\psi_{t}(t)}^2\\\\\n\t+\\frac12c^4 \\nLtwoLtwo{\\nabla \\Delta \\psi_0}^2+\\frac12 \\nLtwoLtwo{\\nabla \\Delta \\psi_1}^2.\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\]\n\tBy the semi-discrete PDE, we know that $\\nabla \\phi \\cdot \\nabla \\psi_t=0$ on $\\partial \\Omega$. Thus\n\t\\[\n\t\\begin{aligned}\n\t\\prodLtwo{\\nabla \\phi \\cdot \\nabla \\psi_t}{\\Delta^2 \\psi_{tt}} =&\\,\\prodLtwo{\\Delta[\\nabla \\phi \\cdot \\nabla \\psi_t]}{\\Delta \\psi_{tt}} \\\\\n\t=&\\, \\prodLtwo{\\nabla \\Delta \\phi\\cdot\\nabla \\psi_t+ 2 D^2 \\phi:D^2 \\psi_t+ \\nabla \\phi\\cdot\\nabla \\Delta \\psi_t}{\\Delta \\psi_{tt}}\n\t\\end{aligned}\n\t\\]\t\n\twhere $D^2 v=(\\partial_{x_i} \\partial_{x_j} v)_{i,j}$ is the Hessian, which satisfies\n\t\\[\n\t\\nLfour{D^2 v}\\leq C_{H^1, L^4} (\\nLtwo{D^3 v}+\\nLtwo{D^2 v})\\leq C_{H^1, L^4} C_{\\textup{H}}(\\nLtwo{\\nabla\\Delta v}+\\nLtwo{\\Delta v}).\n\t\\]\n\tThis further implies that\n\t\\[\n\t\\begin{aligned}\n\t&\\int_0^t \\prodLtwo{\\nabla \\phi \\cdot \\nabla \\psi_t}{\\Delta^2 \\psi_{tt}}\\, \\textup{d} s \\\\\n\t\\lesssim&\\, \\begin{multlined}[t] \\nLtwotLtwo{\\Delta \\psi_{tt}} \\left\\{\\vphantom{C_{H^1, L^4}^2}\\nLinfLtwo{\\nabla \\Delta \\phi}\\nLtwotLinf{\\nabla \\psi_{t}}+\\nLinfLinf{\\nabla \\phi}\\nLtwotLtwo{\\nabla \\Delta \\psi_t}\\right.\\\\ \\left. + (\\nLinfLtwo{\\nabla \\Delta \\phi}+\\nLinfLtwo{\\Delta \\phi})(\\nLtwotLtwo{\\nabla \\Delta \\psi_{t}}+\\nLtwotLtwo{\\Delta \\psi_{t}})\\right\\}.\\end{multlined}\n\t\\end{aligned}\n\t\\]\n\tSince $w \\in B_R$, the function $\\phi$ is uniformly bounded, and so\n\t\\[\n\t\\begin{aligned}\n\t\\int_0^t \\prodLtwo{\\nabla \\phi \\cdot \\nabla \\psi_t}{\\Delta^2 \\psi_{tt}}\\, \\textup{d} s \n\t\\lesssim&\\, \\begin{multlined}[t] R \\nLtwotLtwo{\\Delta \\psi_{tt}} \\left\\{\\vphantom{C_{H^1, L^4}^2}\\nLtwotLinf{\\nabla \\psi_{t}}+\\nLtwotLtwo{\\nabla \\Delta \\psi_t}\\right.\\\\ \\left. +\\nLtwotLtwo{\\nabla \\Delta \\psi_{t}}+\\nLtwotLtwo{\\Delta \\psi_{t}}\\right\\}.\\end{multlined}\n\t\\end{aligned}\n\t\\]\n\tIntegration by parts with respect to time yields\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\int_0^t \\prodLtwo{\\nabla f}{\\nabla \\Delta \\psi_{tt}}\\, \\textup{d} s =&\\, \\prodLtwo{\\nabla f}{\\nabla \\Delta \\psi_{t}}\\big \\vert_0^t-\\int_0^t \\prodLtwo{\\nabla f_t}{\\nabla \\Delta \\psi_{t}}\\, \\textup{d} s \\\\\n\t\\leq&\\, \\begin{multlined}[t]\n\t\\frac{1}{4 \\epsilon}\\nLtwo{\\nabla f(t)}^2+\\epsilon \\nLtwo{\\nabla \\Delta \\psi_{t}(t)}^2-\\prodLtwo{\\nabla f(0)}{\\nabla \\Delta \\psi_1} \\\\\n\t+\\frac12 \\nLtwoLtwo{\\nabla f_t}^2+\\frac12 \\nLtwotLtwo{\\nabla \\Delta \\psi_t}^2. \n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tThe $\\delta$ fractional term can be handled by relying on estimate \\eqref{fractional_est_w} similarly to before. Fixing $\\epsilon>0$ small enough and combining the derived bounds leads to\n\t\\begin{equation} \n\t\\begin{aligned}\n\t&\\begin{multlined}[t]\\frac12 \\tau \\nLtwo{\\Delta \\psi_{tt}(t)}^2 + (\\frac12\\tau c^2-2\\epsilon) \\nLtwo{\\nabla \\Delta \\psi_t(t)}^2\n\t+\\int_0^t \\nLtwo{{\\textup{D}}_t^{2-\\alpha}\\nabla \\Delta \\psi}^2\\, \\textup{d} s \\end{multlined}\\\\\n\t\\lesssim&\\,\\begin{multlined}[t]\\nLtwo{\\Delta \\psi_2}^2 + \\nLtwo{\\nabla \\Delta \\psi_1}^2 +\\nLtwo{\\nabla \\Delta \\psi_0}^2 \\\\\n\t+ R^2(\\|\\psi_{tt}\\|_{L^2_t(H^2)}^2+\\nLtwotLfour{\\nabla \\psi_{tt}}^2) +\\nLtwotLtwo{\\Delta \\psi_{tt}}^2\\\\\n\t+\\left. R^2\\nLtwotLinf{\\nabla \\psi_{t}}^2+(1+R^2+T)\\nLtwotLtwo{\\nabla \\Delta \\psi_t}^2\\right. +R^2\\nLtwotLtwo{\\Delta \\psi_{t}}^2+\\|\\nabla f\\|^2_{H^1(L^2)}. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tNote that by elliptic regularity, we have\n\t\\[\n\t\\nLtwo{\\psi_{tt}(t)} \\leq C_{\\textup{PF}} \\nLtwo{\\nabla \\psi_{tt}(t)} \\leq \\nHtwo{\\psi_{tt}(t)} \\lesssim \\nLtwo{\\Delta \\psi_{tt}(t)}.\n\t\\]\n\tAn application of Gronwall's inequality yields \n\t\\begin{equation} \\label{interim_bound_linKuzn}\n\t\\begin{aligned}\n\t&\\sup_{t \\in (0,\\tilde{T})}\\nLtwo{\\Delta \\psi_{tt}(t)}^2+\\sup_{t \\in (0,\\tilde{T})}\\nLtwo{\\nabla \\Delta \\psi_{t}(t)}^2 +\\int_0^t \\nLtwo{{\\textup{D}}_t^{2-\\alpha}\\nabla \\Delta \\psi}^2\\, \\textup{d} s \\\\\n\t\\leq&\\, C(T, R) (\\|\\nabla f\\|^2_{H^1(H^1)}+\\nLtwo{\\Delta \\psi_2}^2 + \\nLtwo{\\nabla \\Delta \\psi_1}^2 + \\nLtwo{\\nabla \\Delta \\psi_0}^2).\n\t\\end{aligned}\n\t\\end{equation}\n\tThe uniqueness follows by using $\\psi_{tt}$ as the test function in the homogeneous problem. \\\\\n\t\n\n\t\\noindent (II) It is straightforward to check now that $\\TK$ is a well-defined self-mapping. We thus focus on proving strict contractivity. Take $w^{(1)}, w^{(2)} \\in B_R$ and set $\\psi^{(1)}=\\TK \\phi^{(1)}$ and $\\psi^{(2)}=\\TK \\phi^{(2)}$. Then the difference $\\overline{\\psi}=\\psi^{(1)}-\\psi^{(2)}$ solves\n\t\\begin{equation} \\label{Kuzn_contractivity_eq}\n\t\\begin{aligned}\n\t\\begin{multlined}[t]\n\t\\tau \\overline{\\psi}_{ttt}+(1+2 \\tilde{k}w_t^{(1)} )\\overline{\\psi}_{tt}-c^2 \\Delta \\overline{\\psi}-\\tau c^2 \\Delta \\overline{\\psi}_t -\\delta {\\textup{D}}_t^{2-\\alpha} \\Delta \\overline{\\psi} \n\t+ 2 \\tilde{\\ell}\\nabla w^{(2)} \\cdot \\nabla \\overline{\\psi}_t=\\tilde{f}. \n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\twith the right-hand side\n\t\\[\n\t\\tilde{f} = -2\\tilde{k} \\overline{w}_t \\psi^{(2)}_{tt}- 2 \\tilde{\\ell}\\nabla \\overline{w} \\cdot \\nabla \\psi^{(1)}_t\n\t\\]\n\tand satisfies zero initial conditions. Testing with $\\overline{\\psi}_{tt}$ yields, after standard manipulations,\n\t\\begin{equation} \\label{Contractivity_identity}\n\t\\begin{aligned}\n\t\\nLtwo{\\overline{\\psi}_{tt}(t)}^2+ \\nLtwo{\\nabla \\overline{\\psi}_t(t)}^2+\\nLtwo{\\nabla \\overline{\\psi}(t)}^2 \\leq C(\\tilde{T}, R) \\nLtwoLtwo{\\tilde{f}}^2;\n\t\\end{aligned}\n\t\\end{equation}\n\tsee also estimate (5.5) in~\\cite{kaltenbacher2021inviscid}. It remains to bound the source term. By H\\\"older's inequality, we have\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\nLtwoLtwo{\\tilde{f}}\n\t\\lesssim&\\,\\begin{multlined}[t] \\nLinfLfour{\\psi_{tt}^{(2)}}\\nLtwoLfour{\\overline{w}_t} + \\nLinfLinf{\\nabla \\psi_t^{(1)}}\\nLtwoLtwo{\\nabla \\overline{w} }.\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tThe first term on the right can be further bounded as follows:\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\nLinfLfour{\\psi_{tt}^{(2)}}\\nLtwoLfour{\\overline{w}_t}\n\t\\leq\\, C_{H^1, L^4}^2 \\nLinfLtwo{\\nabla \\psi_{tt}^{(2)}}T\\nLinfLtwo{\\nabla \\overline{w}_t}.\n\t\\end{aligned}\n\t\\end{equation}\n\tBy noting that $\\nLtwoLtwo{\\nabla \\overline{w} } \\leq \\tilde{T} \\nLtwoLtwo{\\nabla \\overline{w}_t}$, it further follows that\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\nLtwoLtwo{\\tilde{f}}^2\n\t\\lesssim&\\,\\begin{multlined}[t] \\nLinfLtwo{\\nabla \\psi_{tt}^{(2)}}^2 \\tilde{T}^2\\nLinfLtwo{\\nabla \\overline{w}_t}^2+\\nLinfLinf{\\nabla \\psi_t^{(1)}}^2\\tilde{T}^2\\nLtwoLtwo{\\nabla \\overline{w_t} }^2.\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tEmploying this bound in \\eqref{Contractivity_identity} and relying on Gronwall's inequality leads to\n\t\\begin{equation} \n\t\\begin{aligned}\n\t&\\sup_{t \\in (0,\\tilde{T})}\\nLtwo{\\overline{\\psi}_{tt}(t)}+\\sup_{t \\in (0,\\tilde{T})}\\nLtwo{\\nabla \\overline{\\psi}_t(t)} \\\\\n\t\\leq&\\,\\begin{multlined}[t] C(T, R)\\tilde{T}(\\nLinfLtwo{\\nabla \\psi_{tt}^{(2)}}+\\nLinfLinf{\\nabla \\psi_t^{(1)}})\\sup_{t \\in (0,\\tilde{T})}\\nLtwo{\\nabla \\overline{w}_t(t)}. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tBy the energy estimate for the linear problem, we know that\n\t\\[\n\t\\nLinfLtwo{\\nabla \\psi_{tt}^{(2)}}+\\nLinfLinf{\\nabla \\psi_t^{(1)}} \\leq \\sqrt{\\tilde{C}(\\tilde{T}, R)}\\, r\n\t\\]\n\tfor some $\\tilde{C}(\\tilde{T}, R)>0$, independent of $\\alpha$. Thus we can achieve strict contractivity of $\\TK$ with respect to the $\\|\\cdot\\|_{X^{\\textup{low}}_{\\textup{fMGT III}}}$ norm by reducing the final time $\\tilde{T}$. The claim then follows by the Banach Fixed-point theorem.\n\\end{proof}\nWe next discuss the limit of this equation with respect to the order of differentiation. Given $\\alpha \\in (0,1)$, under the assumptions of Theorem~\\ref{Thm:fJMGT_W_III}, let $\\psi^\\alpha$ be the solution of the fJMGT--K III equation and let $\\psi$ solve the corresponding JMGT--Kuznetsov equation obtained by setting $\\alpha=1$ in \\eqref{ibvp_fJMGT_K_III}. Then the difference $\\overline{\\psi}=\\psi^{\\alpha}-\\psi$ solves \n\\begin{equation} \\label{fJMGT_W_III_diff}\n\\begin{aligned}\n&\\begin{multlined}[t]\\tau \\overline{\\psi}_{ttt} + (1+2\\tilde{k}\\psi_t^{\\alpha})\\overline{\\psi}_{tt} - c^2\\Delta\\overline{\\psi} -\\tau c^2\\Delta \\overline{\\psi}_t - \\delta \\Dt^{1-\\alpha} \\Delta \\overline{\\psi}_t\\\\\\hspace*{2cm}+2\\tilde{k}\\psi_{tt}\\overline{\\psi}_t+2\\tilde{\\ell} \\nabla \\overline{\\psi} \\cdot \\nabla \\psi_t^{\\alpha}+2\\tilde{\\ell} \\nabla \\psi \\cdot \\nabla \\overline{\\psi}_t \n=\\, \\delta (\\Dt^{1-\\alpha}\\Delta \\psi_t-\\Delta \\psi_t).\\end{multlined}\n\\end{aligned}\n\\end{equation}\nSimilarly to the proof of the previous theorem, testing with $\\overline{\\psi}_{tt}$ and using the uniform boundedness of $\\|\\psi^\\alpha\\|_{X_{\\textup{fMGT--K III}}}$ for $\\alpha \\in (0,1]$ leads to the following bound:\n\\begin{equation} \n\\begin{aligned}\n\\|\\overline{\\psi}_{tt}(t)\\|^2_{L^2}+\\|\\nabla\\overline{\\psi}_{t}(t)\\|^2_{L^2}\\lesssim\\,\\nLtwoLtwo{\\Dt^{1-\\alpha}\\nabla \\psi_t-\\nabla \\psi_t}^2.\n\\end{aligned}\n\\end{equation}\nOn account of Lemma~\\ref{Lemma:Limit}, we then have the following result. \n\\begin{proposition}[Limit of the fJMGT--K III equation] Let the assumptions of Theorem~\\ref{Thm:fJMGT_K_III} hold with $\\psi_1=0$. Let $\\{\\psi^\\alpha\\}_{\\alpha \\in (0,1)}$ be the family of solutions to the \\textup{fJMGT--K III} equation and let $\\psi$ solve the corresponding \\textup{JMGT--Kuznetsov} equation. Then $\\psi^{\\alpha}$ converges to $\\psi$ in $W^{1, \\infty}(0,T; H_0^1(\\Omega)) \\cap W^{2, \\infty}(0,T; L^2(\\Omega))$ as $\\alpha \\rightarrow 1^{-}$.\n\\end{proposition}\n\n\\section{Analysis of the equation with the third-order leading term} \\label{Sec:Analysis_fJMGT_W_III}\nWe begin our analytical considerations by looking at the fJMGT--W III equation \n\\begin{equation}\n\\tau \\psi_{ttt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau c^2\\Delta \\psi_{t}- \\delta {\\textup{D}}_t^{2-\\alpha}\\Delta\\psi=f,\n\\end{equation}\n\\noindent which has an integer-order leading term; cf. Table~\\ref{table:fJMGT}. We intend to analyze it by setting up a fixed-point mapping. To this end, we, first of all, study the following linearization:\n\\begin{equation} \\label{fMGT_III_sigma}\n\\tau \\psi_{ttt} + (1+\\sigma(x,t)) \\psi_{tt} - c^2\\Delta\\psi -\\tau c^2\\Delta \\psi_t - \\delta {\\textup{D}}_t^{2-\\alpha} \\Delta \\psi = f, \\quad 0< \\alpha \\leq 1.\n\\end{equation}\nThis is the linear fMGT III equation with a variable coefficient, which we assume is uniformly bounded; cf. Table~\\ref{table:fMGT} below. More precisely, we assume that there exist $\\underline{\\sigma}$, $\\overline{\\sigma}>0$, such that\n\\begin{equation}\\label{nondegeneracy_sigma}\n\\underline{\\sigma} \\leq \\sigma(x,t)\\leq \\overline{\\sigma} \\ \\mbox{ for all } \\ x\\in\\Omega, \\, t\\in(0,T).\n\\end{equation}\nNote that since we study the local-in-time behavior in this work, we do not impose a non-degeneracy condition on $1+\\sigma$. In the upcoming analysis, the crucial estimate involving fractional derivatives will be the following:\n\\begin{equation} \\label{fractional_est_w}\n\\begin{aligned}\n\\int_0^t \\prodLtwo{{\\textup{D}}_t^{2-\\alpha} w}{w_{tt}} \\, \\textup{d} s \n\\geq&\\, \\cos(\\pi \\alpha\/2) \\|w_{tt} \\|_{H^{-\\alpha\/2}(0,t; L^2(\\Omega))}^2,\n\\end{aligned}\n\\end{equation}\nwhich follows by \\eqref{coercivityI}. To formulate the first well-posedness result, we introduce the solution space\n\\begin{equation} \\label{def_X_fMGTIII}\n\\begin{aligned}\n\\quad X^{\\textup{low}}_{\\textup{fMGT III}}=\\,\\left\\{\\vphantom{H^{-\\alpha\/2}(0,T; H_0^1(\\Omega))}\\right.&\\psi \\in L^{\\infty}(0,T; H_0^1(\\Omega)): \\psi_t \\in L^{\\infty}(0,T; H_0^1(\\Omega)),\\\\&\\, \\psi_{tt} \\in L^\\infty(0,T; L^2(\\Omega)) \\cap H^{-\\alpha\/2}(0,T; H_0^1(\\Omega)),\\\\&\\left.\\, \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t\\|^2_{L^2} \\in L^\\infty(0,T), \\ \\psi_{ttt} \\in L^2(0,T; H^{-1}(\\Omega)) \\vphantom{H^{-\\alpha\/2}(0,T; H_0^1(\\Omega))}\\right\\} \n\\end{aligned}\n\\end{equation}\nfor $ \\alpha \\in (0,1)$, and\n\\begin{equation} \\label{def_X_fMGTIII_alphaone}\n\\begin{aligned}\nX^{\\textup{low}}_{\\textup{fMGT III}}= W^{1, \\infty}(0,T; H_0^1(\\Omega)) \\cap W^{2, \\infty}(0,T; L^2(\\Omega))\\cap H^3(0,T; H^{-1}(\\Omega)) \n\\end{aligned}\n\\end{equation}\nin case $\\alpha=1$. We denote by $\\|\\cdot\\|_{X^{\\textup{low}}_{\\textup{fMGT III}}}$ the corresponding norm on this space. We claim that the fMGT III equation \\eqref{fMGT_III_sigma} has a unique solution in this space under suitable assumptions on the data and the variable coefficient.\n\\begin{proposition}[Well-posedness of the fMGT III equation] \\label{Prop:fMGT_III_lower} Let $\\alpha \\in (0,1]$ and $\\sigma \\in L^\\infty(0,T; L^\\infty(\\Omega))$. Given $f \\in L^2(0,T; L^2(\\Omega))$ and \\[(\\psi_0, \\psi_1, \\psi_2) \\in (H_0^1(\\Omega), H_0^1(\\Omega), L^2(\\Omega)),\\] there exists a unique $\\psi$ in $X^{\\textup{low}}_{\\textup{fMGT III}}$, such that\n\t\\begin{equation} \\label{fMGT1_sigma}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] \\langle \\tau \\psi_{ttt}, v\\rangle_{H^{-1}, H_0^1} + \\prodLtwo{(1+\\sigma) \\psi_{tt}}{v} \\\\+ \\prodLtwo{c^2\\nabla\\psi +\\tau c^2\\nabla \\psi_t +\\delta {\\textup{D}}_t^{2-\\alpha} \\nabla \\psi}{\\nabla v} = \\prodLtwo{f}{v} \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tfor all $v \\in H_0^1(\\Omega)$, a.e. in $(0,T)$, with $(\\psi, \\psi_t, \\psi_{tt})\\vert_{t=0}=(\\psi_0, \\psi_1, \\psi_2)$. Furthermore, the solution satisfies the following estimate:\n\t\\begin{equation} \\label{energy_est1_fJMGT_W_III}\n\t\\begin{aligned}\n\t&\\,\\begin{multlined}[t]\n\t\\|\\psi\\|^2_{W^{1,\\infty}(H^1)}+\\nLinfLtwo{\\psi_{tt}}^2 +\\|\\psi_{ttt}\\|^2_{L^2(H^{-1})}\\\\[2mm] \\hspace*{1cm}+\\cos(\\alpha \\pi\/2)\\|\\psi_{tt}\\|^2_{H^{-\\alpha\/2}(H^1)}+ \\sup_{t \\in (0,T)} \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t\\|^2_{L^2}\n\t\\end{multlined} \\\\\n\t\\lesssim&\\, \\nLtwoLtwo{f}^2+\\nLtwo{\\nabla \\psi_0}^2+\\nLtwo{\\nabla \\psi_1}^2+\\nLtwo{\\psi_2}^2,\n\t\\end{aligned}\n\t\\end{equation}\t\nwhere for $\\alpha=1$, the $\\cos(\\alpha \\pi\/2)$ term should be omitted.\t\n\\end{proposition}\n\\begin{proof}\n\tWe focus in the proof on the case $\\alpha \\in (0,1)$ since the case $\\alpha=1$ follows in a more straightforward manner. We perform the analysis by employing the standard Galerkin procedure to discretize the problem in space~\\cite[\\S 7]{evans2010partial}, with alterations needed to accommodate the third-order derivative and the fractional term. We approximate the solution by\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\psi^n(x,t) =\\sum_{i=1}^n \\xi^n_i(t)\\phi_i(x),\n\t\\end{aligned}\n\t\\end{equation}\n\twhere $\\{\\phi_i\\}_{i=1}^\\infty$ are the eigenfunctions of the Dirichlet-Laplacian operator:\n\t\\[\n\t-\\Delta \\phi_i = \\lambda_i \\phi_i \\ \\text{ in } \\Omega, \\qquad \\phi_i=0 \\ \\text{ on } \\partial \\Omega.\n\t\\]\n\tDenote $V_n=\\textup{span}\\{\\phi_1, \\ldots, \\phi_n\\}$. The semi-discrete problem is given by\n\t\\begin{equation} \\label{semi-discrete_III}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] (\\tau \\psi_{ttt}^n, \\phi_j)_{L^2} + \\prodLtwo{(1+\\sigma) \\psi_{tt}^n}{\\phi_j}\\\\ + \\prodLtwo{c^2\\nabla\\psi^n +\\tau c^2\\nabla \\psi_t^n +\\delta {\\textup{D}}_t^{2-\\alpha} \\nabla \\psi^n}{\\nabla \\phi_j} = \\prodLtwo{f}{\\phi_j}, \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tfor all $j=1, \\ldots, n$, with approximate initial conditions $(\\psi^n, \\psi_t^n, \\psi_{tt}^n)\\vert_{t=0}=(\\psi_{0}^n, \\psi_{1}^n, \\psi_{2}^n)$ chosen as $L^2$ projections of $(\\psi_0, \\psi_1, \\psi_2)$ onto $V_n$. In other words,\t\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\psi_{0}^n= \\sum_{i=1}^n \\xi^n_{0, i} \\phi_i(x), \\quad \\psi_{1}^n= \\sum_{i=1}^n \\xi^n_{1, i} \\phi_i(x), \\quad \\psi_{2}^n= \\sum_{i=1}^n \\xi^n_{2, i} \\phi_i(x)\n\t\\end{aligned}\n\t\\end{equation}\n\twith \n\t\\begin{equation}\n\t\\xi^n_{0, i} =(\\psi_0, \\phi_i)_{L^2}, \\quad \\xi^n_{1, i} =(\\psi_1, \\phi_i)_{L^2}, \\quad \\xi^n_{2, i} =(\\psi_2, \\phi_i)_{L^2}.\n\t\\end{equation}\n\tThen we know that\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\|\\psi^n_0\\|_{H^1} \\leq&\\, \\|\\psi_0\\|_{H^1} \\ \\text{ and } \\ &&\\psi^n_0 \\rightarrow \\psi_0 \\ \\text{ in } \\ H_0^1(\\Omega),\\\\\n\t\\|\\psi^n_1\\|_{H^1} \\leq&\\, \\|\\psi_1\\|_{H^1} \\ \\text{ and } \\ &&\\psi^n_1 \\rightarrow \\psi_1 \\ \\text{ in } \\ H_0^1(\\Omega),\\\\\n\t\\|\\psi^n_2\\|_{L^2} \\leq&\\, \\|\\psi_2\\|_{L^2} \\ \\, \\text{ and } \\ &&\\psi^n_2 \\rightarrow \\psi_2 \\ \\text{ in } \\ L^2(\\Omega);\n\t\\end{aligned}\n\t\\end{equation}\n\tcf.~\\cite[\\S 7, Lemma 7.5]{robinson2001infinite}. \\\\\n\t\n\t\\noindent (I) \\emph{Existence of an approximate solution.} We first show that for a given $n$, a unique approximate solution exists. With $\\boldsymbol{\\xi}=[\\xi^n_1 \\ \\xi^n_2 \\ \\ldots \\ \\xi^n_n]^T$, the approximate problem can be rewritten in matrix form\n\t\\begin{equation} \\label{matrix_eq}\n\t\\begin{aligned}\n\t\\tau M\\boldsymbol{\\xi}_{ttt}+M_{\\sigma}(t)\\boldsymbol{\\xi}_{tt}+c^2 K\\boldsymbol{\\xi}+\\tau c^2 K \\boldsymbol{\\xi}_t+\\delta K {\\textup{D}}_t^{2-\\alpha} \\boldsymbol{\\xi}= \\boldsymbol{f}\n\t\\end{aligned}\n\t\\end{equation}\n\twith the entries of matrices $M=[M_{ij}]$, $M_\\sigma(t)=[M_{\\sigma, ij}(t)]$, $K=[K_{ij}]$, and the vector $\\boldsymbol{f}(t)=[f_i(t)]$ given by\n\t\\begin{equation} \\label{matrices}\n\t\\begin{aligned}\n\t&M_{ij}= (\\phi_i, \\phi_j)_{L^2}, \\quad &&M_{\\sigma, ij}(t)= ((1+\\sigma(t))\\phi_i, \\phi_j)_{L^2}, \\\\\n\t&K_{ij}= (\\nabla \\phi_i, \\nabla \\phi_j)_{L^2}, \\quad && f_i(t)=(f(t), \\phi_i)_{L^2}.\n\t\\end{aligned}\n\t\\end{equation}\n\tWe also introduce the vectors of coordinates of the approximate initial data in the basis: \n\t\\[\n\t\\boldsymbol{\\xi}_0=[\\xi^n_{0,1} \\ \\xi^n_{0,2} \\ \\ldots \\ \\xi^n_{0,n}]^T, \\quad \\boldsymbol{\\xi}_1=[\\xi^n_{1,1} \\ \\xi^n_{1,2} \\ \\ldots \\ \\xi^n_{1,n}]^T, \\quad \\boldsymbol{\\xi}_2=[\\xi^n_{2,1} \\ \\xi^n_{2,2} \\ \\ldots \\ \\xi^n_{2,n}]^T. \n\t\\]\n\tThen by setting $\\boldsymbol{\\mu}=\\boldsymbol{\\xi}_{ttt}$, we have\n\t\\begin{equation} \\label{eq_xi}\n\t\\boldsymbol{\\xi}(t)=\\boldsymbol{\\xi}_0 +t \\boldsymbol{\\xi}_1+\\frac{1}{2}t^2\\boldsymbol{\\xi}_2+\\int_0^t \\frac12(t-s)^2 \\boldsymbol{\\mu}(s)\\, \\textup{d} s ,\n\t\\end{equation}\n\tand we can restate the semi-discrete problem as a system of Volterra integral equations given by\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\begin{multlined}[t]\n\t\\tau M \\boldsymbol{\\mu}(t)+M_{\\sigma}(t)\\left(\\int_0^t \\boldsymbol{\\mu}(s)\\, \\textup{d} s +\\boldsymbol{\\xi}_{2}\\right)+c^2K \\left(\\boldsymbol{\\xi}_0 +t \\boldsymbol{\\xi}_1+\\frac{t^2}{2}\\boldsymbol{\\xi}_2+\\int_0^t \\frac12(t-s)^2 \\boldsymbol{\\mu}(s)\\, \\textup{d} s \\right)\\\\+\\tau c^2 K\\left(\\boldsymbol{\\xi}_1 +t \\boldsymbol{\\xi}_2+\\int_0^t (t-s) \\boldsymbol{\\mu}(s)\\, \\textup{d} s \\right)+ \\frac{\\delta}{\\Gamma(\\alpha)}K\\int_0^t(t-s)^{\\alpha-1}\\left(\\int_0^s\\boldsymbol{\\mu}(r)\\, \\textup{d}r+\\boldsymbol{\\xi}_2\\right)\\textup{d}s\\\\ =\\boldsymbol{f}(t)\\hphantom{fill} \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tfor $t \\in (0,T)$. By using Dirichlet's formula \n\t\\begin{align} \\label{exchange_integrals}\n\t{\\frac{\\delta}{\\Gamma(\\alpha)}}\\int_0^t(t-s)^{\\alpha-1}\\left(\\int_0^s\\boldsymbol{\\mu}(r)\\,\\textup{d}r\\right) \\, \\textup{d} s = {\\frac{\\delta}{\\Gamma(\\alpha)}}\\int_0^t \\left(\\int_r^t (t-s)^{\\alpha-1}\\,\\textup{d}s\\right) \\boldsymbol{\\mu}(r) \\textup{d}r,\n\t\\end{align}\n\twe arrive at an equivalent reformulation\n\t\\begin{equation} \\label{eq_mu}\n\t\\begin{aligned}\n\t\\boldsymbol{\\mu}(t) =\\tilde{\\boldsymbol{f}}(t)+\\int_0^t K_{\\alpha}(t,s)\\boldsymbol{\\mu}(s)\\, \\textup{d} s ,\n\t\\end{aligned}\n\t\\end{equation}\n\twhere the first term on the right is defined as\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\tilde{\\boldsymbol{f}}(t)=&\\,\\begin{multlined}[t]-\\frac{1}{\\tau}M^{-1} \\left\\{M_{\\sigma}(t)\\boldsymbol{\\xi}_2+c^2K (\\boldsymbol{\\xi}_0+t \\boldsymbol{\\xi}_1+\\frac12 t^2 \\boldsymbol{\\xi}_2)+\\tau c^2 K(\\boldsymbol{\\xi}_1+t\\boldsymbol{\\xi}_2)-\\boldsymbol{f}(t) \\right.\\\\ \\left.\n\t+\\frac{\\delta}{\\Gamma(\\alpha)}K\\int_0^t (t-s)^{\\alpha-1} \\boldsymbol{\\xi}_2\\, \\textup{d} s \\right\\} \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tand the kernel function is given by\n\t\\begin{equation}\n\t\\begin{aligned} \n\tK_{\\alpha}(t,s)=&\\,\\begin{multlined}[t] -\\frac{1}{\\tau}M^{-1}\\left(M_{\\sigma}(t)+\\frac12 c^2K(t-s)^2+\\tau c^2K (t-s)\\right) \\\\\n\t-\\frac{1}{\\tau} \\frac{\\delta}{\\Gamma(\\alpha+1)}M^{-1} K (t-s)^\\alpha.\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n{To arrive at the kernel expression, we have employed\n\t\\[\n\t\\frac{\\delta}{\\Gamma(\\alpha)}K\\int_s^t (t-r)^{\\alpha-1}\\, \\textup{d}r=\\frac{\\delta}{\\alpha \\Gamma(\\alpha)}K(t-s)^\\alpha.\n\t\\] }\n\\noindent Due to the $L^\\infty(0,T)$ regularity of the kernel $K_\\alpha$ on $D=\\{(t,s): \\ 0 \\leq s \\leq t \\leq T\\}$ and the fact that function $\\tilde{\\boldsymbol{f}}$ belongs to $L^2(0,T)$, vector equation \\eqref{eq_mu} has a unique solution $\\boldsymbol{\\mu} \\in L^2(0,T)$. This claim directly follows by considering (systems of) integral equations in $L^2(0, T)$ instead of $C[0,T]$ in~\\cite[Theorem 2.1.7]{brunner2004collocation}; see also \\cite[Theorem 4.2, p. 24 in \\S 9]{GLS90}. From \\eqref{eq_xi}, taking into account initial data, a unique $\\boldsymbol{\\xi} \\in H^3(0,T)$ and, in turn, $\\psi^n \\in H^3(0,T; V_n)$ exists. \\\\\n\t\n\t\n\t\n\t\\noindent (II) \\emph{A priori energy analysis.} We next focus on the derivation of the energy estimate, which goes through by testing the semi-discrete problem by $\\psi^n_{tt}(t) \\in V_n$. More precisely, we test \\eqref{semi-discrete_III} with $\\xi^n_{tt}(t)$ and sum over $j=1, \\ldots, n$. After integrating over $(0,t)$, we at first obtain the identity\n\t\\begin{equation} \\label{energy_id}\n\t\\begin{aligned}\n\t&\\begin{multlined}[t]\\frac12 \\tau\\nLtwo{\\psi_{tt}^n(t)}^2 \\Big \\vert_0^t\n\t+ \\frac12\\tau c^2\\nLtwo{\\nabla \\psi_t^n(s)}^2 \\Big \\vert_0^t +\\delta \\int_0^t \\prodLtwo{\\Dt^{2-\\alpha}\\nabla \\psi^n}{\\nabla \\psi_{tt}^n} \\, \\textup{d} s \n\t\\end{multlined}\\\\\n\t=&\\begin{multlined}[t] - \\int_0^t(1+\\sigma)\\| \\psi_{tt}^n\\|^2_{L^2}\\, \\textup{d} s +\\int_0^t \\prodLtwo{f}{\\psi_{tt}^n}\\, \\textup{d} s - c^2 \\prodLtwo{\\nabla \\psi^n}{\\nabla \\psi_t^n} \\Big \\vert_0^t \n\t+ c^2 \\int_0^t \\nLtwo{\\nabla \\psi_t^n}^2 \\, \\textup{d} s . \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tBy Young's $\\varepsilon$-inequality, we have\n\t\\begin{equation}\n\t\\begin{aligned}\n\tc^2\\int_0^t \\prodLtwo{\\nabla \\psi^n}{\\nabla \\psi_t^n} \\, \\textup{d} s \\leq \\frac{1}{4\\varepsilon}c^2(C(T)\\|\\nabla \\psi_t^n\\|_{L^2_t(L^2)}+\\nLtwo{\\nabla \\psi^n_0})^2 +\\varepsilon c^2 \\nLtwo{\\nabla \\psi_t^n(t)}^2.\n\t\\end{aligned}\n\t\\end{equation}\n\tFor $0<\\varepsilon < \\tau\/2$, employing estimate \\eqref{fractional_est_w} and Gronwall's inequality leads to \n\t\\begin{equation} \\label{energy_est1_fJMGT_W_III_discrete}\n\t\\begin{aligned}\n\t&\\begin{multlined}[t] \\nLtwo{\\psi_{tt}^n(t)}^2 \n\t+ \\nLtwo{\\nabla \\psi_t^n(t)}^2+ \\cos(\\pi \\alpha\/2) \\| \\nabla \\psi_{tt}^n \\|_{H^{-\\alpha\/2}(0,t; L^2(\\Omega))}^2\n\t\\end{multlined}\\\\\n\t\\lesssim&\\,\\begin{multlined}[t] \\nLtwoLtwo{f}^2+\\nLtwo{\\nabla \\psi_0}^2+\\nLtwo{\\nabla \\psi_1}^2+\\nLtwo{\\psi_2}^2. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tNote that we can also use the estimate\n\t\\[\n\t\\nLtwo{\\psi^n(t)} \\leq C_{\\textup{PF}}\\nLtwo{\\nabla \\psi^n(t)} \\lesssim T \\sup_{t \\in (0,T)}\\nLtwo{\\nabla \\psi_t^n(t)}+{\\nLtwo{\\nabla \\psi^n_0}}.\n\t\\]\t\n\tAdditionally, standard arguments (cf.~\\cite[\\S 7]{evans2010partial}) lead to the bound\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\int_0^t\\|\\psi_{ttt}^n\\|^2_{H^{-1}}\\, \\textup{d} s \\lesssim&\\, \\begin{multlined}[t] \\int_0^t (1+\\overline{\\sigma})\\nLtwo{\\psi_{tt}^n}^2\\, \\textup{d} s +\n\t\\int_0^t \\nLtwo{\\nabla \\psi^n}^2\\, \\textup{d} s \\\\+\\int_0^t \\nLtwo{\\nabla \\psi_t^n}^2\\, \\textup{d} s +\\int_0^t \\nLtwo{{\\textup{D}}_t^{2-\\alpha}\\nabla \\psi^n}^2\\, \\textup{d} s +\\int_0^t \\nLtwo{f}^2\\, \\textup{d} s .\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\t\n\tWe can further estimate the fractional term on the right as follows:\n\t\\begin{equation} \\label{est_Dtal}\n\t\\begin{aligned}\n\t\\|{\\textup{D}}_t^{2-\\alpha}\\nabla \\psi^n\\|^2_{L^2(0,t; L^2)} = \\|\\textup{I}^{\\alpha} \\nabla \\psi_{tt}^n\\|^2_{L^2(0,t; L^2)} \\lesssim&\\, \\|\\nabla \\psi_{tt}^n\\|^2_{X_{-\\alpha}(0,t; L^2)} \\\\\n\t\\lesssim&\\,\\|\\nabla \\psi_{tt}^n\\|^2_{H^{-\\alpha\/2}(0,t; L^2)},\n\t\\end{aligned}\n\t\\end{equation}\n\twhere the first inequality follows by~\\cite[Theorem 1]{gorenflo1999operator}. {Note that $\\{X_{\\beta}\\}_{\\beta \\in \\R}$ represents a scale of Hilbert spaces of functions $(0,t) \\mapsto L^2(\\Omega)$; cf.~\\cite[\\S 5]{baumeister1987stable} and~\\cite[Lemma 8]{gorenflo1999operator}.} The second inequality follows by the fact that \\[X_{\\alpha}(0,t; L^2(\\Omega))\\subseteq X_{\\alpha\/2}(0,t; L^2(\\Omega))= H_0^{\\alpha\/2}(0,t; L^2(\\Omega))\\] for $\\alpha<1$ and therefore $\\alpha\/2<\\frac12$, together with duality. Note also that, on account of estimate \\eqref{eqn:Alikhanov_1}, we know that\n\t\\begin{equation}\\label{Alikhanov_Galerkin}\n\t\\begin{aligned}\n\t(\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi^n_{t}, \\nabla \\psi_{tt}^n)_{L^2}=&\\,\t(\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi^n_{t}, \\Dt^\\alpha\\nabla \\Dt^{1-\\alpha} \\psi_t^n)_{L^2}\\\\\\geq&\\, \\frac12 \\Dt^\\alpha\\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t^n\\|^2_{L^2}\\\\\n\t=&\\, \\frac12\\frac{\\textup{d}}{\\textup{d}t} \\, \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t^n\\|^2_{L^2}.\n\t\\end{aligned}\n\t\\end{equation}\n\tEmploying this estimate instead of \\eqref{fractional_est_w} in the above derivation yields a uniform bound on $\\displaystyle \\sup_{t \\in (0,T)} \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t^n\\|^2_{L^2}$, which will be needed in the proof of uniqueness. \\\\\n\t\n\t\\noindent (III) \\emph{Passing to the limit.} Thanks to the uniform bounds and \\eqref{est_Dtal}, we have weak convergence of a subsequence, which we do not relabel, in the following sense:\n\t\\begin{equation} \\label{weak_limits1}\n\t\\begin{alignedat}{4} \n\t\\psi_{ttt}^n &\\relbar\\joinrel\\rightharpoonup \\psi_{ttt} &&\\ \\text{ weakly} &&\\text{ in } &&L^2(0,T; H^{-1}(\\Omega)), \\\\\n\t\\psi_{tt}^n &\\relbar\\joinrel\\rightharpoonup \\psi_{tt} &&\\ \\text{ weakly-$\\star$} &&\\text{ in } &&L^\\infty(0,T; L^2(\\Omega)), \\\\\n\t\\psi_t^n &\\relbar\\joinrel\\rightharpoonup \\psi_t &&\\ \\text{ weakly-$\\star$} &&\\text{ in } &&L^\\infty(0,T; H_0^1(\\Omega)), \\\\\n\t\\psi^n &\\relbar\\joinrel\\rightharpoonup \\psi &&\\ \\text{ weakly-$\\star$} &&\\text{ in } &&L^\\infty(0,T; H_0^1(\\Omega)).\n\t\\end{alignedat} \n\t\\end{equation}\n\tFurthermore, we know that\n\t\t\\begin{equation} \\label{weak_limits2}\n\t\\begin{alignedat}{4} \n\t\\nabla \\psi_{tt}^n &\\relbar\\joinrel\\rightharpoonup \\nabla \\psi_{tt} &&\\ \\ \\text{ weakly} &&\\text{ in } &&H^{-\\alpha\/2}(0,T; L^2(\\Omega)), \\\\\n\t{\\textup{D}}_t^{2-\\alpha}\\nabla \\psi^n &\\relbar\\joinrel\\rightharpoonup {\\textup{D}}_t^{2-\\alpha}\\nabla \\psi &&\\ \\ \\text{ weakly} &&\\text{ in } &&L^2(0,T; L^2(\\Omega)), \\\\\n\t\\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t^n\\|^2_{L^2} &\\relbar\\joinrel\\rightharpoonup \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t\\|^2_{L^2} &&\\ \\ \\text{ weakly-$\\star$} &&\\text{ in } &&L^\\infty(0,T).\n\t\\end{alignedat} \n\t\\end{equation}\n\tWe can thus pass to the weak limit in the usual way to conclude that $\\psi$ solves \\eqref{fMGT1_sigma}. Further, weak\/weak-$\\star$ lower semi-continuity of norms implies\n\t\\[\t\n\t\\begin{aligned}\n\t\\| \\nabla \\psi_{tt} \\|_{H^{-\\alpha\/2}(0,t; L^2(\\Omega))}^2\\leq&\\, \\liminf_{n \\rightarrow \\infty} \\| \\nabla \\psi_{tt}^n \\|_{H^{-\\alpha\/2}(0,t; L^2(\\Omega))}^2, \\\\\n\t \\sup_{t \\in (0,T)} \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t\\|^2_{L^2} \\leq&\\, \\liminf_{n \\rightarrow \\infty} \\sup_{t \\in (0,T)} \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_t^n\\|^2_{L^2}.\n\t\\end{aligned} \n\t\\]\n\tand thus by passing to the limit in the energy estimate for $\\psi^n$, we conclude that $\\psi$ satisfies \\eqref{energy_est1_fJMGT_W_III}. \\\\\n\t\n\t\n\t\\noindent (IV) \\emph{Attainment of the initial conditions.} We next show that $\\psi$ attains its initial conditions. By \\eqref{weak_limits1} and~\\cite[Lemma 3.1.7]{zheng2004nonlinear}, we know that\n\t\\[\n\t\\psi^n (0) \\relbar\\joinrel\\rightharpoonup \\psi(0) \\text{ weakly} \\text{ in } H_0^1(\\Omega), \n\t\\]\n\tand since $\\psi^n(0) \\rightarrow \\psi_0$ in $H_0^1(\\Omega)$, we have $\\psi(0)=\\psi_0$. Further,\n\t\\[\n\t\\psi_t^n (0) \\relbar\\joinrel\\rightharpoonup \\psi_t(0) \\text{ weakly} \\text{ in } L^2(\\Omega), \n\t\\]\n\tand thus $\\psi_t(0)=\\psi_1$ as an equality in $L^2(\\Omega)$. Similarly, $\\psi_{tt}(0)=\\psi_2$ as an equality in $H^{-1}(\\Omega)$; that is, \n\t\\[\\langle \\psi_{tt}(0 ), v \\rangle_{H^{-1}, H^1} = \\langle \\psi_{2}, v \\rangle_{H^{-1}, H^1} , \\quad \\forall v \\in H_0^1(\\Omega).\\]\n\n\t\t\\noindent (V) \\emph{Uniqueness.} To prove uniqueness, we should show that the only solution of \n\t\\begin{equation} \\label{homogeneous_eq}\n\t\\begin{aligned}\n\t\\tau \\psi_{ttt}+(1+\\sigma)\\psi_{tt}-\\tau c^2\\Delta \\psi_t-c^2 \\Delta \\psi-\\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\psi_t=0\n\t\\end{aligned}\n\t\\end{equation}\n\twith $\\psi_0=\\psi_1=\\psi_2=0$ in $X^{\\textup{low}}_{\\textup{fMGT III}}$ is $\\psi=0$. The issue, however, is that at this point we are not allowed to directly test \\eqref{fMGT1_sigma} with $\\psi_{tt}$ due to its low regularity. Instead, in the spirit of~\\cite[\\S 2.4]{temam2012infinite}, we will prove that such $\\psi$ satisfies\n\t\\begin{equation} \\label{unique_III_low}\n\t\\begin{aligned}\n\t&(\\tau \\psi_{ttt}-\\tau c^2\\Delta \\psi_t-c^2 \\Delta \\psi-\\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\psi_t, \\psi_{tt})_{L^2} \\\\\n\\gtrsim &\\,\\begin{multlined}[t] \\frac{\\textup{d}}{\\textup{d}t} \\left\\{\\frac12\\tau \\nLtwo{\\psi_{tt}}^2+\\frac12\\tau c^2\\nLtwo{\\nabla \\psi_t}^2 +c^2(\\nabla \\psi, \\nabla \\psi_t)_{L^2}\\right. \\\\ \\left.\\hspace*{3cm}+\\frac{\\delta}{2}\\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_{t}\\|^2_{L^2}\\right\\} - c^2\\nLtwo{\\nabla \\psi_t}^2.\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tThis estimate combined with \\eqref{homogeneous_eq} implies that\n\t\\begin{equation} \\label{unique_III_low_b}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] \\frac{\\textup{d}}{\\textup{d}t} \\left\\{\\frac12\\tau \\nLtwo{\\psi_{tt}}^2+\\frac12\\tau c^2\\nLtwo{\\nabla \\psi_t}^2+c^2(\\nabla \\psi, \\nabla \\psi_t)_{L^2} \\right. \\\\ \\left.\\hspace*{3cm}+\\frac{\\delta}{2}\\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\psi_{t}\\|^2_{L^2}\\right\\} - c^2\\nLtwo{\\nabla \\psi_t}^2\n\t\\lesssim -((1+\\sigma \\psi_{tt}, \\psi_{tt})_{L^2} , \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tafter which we can proceed as in the previous energy analysis to arrive at $\\psi=0$. We note that if $\\psi \\in X^{\\textup{low}}_{\\textup{fMGT III}}$ solves \\eqref{homogeneous_eq}, then \n\t\\begin{equation} \\label{reg1}\n\t\\begin{aligned}\n\t\\psi_t \\in L^2(H_0^1(\\Omega)), \\quad \\psi_{tt} \\in L^2(0,T; L^2(\\Omega)) \\cap H^{-\\alpha\/2}(0,T; H_0^1(\\Omega)).\n\t\\end{aligned}\n\t\\end{equation}\n\tFurthermore, a bootstrap argument yields\n\t\\begin{equation} \\label{reg2}\n\t\\begin{aligned}\n\t\\tau \\psi_{ttt}-\\tau c^2\\Delta \\psi_t-c^2 \\Delta \\psi-\\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\psi_t= - (1+\\sigma)\\psi_{tt} \\in L^2(L^2).\n\t\\end{aligned}\n\t\\end{equation}\n\tWe next construct a regularization of $\\psi$ which satisfies \\eqref{unique_III_low}, following~\\cite[\\S 2.4, Lemma 4.1]{temam2012infinite}. Let $\\tilde{\\psi}: \\R \\mapsto H_0^1(\\Omega)$ be defined by\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\tilde{\\psi}= \\begin{cases}\n\t\\theta \\psi, \\ &\\text{ on } (0,T), \\\\\n\t0, \\ &\\text{ on } \\R \\setminus [0,T],\n\t\\end{cases}\n\t\\end{aligned}\n\t\\end{equation}\n\twhere $\\theta: \\R \\mapsto [0,1]$ is a $C^\\infty$ truncation function, equal to $0$ on $\\R \\setminus [0,T]$ and to $1$ on some sub-interval of $(0,T)$. Then\n\t\\begin{equation} \\label{reg3}\n\t\\begin{aligned}\n\t&\\tilde{\\psi}_t \\in L^2(0, \\infty; H_0^1(\\Omega)), \\quad \\tilde{\\psi}_{tt} \\in L^2(0, \\infty; L^2(\\Omega)) \\cap H^{-\\alpha\/2}(0,\\infty; H_0^1(\\Omega)),\\\\\n\t&\\, \\tau \\tilde{\\psi}_{ttt}-\\tau c^2\\Delta \\tilde{\\psi}_t-c^2 \\Delta \\tilde{\\psi}-\\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\tilde{\\psi}_t \\in L^2(0, \\infty; L^2(\\Omega)).\n\t\\end{aligned}\n\t\\end{equation}\n\tWe regularize $\\tilde{\\psi}$ by $\\tilde{\\psi}_\\varepsilon= \\varrho_\\varepsilon * \\tilde{\\psi}$, with $\\varrho_\\varepsilon$ being a $C^\\infty$ mollifier. Then $\\tilde{\\psi}_\\varepsilon: \\R \\mapsto H_0^1(\\Omega)$ is a $C^\\infty$ function, which satisfies\n\t\\begin{equation} \n\t\\begin{aligned}\n\t&(\\tau \\tilde{\\psi}_{\\varepsilon, ttt}-\\tau c^2 \\Delta \\tilde{\\psi}_{\\varepsilon, t}-c^2 \\Delta \\tilde{\\psi}_\\varepsilon- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\tilde{\\psi}_{\\varepsilon, t}, \\tilde{\\psi}_{\\, \\varepsilon, tt})_{L^2} \\\\\n\t=&\\, \\begin{multlined}[t]\\frac{\\textup{d}}{\\textup{d}t} \\left\\{\\frac12\\tau \\nLtwo{\\tilde{\\psi}_{\\varepsilon, tt}}^2+\\frac12\\tau c^2\\nLtwo{\\nabla \\tilde{\\psi}_{\\varepsilon, t}}^2+c^2(\\nabla \\tilde{\\psi}_\\varepsilon, \\nabla \\tilde{\\psi}_{\\varepsilon, t})_{L^2} \\right\\} \\\\- c^2\\nLtwo{\\nabla \\tilde{\\psi}_{\\varepsilon,t}}^2+\\delta (\\nabla {\\textup{D}}_t^{1-\\alpha} \\tilde{\\psi}_{\\varepsilon, t}, \\nabla \\tilde{\\psi}_{\\varepsilon,tt})_{L^2}. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tSimilarly to \\eqref{Alikhanov_Galerkin}, we have\n\t\\[\n\t\\begin{aligned}\n (\\nabla {\\textup{D}}_t^{1-\\alpha} \\tilde{\\psi}_{\\varepsilon, t}, \\nabla \\tilde{\\psi}_{\\varepsilon,tt})_{L^2}=\\,(\\nabla {\\textup{D}}_t^{1-\\alpha} \\tilde{\\psi}_{\\varepsilon, t}, \\Dt^\\alpha\\nabla \\Dt^{1-\\alpha} \\tilde{\\psi}_{\\varepsilon, t})_{L^2}\\geq \\frac12\\frac{\\textup{d}}{\\textup{d}t} \\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\tilde{\\psi}_{\\varepsilon, t}\\|^2_{L^2}.\n \\end{aligned}\n\t\\]\n\t Therefore,\n\t\t\\begin{equation} \\label{ineq_tilde_eps}\n\t\\begin{aligned}\n\t&(\\tau \\tilde{\\psi}_{\\varepsilon, ttt}-\\tau c^2 \\Delta \\tilde{\\psi}_{\\varepsilon, t}-c^2 \\Delta \\tilde{\\psi}_\\varepsilon- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\tilde{\\psi}_{\\varepsilon, t}, \\tilde{\\psi}_{\\, \\varepsilon, tt})_{L^2} \\\\\n\t\\gtrsim&\\, \\begin{multlined}[t] \\frac{\\textup{d}}{\\textup{d}t} \\left\\{\\frac12\\tau \\nLtwo{\\tilde{\\psi}_{\\varepsilon, tt}}^2+\\frac12\\tau c^2\\nLtwo{\\nabla \\tilde{\\psi}_{\\varepsilon, t}}^2 +c^2(\\nabla \\tilde{\\psi}_\\varepsilon, \\nabla \\tilde{\\psi}_{\\varepsilon, t})_{L^2}\\right.\\\\ \\left.+\\frac{\\delta}{2}\\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\tilde{\\psi}_{\\varepsilon, t}\\|^2_{L^2}\\right\\} - c^2\\nLtwo{\\nabla \\tilde{\\psi}_{\\varepsilon,t}}^2.\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tThanks to \\eqref{reg3}, we know that\n\t\\[\n\t\\begin{aligned}\n\t&\\lim_{\\varepsilon \\rightarrow 0}\\ (\\tau \\tilde{\\psi}_{\\varepsilon, ttt}-\\tau c^2 \\Delta \\tilde{\\psi}_{\\varepsilon, t}-c^2 \\Delta \\tilde{\\psi}_\\varepsilon- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\tilde{\\psi}_{\\varepsilon, t}, \\tilde{\\psi}_{\\varepsilon, tt})_{L^2} \\\\\n\t=&\\,(\\tau \\tilde{\\psi}_{ttt}-\\tau c^2 \\Delta \\tilde{\\psi}_{t}-c^2 \\Delta \\tilde{\\psi}- \\delta {\\textup{D}}_t^\\alpha \\Delta \\tilde{\\psi}_{t}, \\tilde{\\psi}_{tt})_{L^2}. \n\t\\end{aligned}\n\t\\]\n\tWe can thus pass to the limit $\\varepsilon \\rightarrow 0$ in \\eqref{ineq_tilde_eps} to arrive at \n\t\\begin{equation}\n\t\\begin{aligned}\n\t&(\\tau \\tilde{\\psi}_{ttt}-\\tau c^2 \\Delta \\tilde{\\psi}_t-c^2 \\Delta \\tilde{\\psi}- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\tilde{\\psi}_t, \\tilde{\\psi}_{tt})_{L^2} \\\\\n\\gtrsim&\\, \\begin{multlined}[t]\\frac{\\textup{d}}{\\textup{d}t} \\left\\{\\frac12\\tau \\nLtwo{\\tilde{\\psi}_{tt}}^2+\\frac12\\tau c^2\\nLtwo{\\nabla \\tilde{\\psi}_{t}}^2 +c^2(\\nabla \\tilde{\\psi}, \\nabla \\tilde{\\psi}_t)_{L^2}+\\frac{\\delta}{2}\\textup{I}^{1-\\alpha} \\|\\nabla {\\textup{D}}_t^{1-\\alpha} \\tilde{\\psi}_{t}\\|^2_{L^2}\\right\\}\\\\ - c^2\\nLtwo{\\nabla \\tilde{\\psi}_{t}}^2.\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tBy restriction to $(0,T)$, the same holds for $\\theta \\psi$, from which the claim follows.\n\\end{proof}\n\nTo formulate the second well-posedness result, we introduce a higher-order solution space:\n\\begin{equation}\n\\begin{aligned}\nX^{\\textup{high}}_{\\textup{fMGT III}}= W^{1, \\infty}(0,T; {H_\\diamondsuit^2(\\Omega)}) \\cap W^{2, \\infty}(0,T; H_0^1(\\Omega))\\cap H^3(0,T; L^2(\\Omega)). \n\\end{aligned}\n\\end{equation}\nWe denote by $\\|\\cdot\\|_{X^{\\textup{high}}_{\\textup{fMGT III}}}$ the corresponding norm on this space. Under stronger regularity assumptions on the data and the coefficient $\\sigma$, the fMGT III equation \\eqref{fMGT_III_sigma} has a unique solution in this space.\n\\begin{proposition}[Higher regularity for the fMGT III equation] \\label{Prop:fMGT_III_higher} Let $\\alpha \\in (0,1]$ and $\\sigma \\in L^\\infty(0,T; L^\\infty(\\Omega)\\cap W^{1, 4}(\\Omega))$. Given $f \\in L^2(0,T; H_0^1(\\Omega))$ and \\[(\\psi_0, \\psi_1, \\psi_2) \\in{H_\\diamondsuit^2(\\Omega)} \\times {H_\\diamondsuit^2(\\Omega)} \\times H_0^1(\\Omega),\\] there exists a unique solution $\\psi \\in X^{\\textup{high}}_{\\textup{fMGT III}}$, which solves \\eqref{fMGT1_sigma} in the $L^2(0,T; L^2(\\Omega))$ sense, and satisfies\n\t\\begin{equation} \\label{energy_est2_fMGT_W_III}\n\t\\begin{aligned}\n\t\\|\\psi\\|^2_{X^{\\textup{high}}_{\\textup{fMGT III}}}\n\t\\lesssim\\,\\begin{multlined}[t] \\nLtwoLtwo{\\nabla f}^2+\\nLtwo{\\Delta \\psi_0}^2+\\nLtwo{\\Delta \\psi_1}^2+\\nLtwo{\\nabla \\psi_2}^2. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\\end{proposition}\n\\begin{proof}\n\tThe statement when $\\alpha=1$ follows analogously to~\\cite[Theorem 3.1]{KaltenbacherNikolic}. The proof in the case $\\alpha \\in (0,1)$ can again be conducted by employing a Galerkin analysis in space. We only outline the derivation of the energy estimate, which follows by testing the semi-discrete problem by $-\\Delta \\psi^n_{tt}$. We omit the superscript $n$ in the notation below. After integrating over $(0,t)$, we first obtain the identity\n\t\\begin{equation} \\label{energy_id}\n\t\\begin{aligned}\n\t&\\begin{multlined}[t]\\frac12 \\tau\\nLtwo{\\nabla \\psi_{tt}(t)}^2 \\Big \\vert_0^t\n\t+ \\frac12\\tau c^2\\nLtwo{\\Delta \\psi_{t}(t)}^2 \\Big \\vert_0^t+\\delta \\int_0^t \\prodLtwo{\\Dt^{2-\\alpha}\\Delta \\psi}{\\Delta \\psi_{tt}} \\, \\textup{d} s \n\t\\end{multlined}\\\\\n\t=&\\,\\begin{multlined}[t] \\int_0^t \\prodLtwo{\\nabla f}{\\nabla \\psi_{tt}}\\, \\textup{d} s - \\int_0^t \\prodLtwo{(1+\\sigma) \\nabla \\psi_{tt}}{\\nabla \\psi_{tt}} \\, \\textup{d} s -\\int_0^t (\\psi_{tt}\\nabla \\sigma, \\nabla \\psi_{tt})\\, \\textup{d} s \\\\ - c^2 \\prodLtwo{\\Delta \\psi}{\\Delta \\psi_t} \\Big \\vert_0^t \n\t+ c^2 \\int_0^t \\nLtwo{\\Delta \\psi_t}^2 \\, \\textup{d} s . \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tWe can rely on the following estimate:\n\t\\begin{equation}\n\t\\begin{aligned}\n\t&\\int_0^t \\prodLtwo{\\nabla f}{\\nabla \\psi_{tt}}\\, \\textup{d} s -\\int_0^t (\\psi_{tt}\\nabla \\sigma, \\nabla \\psi_{tt})\\, \\textup{d} s - c^2 \\prodLtwo{\\Delta \\psi}{\\Delta \\psi_t} \\Big \\vert_0^t \\\\\n\t\\leq&\\, \\begin{multlined}[t] \\nLtwoLtwo{\n\t\t\\nabla\n\t\tf}\\|\\nabla \\psi_{tt}\\|_{L^2_t(L^2)}+\\|\\psi_{tt}\\|_{L^2_t(L^4)}\\|\\nabla \\sigma\\|_{L^\\infty(L^4)}\\|\\nabla \\psi_{tt}\\|_{L_t^2(L^2)}\\\\\n\t+ c^2 \\|\\Delta \\psi(t)\\|_{L^2}\\|\\Delta \\psi_t(t)\\|_{L^2}+ c^2 \\|\\Delta \\psi_0\\|_{L^2}\\|\\Delta \\psi_1\\|_{L^2}; \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tsee also~\\cite[Theorem 3.1]{kaltenbacher2021inviscid}. We further note that\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\|\\psi_{tt}\\|_{L^2_t(L^4)}\\|\\nabla \\sigma\\|_{L^\\infty(L^4)}\\|\\nabla \\psi_{tt}\\|_{L_t^2(L^2)} \\leq C_{H^1, L^4} \\|\\nabla \\sigma\\|_{L^\\infty(L^4)}\\|\\nabla \\psi_{tt}\\|_{L_t^2(L^2)}^2\n\t\\end{aligned}\n\t\\end{equation}\n\tand that\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\|\\Delta \\psi(t)\\|_{L^2}\\|\\Delta \\psi_t(t)\\|_{L^2} \\leq \\frac{1}{\\epsilon}(\\sqrt{T}\\|\\Delta \\psi_t\\|_{L^2_t(L^2)}+\\|\\Delta \\psi_0\\|_{L^2})^2+\\epsilon \\|\\Delta \\psi_t(t)\\|_{L^2}^2.\n\t\\end{aligned}\n\t\\end{equation}\n\tFor fixed, small enough $\\epsilon>0$, an application of Gronwall's inequality thus yields \\eqref{energy_est2_fJMGT_W_III}, at first in a discrete setting. Additionally, we obtain \n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\nLtwotLtwo{\\psi_{ttt}}^2 \\lesssim& \\, \\begin{multlined}[t] \\nLtwotLtwo{\\psi_{tt}}^2 +\\nLtwotLtwo{\\Delta\\psi}^2+ \\nLtwotLtwo{\\Delta \\psi_t}^2\\\\+ \\nLtwotLtwo{{\\textup{D}}_t^{2-\\alpha} \\Delta \\psi}^2 +\\nLtwoLtwo{f}^2,\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\twhere, similarly to \\eqref{est_Dtal}, we can further estimate the fractional term as follows: \n\t\\begin{equation}\n\t\\nLtwotLtwo{{\\textup{D}}_t^{2-\\alpha} \\Delta \\psi} \\lesssim \\|\\Delta \\psi_{tt}\\|_{H_t^{-\\alpha\/2}(L^2)}.\n\t\\end{equation}\n\tThe rest of the arguments follow analogously to the proof of Proposition~\\ref{Prop:fMGT_III_lower}. We point out that in this higher-order setting, we are allowed to test the homogeneous problem ($f=\\psi_0=\\psi_1=\\psi_2=0$) directly with $\\psi_{tt}$ to prove uniqueness. \\\\\n\\indent\tNote that for $\\psi \\in X^{\\textup{high}}_{\\textup{fMGT III}}$, thanks to the embedding $W^{1, \\infty}(0,T; {H_\\diamondsuit^2(\\Omega)}) \\hookrightarrow C([0,T]; {H_\\diamondsuit^2(\\Omega)})$, we know that $\\psi \\in C([0,T]; {H_\\diamondsuit^2(\\Omega)})$. Likewise, we have \\[\\psi_t \\in L^\\infty(0,T; {H_\\diamondsuit^2(\\Omega)}) \\cap C([0,T]; H^1_0(\\Omega)).\\] According to~\\cite[\\S2, Lemma 3.3]{temam2012infinite}, this implies that $ \\psi_t$ is weakly continuous from $[0,T]$ into ${H_\\diamondsuit^2(\\Omega)}$. Similarly, we can prove that $\\psi_{tt} \\in C_{w}([0,T]; H_0^1(\\Omega))$. \n\\end{proof}\n\\indent We are now ready to prove a well-posedness result for the nonlinear fJMGT--W III equation.\n\\begin{theorem}[Local well-posedness of the fJMGT--W III equation] \\label{Thm:fJMGT_W_III} Let $\\alpha \\in (0,1]$, $\\tilde{T}>0$, and $\\varrho>0$. Further, assume that $f \\in L^2(0,\\tilde{T}; H_0^1(\\Omega))$ and that\n\t\\begin{equation}\n\t\\|f\\|^2_{L^2(H^1)}+\\|\\psi_0\\|_{H^2}^2+\\|\\psi_1\\|_{H^2}^2+\\|\\psi_2\\|_{H^1}^2 \\leq \\varrho^2. \n\t\\end{equation}\n\tThen there exists $T=T(\\varrho) \\leq \\tilde{T}$, such that the initial boundary-value problem\n\t\\begin{equation}\\label{ibvp_fJMGT_W_III}\n\t\\left \\{\n\t\\begin{aligned}\n\t\\tau \\psi_{ttt} + &(1+2k \\psi_t) \\psi_{tt} - c^2\\Delta\\psi -\\tau c^2\\Delta \\psi_t - \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\psi_t =f\\hspace*{-2mm}&&\\text{in }\\Omega\\times(0,T), \\\\[1mm]\n\t&\\psi=\\,0&&\\text{on }\\partial\\Omega\\times(0,T),\\\\[1mm]\n\t&(\\psi, \\psi_t, \\psi_{tt})=\\,(\\psi_0, \\psi_1, \\psi_2)&&\\mbox{in }\\Omega\\times \\{0\\},\n\t\\end{aligned} \\right.\n\t\\end{equation}\n\thas a unique solution $\\psi \\in X^{\\textup{high}}_{\\textup{fMGT III}}$, which satisfies\t\n\t\\begin{equation} \\label{energy_est2_fJMGT_W_III}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] \\|\\psi\\|^2_{X^{\\textup{high}}_{\\textup{fMGT III}}}\n\t\\end{multlined}\n\t\\lesssim\\,\\begin{multlined}[t] \t\\|f\\|^2_{L^2(H^1)}+\\|\\psi_0\\|_{H^2}^2+\\|\\psi_1\\|_{H^2}^2+\\|\\psi_2\\|_{H^1}^2. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tThe proof follows by applying the Banach Fixed-point theorem to the mapping $\\mathcal{T}: w \\mapsto \\psi$, where $\\psi$ solves the linearized equation \\eqref{fMGT1_sigma} with $\\sigma= 2k w_t$ and\n\t\\begin{equation} \\label{defBR}\n\tw \\in B_R:=\\{w \\in X^{\\textup{high}}_{\\textup{fMGT III}}\\, : \\|w\\|_{ X^{\\textup{high}}_{\\textup{fMGT III}}}\\leq R, \\ w(0)=\\psi_0, \\, w_t(0)=\\psi_1, \\, w_{tt}(0)=\\psi_2 \\},\n\t\\end{equation}\n\twith $R>0$ specified below. Note that\n\t\\[\n\t\\begin{aligned}\n\t\\nLinfLinf{\\sigma}+\\|\\sigma\\|_{L^\\infty(W^{1,4})} \\leq&\\, 2C_{H^2, L^\\infty} |k| \\|w_t\\|_{L^\\infty(H^2)}+2C_{H^1, L^4} |k|\\|w_t\\|_{L^\\infty(H^2)}\\\\ \\lesssim&\\, R.\n\t\\end{aligned}\n\t\\]\n\tThus by employing estimate \\eqref{energy_est2_fMGT_W_III}, where the hidden constant has the form $C_1 \\exp(C_2 (R+1)\\tilde{T})$, it immediately follows that $\\mathcal{T}$ is a well-defined self-mapping on $B_R^{\\textup{W}}$, provided $R>0$ is chosen so that \n\t\\[\n\t\\sqrt{C_1 \\exp(C_2 (R+1)\\tilde{T})} \\, \\varrho \\leq R. \n\t\\]\n\t\\indent Next, we prove that $\\mathcal{T}$ is strictly contractive. {Note that we will prove contractivity with respect to the weaker norm $\\|\\cdot\\|_{X^{\\textup{low}}_{\\textup{fMGT III}}}$; recall the definition of the space $X^{\\textup{low}}_{\\textup{fMGT III}}$ in \\eqref{def_X_fMGTIII} for $\\alpha \\in (0,1)$ and \\eqref{def_X_fMGTIII_alphaone} for $\\alpha=1$}.\\\\\n\t\\indent We take any $w^{(1)}$ and $w^{(2)}$ in $B_R^W$ and set $\\psi^{(1)}=\\mathcal{T} w^{(1)}$ and $\\psi^{(2)}=\\mathcal{T} w^{(2)} $. We also introduce the short-hand notation for the differences \\[\\overline{\\psi}=\\psi^{(1)} -\\psi^{(2)}, \\qquad \\overline{w}= w^{(1)} -w^{(2)}.\\] Then we know that $\\overline{\\psi}$ solves the linear equation\n\t\\begin{equation} \\label{West_contract_eq}\n\t\\tau\\overline{\\psi}_{ttt}+(1+2k w_t^{(1)})\\overline{\\psi}_{tt}-c^2 \\Delta \\overline{\\psi}-\\tau c^2 \\Delta \\overline{\\psi}_t-\\delta {\\textup{D}}_t^{1-\\alpha} \\Delta \\overline{\\psi}_t +2k \\overline{w}_t\\psi^{(2)} _{tt}=0\n\t\\end{equation}\n\tand has zero initial conditions. Employing the lower-order estimate \\eqref{energy_est1_fJMGT_W_III} with $\\sigma=2k w_t^{(1)}$ and $f= -2k \\overline{w}_t\\psi^{(2)} _{tt}$ yields the bound\n\t\\begin{align}\n\t\\|\\overline{\\psi}\\|_{X^{\\textup{low}}_{\\textup{fMGT III}}}\\leq&\\, \\sqrt{C_1\\exp(C_2(R+1)\\tilde{T})} \\nLtwoLtwo{f}\\\\\n\t\\leq&\\, \n\t\\begin{multlined}[t]2\\sqrt{C_1\\exp(C_2(R+1)\\tilde{T})}|k| \\nLinfLfour{\\psi_{tt}^{(2)}}\\sqrt{\\tilde{T}}\\nLinfLfour{\\overline{w}_t} \\end{multlined}\\\\\n\t\\leq&\\, \\theta \\|\\overline{w}\\|_{X^{\\textup{low}}_{\\textup{fMGT III}}}. \n\t\\end{align}\n\tThus we can guarantee that $\\theta \\in (0,1)$ and obtain strict contractivity of $\\mathcal{T}$ by decreasing $\\tilde{T}$. \\\\\n\t\\indent We note that the space $B_R$ with the metric induced by the norm $\\|\\cdot\\|_{X^{\\textup{low}}_{\\textup{fMGT III}}}$ is a closed subset of a complete normed space; cf.~\\cite[Theorem~4.1]{kaltenbacher2021inviscid}. Existence of a unique solution in $B_R$ then follows by Banach's Fixed-point theorem. \n\\end{proof}\n\n\\subsection{Limiting behavior of the fJMGT--W III equation} \\label{Sec:Limit}\nWe next discuss the limit with respect to the order of differentiation. Given $\\alpha \\in (0,1)$, under the assumptions of Theorem~\\ref{Thm:fJMGT_W_III}, let $\\psi^\\alpha$ be the solution of the fJMGT--W III equation:\n{\\[\n\t\\tau \\psi^\\alpha_{ttt}+(1+2k\\psi^\\alpha_t)\\psi^\\alpha_{tt}-c^2 \\Delta \\psi^\\alpha -\\tau c^2\\Delta \\psi^\\alpha_{t}- \\delta {\\textup{D}}_t^{2-\\alpha}\\Delta\\psi^\\alpha=f.\n\t\\]}\n Let $\\psi$ solve the corresponding JMGT--Westervelt equation obtained by setting $\\alpha=1$ above. Then the difference $\\overline{\\psi}=\\psi^{\\alpha}-\\psi$ solves \n\\begin{equation} \\label{fJMGT_W_III_diff}\n\\begin{aligned}\n&\\tau \\overline{\\psi}_{ttt} + (1+2k\\psi_t^{\\alpha})\\overline{\\psi}_{tt} - c^2\\Delta\\overline{\\psi} -\\tau c^2\\Delta \\overline{\\psi}_t - \\delta \\Dt^{1-\\alpha} \\Delta \\overline{\\psi}_t+2k\\overline{\\psi}_t\\psi_{tt} \\\\\n=&\\, \\delta (\\Dt^{1-\\alpha}\\Delta \\psi_t-\\Delta \\psi_t).\n\\end{aligned}\n\\end{equation}\nSimilarly to the proof of Proposition~\\ref{Prop:fMGT_III_lower}, testing with $\\overline{\\psi}_{tt}$ (which we are allowed to do in this higher-regularity setting) leads to \n\\begin{equation} \n\\begin{aligned}\n\\nLtwo{\\overline{\\psi}_{tt}(t)}^2 \n+ \\nLtwo{\\nabla \\overline{\\psi}_{t}(t)}^2 \\lesssim\\,\\nLtwoLtwo{\\Dt^{1-\\alpha}\\nabla \\psi_t-\\nabla \\psi_t}^2.\n\\end{aligned}\n\\end{equation}\nBy recalling Lemma~\\ref{Lemma:Limit}, we find that if $\\psi_t(0)=0$, then\n\\begin{equation}\n\\lim_{\\alpha \\rightarrow1^-} \\nLtwoLtwo{\\Dt^{1-\\alpha}\\nabla \\psi_t-\\nabla \\psi_t}=0,\n\\end{equation}\nand thus arrive at the following result. \n\\begin{proposition} Let the assumptions of Theorem~\\ref{Thm:fJMGT_W_III} hold with $\\psi_1=0$. Let $\\{\\psi^\\alpha\\}_{\\alpha \\in (0,1)}$ be the family of solutions to the \\textup{fJMGT--W III} equation and let $\\psi$ solve the corresponding \\textup{JMGT} equation with $\\alpha=1$. Then $\\psi^{\\alpha}$ converges to $\\psi$ in $W^{1, \\infty}(0,T; H_0^1(\\Omega)) \\cap W^{2, \\infty}(0,T; L^2(\\Omega))$ as $\\alpha \\rightarrow 1^-$.\n\\end{proposition}\n\n\\section{Well-posedness of the fractional model} \\label{Sec:FixedPoint}\n\n\\section{Introduction} \\label{Sec:Introduction}\n\\begin{equation} \\label{Eq:FracMGT}\n\\begin{aligned}\n\\tau^\\alpha D_t^{\\alpha}\\psi_{tt}+(1- k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi-\\tau^\\alpha c^2 D_t^\\alpha \\Delta \\psi -\\delta D_t^{1-\\alpha} \\Delta \\psi_t= k(|\\nabla \\psi|^2)_t\n\\end{aligned}\n\\end{equation}\n\\section{The limit as $\\alpha \\rightarrow 1$} \\label{Sec:Limit}\n\n\\subsection{Limiting behavior of the fMGT--W and fJMGT--W I equations}\nThe difference $\\overline{\\psi}=\\psi^{\\alpha}-\\psi$ solves \n\\begin{equation} \\label{fJMGT_W_III_diff_lin}\n\\begin{aligned}\n&\\tau^\\alpha \\Dt^{2+\\alpha}\\overline{\\psi} + \\sigma\\overline{\\psi}_{tt} - c^2\\Delta\\overline{\\psi} -\\tau^\\alpha c^2\\Delta \\Dt^\\alpha\\overline{\\psi} - \\delta \\Dt^\\beta \\Delta \\overline{\\psi}_t \\\\\n&=\\,(\\tau\\psi_{ttt}-\\tau^\\alpha\\Dt^{2+\\alpha}\\psi_{tt}) - c^2\\Delta(\\tau\\psi_t-\\tau^\\alpha\\Dt^\\alpha\\psi) - \\delta \\Delta(\\psi_t-\\Dt^\\beta\\psi_t)=:\\tilde{f}\n\\end{aligned}\n\\end{equation}\nin the linear case and \n\\begin{equation} \\label{fJMGT_W_III_diff}\n\\begin{aligned}\n&\\tau^\\alpha \\Dt^{2+\\alpha}\\overline{\\psi} + (1+2k\\psi_t^{\\alpha})\\overline{\\psi}_{tt} - c^2\\Delta\\overline{\\psi} -\\tau^\\alpha c^2\\Delta \\Dt^\\alpha\\overline{\\psi} - \\delta \\Dt^\\beta \\Delta \\overline{\\psi}+2k\\psi_{tt}\\overline{\\psi}_t =\\tilde{f}\n\\end{aligned}\n\\end{equation}\nin the nonlinear case with vanishing initial data and $\\beta=\\beta(\\alpha)$ as in \\eqref{def_beta} (which implies that the $\\beta$ difference term on the right-hand side just vanishes in case of the \\textup{fJMGT--W} equation).\n\nMultiplication with $\\Dt^{1+\\alpha}\\overline{\\psi}$ with the abbreviations \n$\\sigma^\\alpha=\\sigma$, $\\mu=0$ in the linear case \nand $\\sigma^\\alpha=2k\\psi_t^{\\alpha}$, $\\mu=2k\\psi_{tt}\\overline{\\psi}_t$ in the nonlinear case yields, analogously to the proof of uniqueness in Proposition~\\ref{Prop:fMGT_I},\n\\begin{equation} \\label{limit_diff_I}\n\\begin{aligned}\n\\textup{lhs}:=& \\, \\frac{\\tau^\\alpha}{2}\\nLtwo{\\Dt^{1+\\alpha}\\overline{\\psi}(t)}^2\n+ \\frac{\\tau^\\alpha c^2}{2}\\nLtwo{\\nabla\\Dt^{\\alpha}\\overline{\\psi}(t)}^2\\\\\n&\\quad+ \\cos(\\pi(1-\\alpha)\/2)\\|\\overline{\\psi}_{tt}\\|_{H_t^{-(1-\\alpha)}(L^2)}^2\n+ \\delta \\cos(\\pi\\gamma\/2) \\|\\Dt^m \\nabla \\overline{\\psi}\\|_{L_t^2(L^2)}^2\n\\\\\n=& \\int_0^t\\prodLtwo{\\tilde{f}+c^2\\Delta\\overline{\\psi}-\\sigma^\\alpha\\overline{\\psi}_{tt}+\\mu}{\\Dt^{1+\\alpha}\\overline{\\psi}} \\, \\textup{d} s \\\\\n\\leq&\\, \\begin{multlined}[t]\\frac{\\epsilon}{2} \\|\\Dt^{1+\\alpha}\\overline{\\psi}\\|_{L_t^\\infty(L^2)}^2 \n+\\frac{1}{2\\epsilon} \\|\\tilde{f}+2k\\psi_{tt}\\overline{\\psi}_{t}\\|_{L_t^1(L^2)}^2\n+\\frac{\\epsilon}{2} \\|\\Dt^m\\nabla\\overline{\\psi}\\|_{L_t^2(L^2)}^2 \\\\\n+\\frac{1}{2\\epsilon}\\left(\\|c^2\\nabla\\overline{\\psi}\\|_{H_t^\\rho(L^2)}+\\|\\sigma^\\alpha\\overline{\\psi}_{tt}\\|_{H_t^\\rho(H^{-1})}\\right)^2,\\end{multlined}\n\\end{aligned}\n\\end{equation}\nwhere \n\\[\n\\begin{cases}\n\\gamma=2\\alpha-1\\,, \\quad m=3\/2\\,, \\quad \\rho=\\alpha-1\/2&\\mbox{ for fMGT I,}\\\\\n\\gamma=\\alpha\\,, \\quad m=1+\\alpha\/2\\,, \\quad \\rho=\\alpha\/2&\\mbox{ for fMGT.}\n\\end{cases}\n\\]\nWe know that\n\\[\n\\begin{aligned}\n\\|2k\\psi_{tt}\\overline{\\psi}_{t}\\|_{L^1(L^2)}\n&{\\leq 2|k|C_{H^1,L^6}^\\Omega C_{H^\\alpha,L^3}^\\Omega \\|\\nabla\\psi_{tt}\\|_{L_t^2(L^2)} \\|\\overline{\\psi}_t\\|_{L_t^2(H^\\alpha)}}\\\\\n&{\\lesssim \\|\\nabla\\psi_{tt}\\|_{L^2(L^2)}\n\t\\|\\Dt^{1+\\alpha}\\overline{\\psi}\\|_{L_t^2(L^2)}^{1-\\alpha} \\|\\Dt^\\alpha\\nabla\\overline{\\psi}\\|_{L_t^2(L^2)}^\\alpha,}\\\\\n\\|c^2\\Delta\\overline{\\psi}\\|_{H^\\rho(L^2)}&\\lesssim \\|\\nabla\\Dt^{\\alpha}\\overline{\\psi}\\|_{L^2(L^2)}.\n\\end{aligned}\n\\]\nwhere we have used interpolation; cf.~\\cite[Chapter 7]{Adams}. Let $X^\\sigma$ be either $X^\\sigma_\\textup{fMGT I}$ or $X^\\sigma_\\textup{fMGT}$, depending on the equation. By proceeding similarly to \\eqref{estsigmapsitt_I}--\\eqref{Dt2plusalphapn}, we find that\n\\[\n\\begin{aligned}\n\\|\\sigma^\\alpha\\overline{\\psi}_{tt}\\|_{H^\\rho(H^{-1})}\\lesssim&\\, \\begin{multlined}[t] \\|\\sigma^\\alpha\\|_{X^\\sigma} \\left(\\|\\sigma^\\alpha\\overline{\\psi}_{tt}\\|_{H_t^{\\rho}(H^{-1})} \n+ \\textup{rhs}\\right), \\end{multlined}\n\\end{aligned}\n\\]\nwhere \n\\[\n\\begin{aligned}\n\\textup{rhs}:=&\n c^2\\|\\nabla\\overline{\\psi}\\|_{L^2_t(L^2)} +\\tau^\\alpha c^2\\|\\Dt^\\alpha\\nabla \\overline{\\psi}\\|_{L^2_t(L^2)}\n\t+\\|\\tilde{f}\\|_{L^2(H^{-1})} \\\\\n&\\qquad+ \\delta\n\\begin{cases}\n\\mathfrak{g}_{3\/2-\\alpha}*\\|\\Dt^m\\nabla\\overline{\\psi}\\|_{L^2_t(L^2)}^2&\\mbox{ for fMGT I,}\\\\\n\\|\\Dt^m\\nabla\\overline{\\psi}\\|_{L^2_t(L^2)}^2&\\mbox{ for fMGT.}\n\\end{cases}\n\\end{aligned}\n\\]\nWe can therefore tackle all terms by generalized Gronwall in the \\textup{fMGT--I} case, \nwhereas for \\textup{fMGT} we need to absorb the $\\delta$ term by the lhs $\\delta$ term in \\eqref{limit_diff_I} and therefore need to impose smallness of $\\|\\sigma^\\alpha\\|_{X^\\sigma}$ with an $\\alpha$ dependent bound. \n\nIt remains to estimate the contribution arising from \\[\\tilde{f}= \\tau\\psi_{ttt}-\\tau^\\alpha\\Dt^{2+\\alpha}\\psi - c^2\\Delta(\\tau\\psi_t-\\tau^\\alpha\\Dt^\\alpha\\psi) - \\delta \\Delta(\\psi_t-\\Dt^\\beta\\psi),\\] which we do for each of the difference terms separately, \n\\[\n\\begin{aligned}\n\\|\\tau\\psi_{ttt}-\\tau^\\alpha\\Dt^{2+\\alpha}\\psi_{tt}\\|_{L^1(L^2)}\n\\leq&\\,|\\tau-\\tau^\\alpha|\\, \\|\\psi_{ttt}\\|_{L^1(L^2)}+\\tau^\\alpha \\|(\\Dt-\\Dt^\\alpha)\\psi_{tt}\\|_{L^1(L^2)},\n\\\\\n\\|\\Delta(\\tau\\psi_t-\\tau^\\alpha\\Dt^\\alpha\\psi)\\|_{L^1(L^2)}\n\\leq&\\,|\\tau-\\tau^\\alpha|\\,\\|\\Delta\\psi_t\\|_{L^1(L^2)}+\\tau^\\alpha \\|(\\Dt-\\Dt^\\alpha)\\Delta\\psi\\|_{L^1(L^2)},\n\\end{aligned}\n\\]\nand, in case of \\textup{fJMGT--W I} with $\\beta=2-\\alpha$,\n\\[\n\\begin{aligned}\n\\|\\Delta(\\psi_t-\\Dt^{2-\\alpha}\\psi)\\|_{L^1(L^2)}\n=\\|(\\mbox{id}-\\Dt^{1-\\alpha})\\Delta\\psi_t\\|_{L^1(L^2)}.\n\\end{aligned}\n\\]\nThus, to be able to apply the limits \\eqref{limitalphaL2_1} and \\eqref{limitalphaL2}, we need \n\\[\\psi_{tt},\\, \\Delta\\psi\n{\\in W^{2,1}(0,T;L^2(\\Omega)) \\,, \\ }\n \\Delta\\psi_t \\in W^{1,1}(0,T;L^2(\\Omega)),\\] and, in case of \\textup{fJMGT--W I} additionally $\\psi_t(0)=0$.\\\\\n\\indent Note that the required smoothness of $\\psi$ follows, e.g., from Theorem~\\ref{Thm:fJMGT_K_III} below under restrictive regularity conditions on the initial data. We expect that these assumptions might be relaxed in view of the fact that the $W^{1, \\infty}(0,T; {H_\\diamondsuit^2(\\Omega)}) \\cap W^{2, \\infty}(0,T; H_0^1(\\Omega))\\cap W^{3, \\infty}(0,T; L^2(\\Omega))$ regularity that we obtain from Theorem~\\ref{Thm:fJMGT_W_III} is already very close to what is needed here. Altogether, with \\[\n\\begin{aligned}\nX^{\\textup{low}} = \\left\\{ \\psi\\in H^{1\/2}(0,T; H_0^1(\\Omega)): \n\\Dt^{3\/2}\\psi \\in L^\\infty(0,T; L^2(\\Omega)) \\right\\}\n\\end{aligned}\n\\]\n{and the corresponding norm denoted by $\\|\\cdot\\|_{X^{\\textup{low}}}$}, we have the following results.\n\\begin{proposition}[Limit of the fMGT--I equation] \n\tLet $f \\in H^{1\/2}(0,T;L^2(\\Omega))$, $\\sigma \\in X^\\sigma_\\textup{fMGT I}$, and \\[(\\psi_0, \\psi_1, \\psi_2) \\in ({H_\\diamondsuit^2(\\Omega)}, \\{0\\}, H_0^1(\\Omega)).\\]\n\tFurther, let $\\varrho>0$ be as in Proposition ~\\ref{Prop:fMGT_I} and \n\t$\\|\\sigma\\|_{X^\\sigma_\\textup{fMGT I}} \\leq \\varrho$; let $\\{\\psi^\\alpha\\}_{\\alpha \\in (0,1)}$ be the family of solutions to the \\textup{fMGT--I} equation, let $\\psi$ solve the corresponding equation with $\\alpha=1$ and assume that \n\t\\begin{equation}\\label{assumedregularity}\n\t\\begin{aligned}\n&\\nabla\\psi_{tt}\\in L^2(0,T;L^2(\\Omega)),\\ \\psi_{tt}, \\Delta\\psi \n{\\in W^{2,1}(0,T;L^2(\\Omega)), }\\\\\n&\\Delta\\psi_t \\in W^{1,1}(0,T;L^2(\\Omega)).\n\\end{aligned}\n\\end{equation} \nThen $\\psi^{\\alpha}$ converges to $\\psi$ in the $\\|\\cdot\\|_{X^{\\textup{low}}}$ norm as $\\alpha \\rightarrow 1^{-}$.\n\\end{proposition}\n\\begin{proposition}[Limit of the fJMGT--W I equation] \nLet $f \\in H^{1\/2}(0,T;L^2(\\Omega))$ and $(\\psi_0, \\psi_1, \\psi_2) \\in ({H_\\diamondsuit^2(\\Omega)}, \\{0\\}, H_0^1(\\Omega))$.\n\tFurthermore, let $\\varrho>0$ be as in Theorem~\\ref{Thm:fJMGT_W_I} and\n\t\\[\\|f\\|^2_{H^{1\/2}(H^1)}+\\|\\psi_0\\|_{H^2}^2\n\t+\\|\\psi_2\\|_{H^1}^2 \\leq \\varrho^2;\\] let $\\{\\psi^\\alpha\\}_{\\alpha \\in (0,1)}$ be the family of solutions to the \\textup{fJMGT--W I} equation, let $\\psi$ solve the corresponding equation with $\\alpha=1$, and assume that \\eqref{assumedregularity} holds. Then $\\psi^{\\alpha}$ converges to $\\psi$ in the $\\|\\cdot\\|_{X^{\\textup{low}}}$ norm as $\\alpha \\rightarrow 1^{-}$.\n\\end{proposition}\n\\begin{proposition}[Limit of the fMGT equation] \n\tAssume that $f \\in H^{1\/2}(0,T;L^2(\\Omega))$, $\\sigma=0$, and $(\\psi_0, \\psi_1, \\psi_2) \\in ({H_\\diamondsuit^2(\\Omega)}, {H_\\diamondsuit^2(\\Omega)}, H_0^1(\\Omega))$. Let $\\{\\psi^\\alpha\\}_{\\alpha \\in (0,1)}$ be the family of solutions to the \\textup{fMGT} equation, let $\\psi$ solve the corresponding equation with $\\alpha=1$ and assume that \\eqref{assumedregularity} holds. Then $\\psi^{\\alpha}$ converges to $\\psi$ in the $\\|\\cdot\\|_{X^{\\textup{low}}}$ norm as $\\alpha \\rightarrow 1^{-}$.\n\\end{proposition}\n\n\\subsection{Analysis of the fMGT II equation with $\\boldsymbol{\\sigma=0}$} \\label{Sec:LinearAnalysis}\nWe carry out an analysis of the fMGT II equation\n\\[\n\\tau^\\alpha \\Dt^{2+\\alpha}\\psi + \\psi_{tt} - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi - \\delta \\Dt^\\alpha \\Delta \\psi = f.\n\\]\nIt can be rewritten in terms of \\[z=\\tau^\\alpha \\Dt^\\alpha\\psi+\\psi\\] as\n\\begin{equation}\\label{zform_fMGT4}\nz_{tt}-(c^2+\\tfrac{\\delta}{\\tau^\\alpha})\\Delta z + \\tfrac{\\delta}{\\tau^\\alpha} \\, \\mathfrak{k}_\\alpha*\\Delta z =f\n{-\\delta\\tau^{-\\alpha} E_{\\alpha,1}(-(\\tfrac{t}{\\tau})^{\\alpha})\\Delta\\psi_0}\n\\end{equation}\nwith the kernel function\n\\begin{equation}\\label{kalpha}\n\\mathfrak{k}_\\alpha(t)= \\tau^{-\\alpha}t^{\\alpha-1}E_{\\alpha,\\alpha}(-(\\tfrac{t}{\\tau})^{\\alpha}) = - \\frac{\\textup{d}}{\\textup{d}t} E_{\\alpha,1}(-(\\tfrac{t}{\\tau}^\\alpha)).\n\\end{equation}\nWe observe that this kernel has the following properties:\n\\begin{equation}\\label{kerprop}\n\\mathfrak{k}_\\alpha(t)\\geq0\\,, \\quad \n\\lim_{t\\to0+} \\mathfrak{k}_\\alpha(t) = +\\infty\\,, \\quad \n\\int_0^\\infty \\mathfrak{k}_\\alpha(t)\\, \\textup{d} t =1\\,, \\quad \n\\mathfrak{k}_\\alpha'(t)\\leq 0.\n\\end{equation}\n\\begin{table}[h]\n\t\\captionsetup{width=.9\\linewidth}\n\t\\begin{center} \\small\n\t\t\\begin{tabular}{|m{1.14cm}||m{10.6cm}|}\n\t\t\t\\hline \\vspace*{2mm}\n\t\t\t{fMGT} & \\vspace*{2mm} \\textbf{Linear time-fractional acoustic equations} \\\\[6pt]\n\t\t\t\\Xhline{2\\arrayrulewidth} \\hline \\vspace*{4mm}\n\t\t\t\\centering\t\\hphantom{I} & $(1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)(\\psi_{tt}-c^2 \\Delta \\psi)- \\delta \\Delta\\psi_{t}=f$ \\\\[2mm]\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{5mm}\n\t\t\t\\centering\t$z$ form & {$z_{tt}-c^2\\Delta z -\\dfrac{\\delta}{\\tau^\\alpha}\\Dt^{1-\\alpha}\\Delta z+ \\dfrac{\\delta}{\\tau^\\alpha} \\displaystyle \\int_0^t \\mathfrak{k}(t-s) \\Delta z \\, \\textup{d} s =\\tilde{f}$, \\ \\ $\\mathfrak{k}=\\frac{\\textup{d}}{\\textup{d}t}(\\mathfrak{g}_{1-\\alpha}*\\mathfrak{k}_\\alpha)$} \\\\[5mm]\n\t\t\t\\hline \\hline \\vspace*{3mm}\t\n\t\t\t\\centering\tI & $(1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)(\\psi_{tt}-c^2 \\Delta \\psi)- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta\\psi_{t}=f$\\\\[2mm]\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{5mm}\n\t\t\t\\centering\t$z$ form & {$z_{tt}-c^2\\Delta z -\\dfrac{\\delta}{\\tau^\\alpha}\\Dt^{2-2\\alpha}\\Delta z+ \\dfrac{\\delta}{\\tau^\\alpha} \\displaystyle \\int_0^t \\mathfrak{k}(t-s) \\Delta z \\, \\textup{d} s =\\tilde{f}$, \\ \\ $\\mathfrak{k}=\\frac{\\textup{d}}{\\textup{d}t}(\\mathfrak{g}_{2(1-\\alpha)}*\\mathfrak{k}_\\alpha)$}\\\\[5mm]\n\t\t\t\\hline \\hline \\vspace*{3mm}\n\t\t\t\\centering\tII & $(1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)(\\psi_{tt}-c^2 \\Delta \\psi)-\\delta {\\textup{D}}_t^\\alpha \\Delta\\psi=f$ \\\\[2mm]\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{5mm}\n\t\t\t\\centering\t$z$ form & $z_{tt}-\\left(c^2+\\dfrac{\\delta}{\\tau^\\alpha}\\right)\\Delta z + \\dfrac{\\delta}{\\tau^\\alpha} \\displaystyle \\int_0^t \\mathfrak{k}(t-s) \\Delta z \\, \\textup{d} s =f$, \\ \\ $\\mathfrak{k}=\\mathfrak{k}_\\alpha$\\\\[4mm]\n\t\t\t\\hline \\hline \\vspace*{3mm}\n\t\t\t\\centering\tIII & $(1+\\tau \\partial_t)(\\psi_{tt}-c^2 \\Delta \\psi)- \\delta {\\textup{D}}_t^{1-\\alpha}\\Delta\\psi_{t}=f$ \\\\[2mm]\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{5mm}\t\n\t\t\t\\centering\t$z$ form & {$z_{tt}-c^2\\Delta z -\\dfrac{\\delta}{\\tau}\\Dt^{1-\\alpha}\\Delta z+ \\dfrac{\\delta}{\\tau} \\displaystyle \\int_0^t \\mathfrak{k}(t-s) \\Delta z\\, \\textup{d} s =\\tilde{f}$, \\ \\ $\\mathfrak{k}=\\frac{\\textup{d}}{\\textup{d}t}(\\mathfrak{g}_{1-\\alpha}*\\mathfrak{k}_1)$} \\\\[5mm]\n\t\t\t\\hline \t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\small Linear time-fractional models with $\\mathfrak{k}_\\alpha(t)= \\tau^{-\\alpha}t^{\\alpha-1}E_{\\alpha,\\alpha}(-(\\tfrac{t}{\\tau})^{\\alpha})$, \n\t\t$\\mathfrak{k}_1(t)=\\tau^{-1}\\exp(-(\\tfrac{t}{\\tau}))$, and $\\mathfrak{g}_\\gamma(t)=t^{-\\gamma}$.\n\t\t\\label{table:fMGT}}\n\\end{table}\n~\\\\\nMoreover, since the function $t\\mapsto E_{\\alpha,1}(-(\\tfrac{t}{\\tau}^\\alpha))$ is completely monotone, by Schoenberg's theorem \\cite[Theorem 7.13]{wendland2004} we conclude that the kernel $\\mathfrak{k}_\\alpha$ itself and also the kernel $t\\mapsto \\int_t^\\infty \\mathfrak{k}_\\alpha(s)\\, \\textup{d} s $ is positive definite.\nTherefore, the next result is a straightforward consequence of~\\cite[Theorem 4.5]{cannarsaSforza2008}, where the regularity of $\\psi$ as the unique solution of \n\t\\begin{equation} \\label{ODEpsifromz}\n\\tau^\\alpha \\Dt^\\alpha\\psi+\\psi=z\n\t\\end{equation}\na.e. in $(0,T)$ with $\\psi(0)=\\psi_0$ follows from the fact that $\\textup{I}^\\alpha$ maps $L^\\infty(0,T)$ to $C^{0,\\alpha}(0,T)$; see~\\cite[Corollary 2, p.\\ 56]{samko1993fractional}.\n\\begin{proposition}[Well-posedness of the fMGT II equation] \\label{Prop:fMGT_4} \nLet $\\alpha \\in (0,1]$. Given $f \\in L^1(0,T; L^2(\\Omega))$, $(z_0,z_1) \\in H_0^1(\\Omega)\\times L^2(\\Omega)$, and $\\psi_0\\in{H_\\diamondsuit^2(\\Omega)}$, there exists a unique mild solution \n\\[z\\in W^{2,\\infty}(0,T; H^{-1}(\\Omega))\\cap W^{1,\\infty}(0,T; L^2(\\Omega))\\cap L^\\infty(0,T; H_0^1(\\Omega))\\] of \\eqref{zform_fMGT4} with $(z, z_t)\\vert_{t=0}=(z_0, z_1)$.\nCorrespondingly, for inital data\n\\[\n(\\psi_0, \\psi_1, \\psi_2) \\in \n\\begin{cases}\n{H_\\diamondsuit^2(\\Omega)}\\times \\{0\\}\\times H^{-1}(\\Omega) \\hphantom{~} \\ \\mbox{ if }\\alpha<1, \\\\\n{H_\\diamondsuit^2(\\Omega)}\\times H_0^1(\\Omega)\\times L^2(\\Omega) \\ \\mbox{ if }\\alpha=1,\n\\end{cases}\n\\] \nthere exists a unique solution \n\\[\\psi\n\\in C^{2,\\alpha}(0,T; H^{-1}(\\Omega))\\cap C^{1,\\alpha}(0,T; L^2(\\Omega))\\cap C^{0,\\alpha}(0,T; H_0^1(\\Omega))\\] \naccording to \\eqref{psifromz}; that is, the unique solution of \\eqref{ODEpsifromz} with $\\psi(0)=\\psi_0$ and thus of \n\t\\begin{equation} \\label{fMGT4}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] \\langle \\tau^\\alpha \\Dt^{2+\\alpha}\\psi + \\psi_{tt}, v\\rangle_{H^{-1}, H_0^1} \\\\+ \\prodLtwo{c^2\\nabla\\psi +\\tau^\\alpha c^2\\nabla \\Dt^\\alpha\\psi +\\delta \\Dt^\\alpha \\nabla \\psi}{\\nabla v} = \\prodLtwo{f}{v}, \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\ta.e. in $(0,T)$, with $(\\psi, \\psi_t, \\psi_{tt})\\vert_{t=0}=(\\psi_0, \\psi_1, \\psi_2)$.\nFurthermore, the solution satisfies the estimate\n\t\\begin{equation} \\label{energy_est1_fMGT4}\n\t\\begin{aligned}\n&\\begin{multlined}[t]\\|\\tau^\\alpha D_{tt}\\Dt^\\alpha \\psi+\\psi_{tt}\\|^2_{L^\\infty(H^{-1})}\n+\\|\\tau^\\alpha D_{t}\\Dt^\\alpha\\psi+\\psi_t\\|_{L^\\infty(L^2)}^2 \\\\ \\hspace*{6cm}+ \\|\\tau^\\alpha \\Dt^\\alpha\\nabla\\psi+\\nabla\\psi\\|_{L^\\infty(L^2)}^2 \\end{multlined} \\\\\n\\lesssim&\\, \\begin{cases}\n\\|\\psi_2\\|_{H^{-1}(\\Omega)}^2 + \\|\\nabla\\psi_0\\|_{L^2(\\Omega)}^2\n+\\|f\\|_{\\LoneLtwo}^2 \\hphantom{\\hspace*{2.8cm}} &\\mbox{ for }\\alpha<1,\\\\\n\\|\\tau\\psi_2+\\psi_1\\|_{L^2(\\Omega)}^2 +\\|\\tau\\nabla\\psi_1+\\nabla\\psi_0\\|_{L^2(\\Omega)}^2\n+\\|f\\|_{\\LoneLtwo}^2\\quad & \\mbox{ for }\\alpha=1.\n\\end{cases}\n\t\\end{aligned}\n\t\\end{equation}\t\n\\end{proposition}\nNote that the condition $\\psi_1=0$ in case $\\alpha<1$ is enforced by the singularity at zero in the identity \n\\[z_t=\\Dt\\left(\\tau^\\alpha \\Dt^\\alpha\\psi+\\psi\\right)= \\tau^\\alpha \\left(\\Dt^{1+\\alpha}\\psi + \\psi_t(0)\\frac{t^{-\\alpha}}{\\Gamma(1-\\alpha)}\\right) + \\psi_t\\,.\n\\]\nSpatially higher-order regularity can be obtained with more regular initial data and the source term by using the multiplier $(-\\Delta)^m z_t$ in place of $z_t$ (which led to the energy estimate \\eqref{energy_est1_fMGT4}). To study the limiting behavior, we will make use of the following resulting estimate in case $\\alpha=1$, $m=1$:\n\\begin{equation} \\label{energy_est2_fMGT4_alpha1}\n\\begin{aligned}\n&\\|\\tau \\psi_{ttt}+\\psi_{tt}\\|^2_{L^\\infty(L^2)}\n+\\|\\tau \\nabla\\psi_{tt}+\\nabla\\psi_t\\|^2_{L^\\infty(L^2)} + \\|\\tau\\Delta\\psi_t+\\Delta\\psi\\|^2_{L^\\infty(L^2)}\\\\\n\\lesssim&\\, \\|\\tau\\nabla\\psi_2+\\nabla\\psi_1\\|_{L^2(\\Omega)}^2 +\\|\\tau\\Delta\\psi_1+\\Delta\\psi_0\\|_{L^2(\\Omega)}^2\n+\\|\\nabla f\\|_{\\LoneLtwo}^2.\n\\end{aligned}\n\\end{equation}\t\n\\begin{remark}[On the $z$-form of the other linear models]\\label{rem:fMGT123}\nAs already mentioned, a reformulation of the type \\eqref{zform_fMGT4} is available for the other linear models \\textup{fMGT}, \\textup{fMGT I}, and \\textup{fMGT III} as well; see Table~\\ref{table:fMGT}. However, it is not clear whether properties \\eqref{kerprop} still hold for the corresponding kernels. \\\\\n\\indent Due to the term $\\Dt^{\\beta-\\gamma} ( z-\\mathfrak{k}_\\gamma*z)$ present in these models with \\[\\epsilon=\\beta-\\gamma>0\\in\\{1-\\alpha,2-2\\alpha\\},\\] one might consider using the adjoint of $(\\Dt^\\epsilon)^{-1}$ in the multiplier; that is, test with $((\\Dt^\\epsilon)^{-1})^*z_t$ in place of $z_t$. Indeed, this leads to tractable terms\n\\begin{equation}\\label{testIepsilon}\n\\begin{aligned}\n&-\\tfrac{\\delta}{\\tau^\\alpha}\\int_0^t\\prodLtwo{\\Dt^{\\epsilon} \\Delta( z-\\mathfrak{k}_\\gamma*z)}{((\\Dt^\\epsilon)^{-1})^*z_t}\\, \\textup{d} t\n= \\tfrac{\\delta}{\\tau^\\alpha}\\int_0^t\\prodLtwo{\\nabla( z-\\mathfrak{k}_\\gamma*z)}{z_t}\\, \\textup{d} t,\\\\\n&\\int_0^t\\prodLtwo{z_{tt}}{((\\Dt^\\epsilon)^{-1})^*z_t}\\, \\textup{d} t \n= \\int_0^t\\prodLtwo{(\\Dt^\\epsilon)^{-1} \\Dt^\\epsilon \\textup{I}^{\\epsilon} z_{tt}}{z_t}\\, \\textup{d} t\n= \\int_0^t\\prodLtwo{\\Dt^{2-\\epsilon} z}{z_t}\\, \\textup{d} t\\,.\n\\end{aligned}\n\\end{equation}\nHowever, the $c^2$ term does not appear to be amenable to useful estimates since in \n\\[\n- c^2\\int_0^t\\prodLtwo{\\Delta z}{((\\Dt^\\epsilon)^{-1})^*z_t}\\, \\textup{d} t\n= c^2\\int_0^t\\prodLtwo{\\nabla \\textup{I}^\\epsilon }{\\nabla z_t}\\, \\textup{d} t\n\\]\nthe difference between the time differentiation orders of the two factors is $1-(-\\epsilon)=1+\\epsilon>1$, which leads to an adverse sign, while the norm of $\\nabla z_t$ is not controllable by any of the other left-hand side terms resulting from \\eqref{testIepsilon}.\n\\end{remark}\n\\subsection{Limiting behavior of the fMGT II equation} \nFor $\\alpha\\in(0,1]$, we denote by $\\psi^\\alpha$ the solution according to Proposition~\\ref{Prop:fMGT_4} under the assumptions \\[(\\psi_0, \\psi_1, \\psi_2) \\in {H_\\diamondsuit^2(\\Omega)}\\times \\{0\\}\\times H_0^1(\\Omega),\\quad f, \\nabla f\\in L^1(0,T; L^2(\\Omega)).\\]\nLet $\\psi$ be the solution of the corresponding MGT equation. Note that then the corresponding functions $z^\\alpha$ and $z$ satisfy the initial conditions \n\\[z^\\alpha(0)=z(0)=\\psi_0,\\quad z^\\alpha_t(0)=0, \\quad z_t(0)=\\tau\\psi_2.\\] Hence to achieve compatibility, besides $\\psi_1=0$ (see the proof of Proposition~\\ref{Prop:fMGT_4}), we also have to assume $\\psi_2=0$.\nThen the difference $\\overline{z}=z^\\alpha-z$ solves \n\\begin{equation} \\label{fMGT_4_diff}\n\\begin{aligned}\n&\\overline{z}_{tt}-(c^2+\\tfrac{\\delta}{\\tau^\\alpha})\\Delta \\overline{z} + \\delta \\, \\mathfrak{k}_\\alpha*\\Delta \\overline{z} \n&=\\delta \\left( (\\tau^{-\\alpha}-\\tau^{-1})\\Delta z - (\\mathfrak{k}_\\alpha-\\mathfrak{k}_1)*\\Delta z\\right) \n\\end{aligned}\n\\end{equation}\nwith homogeneous initial data $\\overline{z}(0)=\\overline{z}_t(0)=0$. Testing with $\\overline{z}_t$ leads to \n\\begin{equation}\\label{fMGT_4_estdiffz}\n\\|\\overline{z}_t\\|_{L^\\infty(L^2)}^2 + 2c^2 \\|\\nabla \\overline{z}\\|_{L^\\infty(L^2)}^2\t\n\\leq 4 \\delta\\Bigl(|\\tau^{-\\alpha}-\\tau^{-1}| + \\|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1\\|_{L^1(0,T)}\\Bigr) \\|\\Delta z\\|_{\\LoneLtwo}^2,\n\\end{equation}\nwhere we can estimate $\\|\\Delta z\\|_{\\LoneLtwo}^2$ according to \\eqref{energy_est2_fMGT4_alpha1} and \n\\[\n\\|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1\\|_{L^1(0,T)}\n=\\int_0^T \\left| \\tau^{-\\alpha}t^{\\alpha-1}E_{\\alpha,\\alpha}(-(\\tfrac{t}{\\tau})^{\\alpha}) \n- \\tau^{-1}\\exp(-(\\tfrac{t}{\\tau}))\\right|\\, \\textup{d} t\\to0\\ \\mbox{ as }\\alpha\\to1^{-},\n\\]\nby Lebesgue's Dominated Convergence theorem.\nThus, with \n\\[\n\\begin{aligned}\n\\psi^\\alpha-\\psi &= \\mathfrak{k}_\\alpha*z^\\alpha-\\mathfrak{k}_1*z =(\\mathfrak{k}_\\alpha-\\mathfrak{k}_1)*z+\\mathfrak{k}_\\alpha*\\overline{z}\\\\\n(\\psi^\\alpha-\\psi)_t(t) &=(\\mathfrak{k}_\\alpha-\\mathfrak{k}_1)(t)\\,z^\\alpha(0)+ ((\\mathfrak{k}_\\alpha-\\mathfrak{k}_1)*z_t)(t)\n+\\mathfrak{k}_\\alpha(t)\\overline{z}(0)+(\\mathfrak{k}_\\alpha*\\overline{z}_t)(t)\\\\\n\\end{aligned}\n\\]\nwe obtain, using $\\overline{z}(0)=0$, $z^\\alpha(0)=\\psi_0$, and Young's Convolution inequality,\n\\begin{equation} \n\\begin{aligned}\n&\\|(\\psi^\\alpha-\\psi)_t\\|_{L^\\infty(L^2)}^2 + \\|\\nabla (\\psi^\\alpha-\\psi)\\|_{L^\\infty(L^2)}^2\\\\\n\\lesssim&\\,\n\\|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1\\|_{L^p(0,T)}^2\\|\\psi_0\\|_{L^2(\\Omega)}^2 \n+ \\|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1\\|_{L^1(0,T)}^2\\|z_t\\|_{L^p(L^2)}^2\n+\\|\\mathfrak{k}_\\alpha\\|_{L^1(0,T)}^2 \\|\\overline{z}_t\\|_{L^p(L^2)}^2\\\\\n&\\quad+ \\|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1\\|_{L^1(0,T)}^2\\|\\nabla z_t\\|_{L^\\infty(L^2)}^2\n+ \\|\\mathfrak{k}_\\alpha\\|_{L^1(0,T)}^2 \\|\\nabla\\overline{z}\\|_{L^\\infty(L^2)}^2\\,,\n\\end{aligned}\n\\end{equation}\nwhere we can use Proposition~\\ref{Prop:fMGT_4} and estimate \\eqref{fMGT_4_estdiffz} to further bound the right-hand side terms.\nNote that since \\[\\|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1\\|_{L^\\infty(0,T)}=\\lim_{t\\to0}|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1|(t)=+\\infty,\\] we only get an estimate of $\\|(\\psi^\\alpha-\\psi)_t\\|_{L^\\infty(L^2)}^2$ if we additionally assume $\\psi_0=0$. Furthermore,\n\\[\n\\begin{aligned}\n\\|\\mathfrak{k}_\\alpha-\\mathfrak{k}_1\\|_{L^p(0,T)}\n\\leq& \n|\\tau^{-\\alpha}-\\tau^{-1}|\\|\\cdot^{\\alpha-1}E_{\\alpha,\\alpha}(-(\\tfrac{\\cdot}{\\tau})^{\\alpha})\\|_{L^p(0,T)}\\\\\n&+\\tau^{-1}\\|\\cdot^{\\alpha-1}-1\\|_{L^p(0,T)}\\|E_{\\alpha,\\alpha}(-(\\tfrac{\\cdot}{\\tau})^{\\alpha})\\|_{L^\\infty(0,T)}\\\\\n&+\\tau^{-1}\\|E_{\\alpha,\\alpha}(-(\\tfrac{\\cdot}{\\tau})^{\\alpha})-\\exp(-(\\tfrac{\\cdot}{\\tau}))\\|_{L^p(0,T)}\n\\to0 \\ \\ \\mbox{ as }\\alpha\\to1^{-}\\,,\n\\end{aligned}\n\\]\nwhere the only critical term is the one containing the singularity, $\\int_0 |t^{-(1-\\alpha)} -1|^p\\, \\textup{d} t$. Its convergence to zero follows from Lebesgue's Dominated Convergence theorem with the $L^1$ bound \\[2^{p-1}(t^{-p(1-\\alpha_0)}+1)\\] for $1\\geq\\alpha\\geq\\alpha_0>1-\\frac{1}{p}$. Thus we arrive at the following result. \n\\begin{proposition} Let $\\psi_0 \\in H_0^1(\\Omega)\\cap H^2(\\Omega)$ and $f$, $\\nabla f\\in L^1(0,T; L^2(\\Omega))$. Further, let $\\{\\psi^\\alpha\\}_{\\alpha \\in (0,1)}$ be the family of solutions to the \\textup{fMGT II} equation and let $\\psi$ solve the corresponding equation with $\\alpha=1$, where the initial data is in both cases given by \\[(\\psi^\\alpha, \\psi^\\alpha_t, \\psi^\\alpha_{tt})\\vert_{t=0}=(\\psi, \\psi_t, \\psi_{tt})\\vert_{t=0}=(\\psi_0,0,0).\\] Then for any $p\\in[1,\\infty)$, $\\psi^{\\alpha}$ converges to $\\psi$ in the $W^{1,p}(0,T;L^2(\\Omega))\\cap L^\\infty(0,T;H_0^1(\\Omega))$ norm as $\\alpha \\rightarrow 1^{-}$.\nIf additionally $\\psi_0=0$, then we also have convergence in $W^{1,\\infty}(0,T;L^2(\\Omega))\\cap L^\\infty(0,T;H_0^1(\\Omega))$.\n\\end{proposition}\n\n\\section{Reformulation of linear models as wave equations with memory} \\label{Sec:WaveEq_Memory}\nBy neglecting the nonlinear terms in the equations given in Table~\\ref{table:fJMGT}, we arrive at their linear counterparts, which are listed separately in Table~\\ref{table:fMGT}. A possible idea to facilitate the linear analysis, which we wish to explore here, is to re-formulate these equations in terms of \n$$z=\\tau^\\gamma \\Dt^\\gamma\\psi+\\psi,$$ \nwhere $\\gamma=1$ in case of fMGT III and $\\gamma=\\alpha$ otherwise. These linear models can be rewritten as second-order wave equations with memory\n\\[\nz_{tt}-c^2\\Delta z -\\delta \\Dt^\\beta\\Delta \\psi = f\n\\]\nwhere $\\beta=2-\\alpha$ for the fMGT III equation; otherwise it is given by \\eqref{def_beta}. By using the Mittag-Leffler functions $E_{\\gamma,\\gamma}$ and $E_{\\gamma,1}$ we can express $\\psi$ as\n\\begin{equation}\\label{psifromz}\n\\begin{aligned}\n\\psi(t)=&\\, {E_{\\gamma,1}(-(\\tfrac{t}{\\tau})^{\\gamma})\\psi_0+}\n\\tau^{-\\gamma}\\int_0^t(t-s)^{\\gamma-1}E_{\\gamma,\\gamma}(-(\\tfrac{t-s}{\\tau})^{\\gamma})z(s)\\, \\textup{d} s \\\\\n=&\\,{E_{\\gamma,1}(-(\\tfrac{t}{\\tau})^{\\gamma})\\psi_0+}\\mathfrak{k}_\\gamma*z.\n\\end{aligned}\n\\end{equation}\nwith $\\mathfrak{k}_\\gamma$ as in \\eqref{kalpha} below.\nThat is, \n$$\n\\Dt^\\beta\\psi = \\Dt^{\\beta-\\gamma}\\Dt^\\gamma\\psi = \\tau^{-\\gamma}\\Dt^{\\beta-\\gamma} (z-\\psi)= \n\\tau^{-\\gamma}\\Dt^{\\beta-\\gamma} \\left( z-\\mathfrak{k}_\\gamma*z {-E_{\\gamma,1}(-(\\tfrac{t}{\\tau})^{\\gamma})\\psi_0}\\right).\n$$\nThe $z$ forms of each of the linear models are also listed in Table~\\ref{table:fMGT}, where \\[\\tilde{f}=f-\\delta\\tau^{-\\gamma}\\Dt^{\\beta-\\gamma} E_{\\gamma,1}(-(\\tfrac{t}{\\tau})^{\\gamma})\\Delta\\psi_0\\] for the fMGT III equation. Note that for $\\delta=0$ they are all the same and their analysis can be performed as in Section~\\ref{Sec:LinearAnalysis}. We thus focus here on the more challenging case of $\\delta>0$.\n\n\n\n\n\\section{Introduction}\nIt is well-known that using the Fourier temperature flux law, given by\n\\begin{equation} \\label{fourier_law}\n\t\\boldsymbol{q}= -\\kappa \\nabla \\theta,\n\\end{equation}\nin the derivation of second-order models of nonlinear acoustics may lead to the so-called paradox of infinite speed of propagation; see~\\cite{kuznetsov1971equations, kaltenbacher2009global, kaltenbacher2007numerical, jordan2008nonlinear, jordan2016survey}. As a remedy, the Maxwell--Cattaneo law may be used instead\n\\[\n\\boldsymbol{q}+\\tau \\boldsymbol{q}_t = - \\kappa \\nabla \\theta,\n\\]\nwhereby a time lag $\\tau>0$ is introduced between the heat flux and the temperature induced by it. This change within the governing equations leads to the third-order in time sound propagation described by a family of Moore--Gibson--Thompson (MGT) equations in linear acoustics:\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\tau \\psi_{ttt}+\\psi_{tt}-c^2 \\Delta \\psi- (\\tau c^2+\\delta) \\Delta \\psi_t = 0\n\t\\end{aligned}\n\\end{equation}\nor Jordan--Moore--Gibson--Thompson (JMGT) equations in nonlinear acoustics:\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\tau \\psi_{ttt}+\\psi_{tt}-c^2 \\Delta \\psi- (\\tau c^2+\\delta) \\Delta \\psi_t = f(\\psi_t, \\psi_{tt}, \\nabla \\psi, \\nabla \\psi_{t});\n\t\\end{aligned}\n\\end{equation}\nsee the works of Moore and Gibson~\\cite{moore1960propagation}, Thompson~\\cite{thompson}, and Jordan~\\cite{jordan2014second, jordan2008nonlinear} for a detailed insight into their derivation and physical background and~\\cite{kaltenbacher2011wellposedness, KaltenbacherNikolic, bongarti2020vanishing, kaltenbacher2012well, pellicer2020uniqueness} for a selection of results on their mathematical analysis.\\\\\n\\indent However, a drawback of using the hyperbolic heat equation is that it may violate the second law\nof thermodynamics; see, for example,~\\cite{zhang2014time, fabrizio2017modeling, ferrillo2018comparing}. Fractional generalizations of the heat flux law have emerged in the literature as a way of interpolating between the properties of the two flux laws; see, e.g.,~\\cite{povstenko2011fractional, compte1997generalized, fabrizio2015some, atanackovic2012cattaneo} and the references contained therein. In~\\cite{compte1997generalized}, Compte and Metzler proposed several generalized time-fractional heat-flux laws in the following form:\n\\begin{equation} \\label{fractional_law}\n\t\\begin{aligned}\n\t\t(1+\\tau^{\\alpha_1} \\Dt^{\\alpha_1})\\boldsymbol{q}(t) = -\\kappa \\Dt^{\\alpha_2} \\nabla \\theta,\n\t\\end{aligned}\n\\end{equation} \nwhere the choice of $(\\alpha_1, \\alpha_2)$ arises from a particular anomalous diffusion process in complex media. In the present work, we derive and analyze the time-fractional (J)MGT equations that arise from the use of fractional temperature laws \\eqref{fractional_law} in place of the standard heat-flux law within the governing equations.\\\\\n\\indent One such model coming from the choice of fractional orders $(\\alpha_1, \\alpha_2)=(1, 1-\\alpha)$ in the generalized Cattaneo law \\eqref{fractional_law} is given by\n\\begin{equation} \\label{general_fJMGT_III}\n\t\\tau \\psi_{ttt}+\\psi_{tt}-c^2 \\Delta \\psi -(\\tau c^2+\\delta {\\textup{D}}_t^{1-\\alpha})\\Delta \\psi_{t}=f(\\psi_t, \\psi_{tt}, \\nabla \\psi, \\nabla \\psi_{t}), \\ 0< \\alpha \\leq 1,\n\\end{equation}\nwhereas the choice $(\\alpha_1, \\alpha_2)=(\\alpha, 1-\\alpha)$ leads to \n\\begin{equation} \\label{general_fJMGT_I}\n\t\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta {\\textup{D}}_t^{2-\\alpha} \\Delta\\psi=f(\\psi_t, \\psi_{tt}, \\nabla \\psi, \\nabla \\psi_{t}),\n\\end{equation}\nwhere we assume that $\\alpha \\in (1\/2, 1)$. We refer to Section~\\ref{Sec:Modeling} below for the definition of $\\textup{D}_t^\\alpha$ and details on the modeling and to Tables~\\ref{table:fJMGT} and \\ref{table:fMGT} for a complete list of the fractional models that are considered in this work. In particular, we analyze the time-fractional JMGT equations in terms of local-in-time solvability and the limiting behavior of their solutions as $\\alpha \\rightarrow 1^{-}$.\\\\\n\\indent To the best of our knowledge, this is the first work dealing with the mathematical analysis of time-fractional MGT models. We point out that, on the other hand, (J)MGT equations with memories that involve smooth kernel functions represent an active field of research; see, e.g.,~\\cite{lasiecka2017global, bucci2019regularity, dell2017moore, alves2018moore, dell2016moore, bounadja2020decay} and the references contained therein. \\\\\n\\indent Our exposition is organized as follows. In Section~\\ref{Sec:Modeling} we derive four fractional versions of JMGT based on the four instances of \\eqref{fractional_law} elaborated on in~\\cite{compte1997generalized}. After a short Section~\\ref{Sec:Preliminaries} with mathematical notation and tools, we first in Section~\\ref{Sec:Analysis_fJMGT_W_III} focus on the version \\eqref{general_fJMGT_III} of fixed highest order three. We prove its well-posedness in the linear as well as in the nonlinear case without gradient nonlinearity and justify the limit $\\alpha\\to 1^{-}$. Next, in Section~\\ref{Sec:Analysis_fJMGT_W_others} we provide a similar analysis for the other models, that have in common a $2+\\alpha$ leading derivative. This analysis works out with one exception, where the damping term is too weak to allow for varying coefficients or nonlinearities and whose linear version is analyzed separately in Section~\\ref{Sec:WaveEq_Memory} based on a reformulation as a second-order wave equation. Before doing so, we return to \\eqref{general_fJMGT_III} in Section~\\ref{Sec:Analysis_fJMGT_K_III} and provide well-posedness and the limit $\\alpha\\to1^{-}$ in its full version, including the gradient nonlinearity, which requires higher-order energy estimates.\n\\section{Modeling with generalized heat-flux equations} \\label{Sec:Modeling}\n\nIn this section, we consider the four general versions of the constitutive equation \\eqref{fourier_law} proposed by Compte and Metzler in~\\cite{compte1997generalized} and discuss the resulting acoustic equations. These time-fractional general flux equations (GFE) are as follows:\n\\begin{alignat}{3}\n\t\\hspace*{-1.5cm}\\text{\\small (GFE)}\\hphantom{III}&&\\qquad \\qquad (1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)\\boldsymbol{q}(t) =&&\\,-\\kappa \\nabla \\theta;\\hphantom{{\\textup{D}}_t^{1-\\alpha}}\\\\[1mm]\n\t\\hspace*{-1.5cm}\\text{\\small(GFE I)}\\hphantom{II}&& \\qquad \\qquad(1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)\\boldsymbol{q}(t) =&&\\, -\\kappa {\\textup{D}}_t^{1-\\alpha} \\nabla \\theta;\\\\[1mm]\n\t\\hspace*{-1.5cm}\\text{\\small(GFE II)}\\, \\hphantom{I}&&\\qquad \\qquad (1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)\\boldsymbol{q}(t) =&&\\, -\\kappa \\Dt^{\\alpha-1} \\nabla \\theta;\\\\[1mm]\n\t\\hspace*{-1.5cm}\\text{\\small(GFE III)}\\,\\, && \\qquad \\qquad (1+\\tau \\partial_t)\\boldsymbol{q}(t) =&&\\, -\\kappa {\\textup{D}}_t^{1-\\alpha} \\nabla \\theta,\n\\end{alignat}\nwhere $\\boldsymbol{q}$ denotes the flux vector, $\\theta$ the absolute temperature, and $\\kappa$ is the thermal conductivity. A numerical study and comparison of the four resulting fractional heat equations has been performed in~\\cite{zhang2014time} in a one-dimensional setting. Although they can all predict negative temperatures, the fractional heat equation based on using (GFE I) appears to avoid this nonphysical behavior for $\\alpha \\in (1\/2, 1)$ close enough to $1\/2$. \\\\\n\\indent Note that while Compte and Metzler~\\cite{compte1997generalized} state the equations using the Riemann--Liouville fractional derivative, in the present work $\\Dt^{\\gamma}$ always denotes the Caputo--Djrbashian fractional derivative:\n\t\\[\n\t\\Dt^{\\gamma}w(t)=\\frac{1}{\\Gamma(1-\\gamma)}\\int_0^t (t-s)^{-\\gamma}\\Dt^{\\lceil\\gamma\\rceil} w(s) \\, \\textup{d} s, \\quad -1<\\gamma <1;\n\t\\]\n\tsee, for example,~\\cite[\\S 1]{kubica2020time} and~\\cite[\\S 2.4.1]{podlubny1998fractional} for its definition.\n\tHere $n=\\lceil\\gamma\\rceil$, $n\\in\\{0,1\\}$ is the integer obtained by rounding up $\\gamma$ and $\\Dt^n$ is the zeroth or first derivative operator. \\\\\n\n\n\\noindent {\\small \\bf (fJMGT)} We begin by discussing the modeling with the first option; that is\n\\begin{equation} \\label{heat_flux_fractional_GFE}\n\t\\begin{aligned}\n\t\t(1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)\\boldsymbol{q}(t) = -\\kappa \\nabla \\theta;\n\t\\end{aligned}\n\\end{equation} \ncf.~\\cite[Eq. (9)]{zhang2014time}. We note that this modification of the heat-flux law is introduced \\emph{ad hoc} in~\\cite{compte1997generalized} and then disregarded, however numerical studies of the resulting heat equation in~\\cite{zhang2014time} incorporate it as well, and so we include it here. \\\\\n\\indent The derivation of the acoustic equation follows the steps taken in~\\cite[\\S 4]{jordan2014second} with now \\eqref{heat_flux_fractional_GFE} in place of the Maxwell--Cattaneo law. This derivation employs a weakly-nonlinear approximation, which for our purposes can be restated as\n\\begin{equation} \\label{weakly_nl_assumptions_alpha}\n\t\\epsilon <<1,\\quad \\theta=O(\\epsilon),\\quad \\tilde{K}=O(\\epsilon), \\quad \\tau^\\alpha=O(\\epsilon), \\quad |\\mathfrak{e}|=O(\\epsilon^2).\n\\end{equation}\nHere $\\epsilon$ is the Mach number, $\\tilde{K}$ is the dimensionless thermal diffusivity, and $\\mathfrak{e}$ the dimensionless entropy. Note that, compared to~\\cite{jordan2014second}, the condition $\\tau^\\alpha=O(\\epsilon)$ replaces $\\tau=O(\\epsilon)$ here. \\\\\n\\indent It is assumed that the sound wave propagates through a thermally conductive and relaxing liquid or gas with negligible viscosity. Starting from a one-dimensional setting, the governing system is first approximated by\n\\begin{equation} \\label{eq_1_GFE}\n\t\\begin{aligned}\n\t\t\\psi_{tt}+\\tfrac12 \\epsilon \\partial_t(\\psi_x)^2-(1+(\\gamma-2)s)[\\psi_xs_x+(1+s)\\psi_{xx}]=-\\epsilon^{-1}\\mathfrak{e}_t,\n\t\\end{aligned}\n\\end{equation}\nwhere $\\psi$ is the acoustic velocity potential, $\\gamma$ the adiabatic index, and $s$ is known as the condensation; see~\\cite[Eq. (44)--(49) and Eq. (53)]{jordan2014second}. Upon employing $s \\approx -\\epsilon \\psi_t$, one arrives at\n\\begin{equation} \\label{eq_1}\n\t\\begin{aligned}\n\t\t\\psi_{tt}+\\tfrac12 \\epsilon \\partial_t(\\psi_x)^2-(1-(\\gamma-2)\\epsilon\\psi_t)[-\\epsilon \\psi_x \\psi_{tx}+(1-\\epsilon \\psi_t)\\psi_{xx}]=-\\epsilon^{-1}\\mathfrak{e}_t;\n\t\\end{aligned}\n\\end{equation}\ncf. \\cite[Eq. (49)]{jordan2014second}. From the entropy production law\n$$\\tilde{\\kappa}{\\mathfrak{e}_t}=-\\tilde{K} \\boldsymbol{q}_x,$$\nwith $\\tilde{\\kappa}$ being the dimensionless thermal conductivity, and the general heat flux law \\eqref{heat_flux_fractional_GFE} in a dimensionless version\n\\[\n(1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha)\\boldsymbol{q}(t) = -\\tilde{\\kappa} \\nabla \\theta,\n\\]\nwe then have the following entropy equation:\n$$ (1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha) \\mathfrak{e}_t=\\tilde{K} \\theta_{xx}.$$\nAfter utilizing that $\\theta \\approx - \\epsilon (\\gamma-1) \\psi_t$, we can rewrite it as\n\\begin{equation} \\label{e_t_GFE}\n\t\\begin{aligned}\n\t\t(1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha)\\mathfrak{e}_t=-\\epsilon\\tilde{K}(\\gamma-1)\\psi_{txx};\n\t\\end{aligned}\n\\end{equation}\ncf.~\\cite[Eq. (57) and (58)]{jordan2008nonlinear}. Applying the relaxation operator $(1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha)$ to \\eqref{eq_1} and using \\eqref{e_t_GFE} to eliminate $\\mathfrak{e}$ then leads to\n\\begin{equation} \\label{eq_2_I}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t] (1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha)\\left\\{\\psi_{tt}+\\tfrac12 \\epsilon \\partial_t(\\psi_x)^2-(1-(\\gamma-2)\\epsilon \\psi_t)[-\\epsilon \\psi_x \\psi_{tx}+(1-\\epsilon \\psi_t)\\psi_{xx}]\\right\\}\\\\\n\t\t\t= \\, \\tilde{K}(\\gamma-1) \\psi_{txx}. \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nWith $\\lambda_{\\alpha} =O(\\epsilon)$, by neglecting the $O(\\epsilon^2)$ terms in the equation above, we arrive at\n\\begin{equation} \\label{eq_3}\n\t\\begin{aligned}\n\t\t(1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha)\\psi_{tt}-\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha \\psi_{xx}-(1-\\epsilon (\\gamma-1)\\psi_t)\\psi_{xx}+\\epsilon \\partial_t(\\psi_x)^2= \\tilde{K}(\\gamma-1) \\psi_{txx}.\n\t\\end{aligned}\n\\end{equation}\nDividing this equation by $(1-\\epsilon (\\gamma-1)\\psi_t)$, using $(1-\\epsilon (\\gamma-1)\\psi_t)^{-1} \\approx 1+\\epsilon (\\gamma-1)\\psi_t$ for $\\epsilon<<1$ and neglecting all $O(\\epsilon^2)$ terms, yields\n\\begin{equation} \\label{final_eq_2}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+\\epsilon (\\gamma-1)\\psi_t)\\psi_{tt}-\\psi_{xx}-\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha \\psi_{xx}\\\\\\hspace*{5cm}-\\tilde{K}(\\gamma-1) \\psi_{txx}+\\epsilon \\partial_t(\\psi_x)^2= 0. \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nExtrapolating to a dimensionalized 3D model in a mathematically general form gives\n\\begin{equation} \\label{fJMGTK}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta \\Delta\\psi_{t}+ \\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=0. \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nSince the quadratic gradient nonlinearity present in this model corresponds to the one in the second-order Kuznetsov equation~\\cite{kuznetsov1971equations}, we will henceforth refer to \\eqref{fJMGTK} as the fractional JMGT--Kuznetsov equation, or the fJMGT--K equation for short. \\\\\n\\indent Assuming local nonlinear effects can be neglected so that \n\\begin{equation}\n\t|\\nabla \\psi|^2 \\approx c^{-2}\\psi_t^2,\n\\end{equation}\nwe obtain\n\\begin{equation} \\label{fJMGTW}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\n\t\t\t\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta \\Delta\\psi_{t}=0.\n\t\t\\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nThe above approximation corresponds to the one commonly used when reducing the Kuznetsov equation to the Westvervelt second-order model of nonlinear acoustics; cf.~\\cite[\\S 2.3]{jordan2016survey}. For this reason, we will refer to \\eqref{fJMGTW} as the fractional Jordan--Moore--Gibson--Thompson--Westervelt equation, or the fJMGT--W equation for short. This approximation is appropriate when cumulative nonlinear effects dominate the local ones, which is the case, e.g., for sound propagation sufficiently far from the source in terms of wavelengths; see the discussion in \\cite[Ch. 3, Section 6]{hamilton1998nonlinear}.\\\\\n\n\\noindent {\\small \\bf (fJMGT I)} As a second option, we employ the general heat-flux model given by\n\\begin{equation} \\label{heat_flux_fractional}\n\t\\begin{aligned}\n\t\t(1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)\\boldsymbol{q}(t) = -\\kappa {\\textup{D}}_t^{1-\\alpha} \\nabla \\theta;\n\t\\end{aligned}\n\\end{equation} \nsee~\\cite[Eq. (14)]{compte1997generalized} and~\\cite[Eq. (10)]{zhang2014time}. The use of this flux law is motivated in~\\cite{compte1997generalized} stochastically by fractal time random walks. Retracing the derivation steps from before leads to the following equation:\n\\begin{equation} \\label{eq_2_II}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t] (1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha)\\left\\{\\psi_{tt}+\\tfrac12 \\epsilon \\partial_t(\\psi_x)^2-(1-(\\gamma-2)\\epsilon \\psi_t)[-\\epsilon \\psi_x \\psi_{tx}+(1-\\epsilon \\psi_t)\\psi_{xx}]\\right\\}\\\\\n\t\t\t= \\, \\tilde{K}(\\gamma-1) {\\textup{D}}_t^{1-\\alpha}\\psi_{txx} \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nin place of \\eqref{eq_2_I}. Neglecting the $O(\\epsilon^2)$ terms then yields\n\\begin{equation} \\label{eq_3}\n\t\\begin{aligned}\n\t\t(1+\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha)\\psi_{tt}-(1-\\epsilon (\\gamma-1)\\psi_t)\\psi_{xx}- \\lambda_{\\alpha} {\\textup{D}}_t^\\alpha \\psi_{xx}+\\epsilon \\partial_t(\\psi_x)^2= \\tilde{K}(\\gamma-1) {\\textup{D}}_t^{1-\\alpha}\\psi_{txx}.\n\t\\end{aligned}\n\\end{equation}\nAnalogously to before, dividing by $(1-\\epsilon (\\gamma-1)\\psi_t)$ leads to\n\\begin{equation} \n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+\\epsilon (\\gamma-1)\\psi_t)\\psi_{tt}-\\psi_{xx}-\\lambda_{\\alpha} {\\textup{D}}_t^\\alpha \\psi_{xx}-\\tilde{K} (\\gamma-1) {\\textup{D}}_t^{1-\\alpha}\\psi_{txx}\\\\+\\epsilon \\partial_t(\\psi_x)^2= 0 \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nin place of \\eqref{final_eq_2}. Then extrapolating to a general 3D equation gives\n\\begin{equation} \\label{fJMGTK_I}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta\\psi_{t}+ \\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=0, \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nwhich we will call the fractional Jordan--Moore--Gibson--Thompson--Kuznetsov equation of type I, or the fJMGT--K I equation for short. By assuming local nonlinear effects can be neglected as before, we arrive at\n\\begin{equation} \\label{fJMGTW_I}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\n\t\t\t\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta {\\textup{D}}_t^{1-\\alpha} \\Delta\\psi_{t}=0,\n\t\t\\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nwhich we will refer to as the fractional Jordan--Moore--Gibson--Thompson--Westervelt equation of type I, or just the fJMGT--W I equation. \\\\\n\n\\noindent {\\small \\bf (fJMGT II)}\nThirdly, we employ the heat-flux model given by\n\\begin{equation} \\label{heat_flux_fractional}\n\t\\begin{aligned}\n\t\t(1+\\tau^\\alpha {\\textup{D}}_t^\\alpha)\\boldsymbol{q}(t) = -\\kappa \\Dt^{\\alpha-1} \\nabla \\theta;\n\t\\end{aligned}\n\\end{equation} \ncf.~\\cite[Eq. (14)]{compte1997generalized} and \\cite[Eq. (11)]{zhang2014time}. This flux law is motivated in~\\cite{compte1997generalized} by nonlocal transport theory with memory effects; that is, a nonlocal relation between the flux $\\boldsymbol{q}$ and temperature $\\theta$:\n\\[\n\\boldsymbol{q}(x,t)= \\int_0^t \\mathfrak{k}(t-s)\\nabla \\theta (x,s)\\, \\textup{d} s ,\n\\]\nwith a suitable choice of the kernel. Analogously to before, we can derive the following general fractional model:\n\\begin{equation} \\label{fJMGTK_II}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta {\\textup{D}}_t^\\alpha \\Delta\\psi+\\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=0, \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nwhich we will from now on refer to as the fractional Jordan--Moore--Gibson--Thompson--Kuznetsov equation of type II, or the fJMGT--K II equation for short. If the local nonlinear effects can be neglected, we obtain\n\\begin{equation} \\label{fJMGTW_II}\n\t\\begin{aligned}\n\t\t\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -(\\tau^\\alpha c^2 +\\delta){\\textup{D}}_t^\\alpha \\Delta \\psi=0,\n\t\\end{aligned}\n\\end{equation}\nwhich we will refer to as the fractional Jordan--Moore--Gibson--Thompson--Westervelt equation of type II, or the fJMGT--W II equation for short.\\\\\n\n\\noindent {\\small \\bf (fJMGT III)} Finally, we consider the wave-like acoustic models resulting from using the following flux law:\n\\begin{equation} \\label{heat_flux_fractional}\n\t\\begin{aligned}\n\t\t(1+\\tau \\partial_t)\\boldsymbol{q}(t) = -\\kappa {\\textup{D}}_t^{1-\\alpha} \\nabla \\theta;\n\t\\end{aligned}\n\\end{equation} \nsee~\\cite[Eq. (18)]{compte1997generalized} and~\\cite[Eq. (12)]{zhang2014time}. In~\\cite{compte1997generalized}, this law is motivated by a delayed equation that may connect the flux to a generalized force\n\\[\n\\boldsymbol{q}(t+\\tau) = -\\kappa {\\textup{D}}_t^{1-\\alpha} \\nabla \\theta.\n\\]\nHere, weakly-nonlinear acoustic approximation is based on assuming that \n\\begin{equation} \\label{weakly_nl_assumptions}\n\t\\epsilon <<1,\\quad \\theta=O(\\epsilon),\\quad \\tilde{K}=O(\\epsilon), \\quad \\tau=O(\\epsilon), \\quad |\\mathfrak{e}|=O(\\epsilon^2).\n\\end{equation}\nRetracing our previous derivation steps then quickly leads to\n\\begin{equation} \n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\\tau \\psi_{ttt}+(1+\\epsilon (\\gamma-1)\\psi_t)\\psi_{tt}-\\psi_{xx}-\\tau \\psi_{txx}-\\tilde{K} (\\gamma-1) {\\textup{D}}_t^{1-\\alpha}\\psi_{txx}\\\\+\\epsilon \\partial_t(\\psi_x)^2= 0. \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nExtrapolating to a dimensionalized 3D model yields\n\\begin{equation} \\label{fJMGTK_III}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\\tau \\psi_{ttt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau c^2 \\Delta \\psi_{t}- \\delta {\\textup{D}}_t^{1-\\alpha}\\Delta\\psi_{t}+ \\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=0, \\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nwhich we will henceforth refer to it as the fractional Jordan--Moore--Gibson--Thompson--Kuznetsov equation of type III, or the fJMGT--K III equation for short. If the local nonlinear effects can be neglected, we obtain\n\\begin{equation} \\label{fJMGTW_III}\n\t\\begin{aligned}\n\t\t\\begin{multlined}[t]\n\t\t\t\\tau \\psi_{ttt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau c^2\\Delta \\psi_{t}- \\delta {\\textup{D}}_t^{1-\\alpha}\\Delta\\psi_{t}=0.\n\t\t\\end{multlined}\n\t\\end{aligned}\n\\end{equation}\nWe will refer to this model as the fractional Jordan--Moore--Gibson--Thompson--Westervelt equation of type III, or the fJMGT--W III equation for short. \\\\\n\n\\indent We collect all discussed time-fractional acoustic equations in Table~\\ref{table:fJMGT} for convenience and state them with a general source function $f$. Note that the constant $\\delta>0$ for models I--III no longer has the dimension of usual sound diffusivity. \\\\\n\\indent We assume that $\\alpha \\in (0,1]$ in the fJMGT II and III equations, whereas we perform the analysis of the fJMGT and fJMGT I equations under the assumption that $\\alpha \\in (1\/2,1]$. Formally letting $\\alpha \\rightarrow 1^{-}$ in these equations leads to the Jordan--Moore--Gibson--Thompson equations, either in the Westervelt or Kuznetsov forms; cf.~\\cite{jordan2014second}.\n\n\\begin{table}[h]\n\t\\captionsetup{width=.94\\linewidth}\n\t\\begin{center} \\small\n\t\t\\begin{tabular}{|m{1.3cm}||m{10.7cm}|}\n\t\t\t\\hline\n\t\t\t\\vspace*{2mm}\n\t\t\t{fJMGT--} & \\vspace*{2mm} \\textbf{Nonlinear time-fractional acoustic equations} \\\\[6pt]\n\t\t\t\\Xhline{2\\arrayrulewidth} \\hline \\vspace*{4mm}\n\t\t\t\\centering\tK & $\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta \\Delta\\psi_{t}+ \\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=f$ \\\\[3mm]\n\t\t\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{3mm}\n\t\t\t\\centering\tW & $\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta \\Delta\\psi_{t}=f$ \\\\[3mm] \\hline\\hline\n\t\t\t\\vspace*{4mm}\n\t\t\t\\centering\tK I & $\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta {\\textup{D}}_t^{2-\\alpha} \\Delta\\psi+ \\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=f$ \\\\[3mm]\n\t\t\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{3mm}\n\t\t\t\\centering\tW I & $\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta {\\textup{D}}_t^{2-\\alpha} \\Delta\\psi=f$ \\\\[3mm] \\hline\\hline\n\t\t\t\\vspace*{4mm}\n\t\t\t\\centering\tK II & $\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta {\\textup{D}}_t^\\alpha \\Delta\\psi+\\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=f$ \\\\[2mm]\n\t\t\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{3mm}\n\t\t\t\\centering\tW II & $\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -(\\tau^\\alpha c^2 +\\delta){\\textup{D}}_t^\\alpha \\Delta \\psi=f$ \\\\[3mm] \\hline\\hline\n\t\t\t\\vspace*{4mm}\n\t\t\t\\centering K III & $\\tau \\psi_{ttt}+(1+2\\tilde{k}\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau c^2 \\Delta \\psi_{t}- \\delta {\\textup{D}}_t^{2-\\alpha}\\Delta\\psi+ \\tilde{\\ell}\\partial_t |\\nabla \\psi|^2=f$ \\\\[3mm]\n\t\t\t\\Xhline{0.02\\arrayrulewidth} \\vspace*{3mm}\n\t\t\t\\centering\tW III & $\\tau \\psi_{ttt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau c^2\\Delta \\psi_{t}- \\delta {\\textup{D}}_t^{2-\\alpha}\\Delta\\psi=f$ \\\\[3mm] \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Nonlinear fJMGT models in the Kuznetsov and Westervelt forms.} \\label{table:fJMGT}\n\\end{table} \nWe will also study the linearizations of these equations (obtained by setting $k=\\tilde{k}=\\tilde{\\ell}=0$), which we will refer to as fractional Moore--Gibson--Thompson (fMGT) equations; cf. Table~\\ref{table:fMGT} below.\n\\input{Preliminaries}\n\\input{Analysis_fJMGT_W1}\n\\input{energyestimates}\n\\input{LimitOther}\n\\input{Analysis_fJMGT_K1}\n\\input{LinearModeling.tex}\n\\input{LinearAnalysis}\n\\section*{Conclusion and Outlook}\nIn this work, based on physical balance and constitutive laws, we have derived four different fractional-order versions of a well-known third-order in time model of nonlinear acoustics, the JMGT equation. The fractional-order of differentiation $\\alpha\\in(0,1]$ (sometimes restricted to $\\alpha\\in(1\/2,1]$) appears as a parameter in each of these models. We have studied the well-posedness of these equations and their linearizations in appropriate spaces and justified the respective limits as $\\alpha\\to1^{-}$, leading to the (J)MGT equation. \\\\\n\\indent Formally taking the limit of these equations as the relaxation time $\\tau$ vanishes would lead to time-fractional second-order acoustic equations, which are of independent interest as well. An analysis of this limit will be the subject of future research.\n\\section*{Acknowledgments}\nThe work of the first author was supported by the Austrian Science Fund {\\sc fwf} under the grants P30054 and DOC 78.\n\n\\section{Theoretical preliminaries} \\label{Sec:Preliminaries}\nIn this section, we gather several theoretical results from fractional calculus that will be useful later on. To simplify the notation, we often omit the spatial domain and the time interval when writing norms; for example, $\\|\\cdot\\|_{W_t^{p,q} (L^r)}$ denotes the norm on $W^{p,q}(0,t;L^r(\\Omega))$ and $\\|\\cdot\\|_{W^{p,q} (L^r)}$ denotes the norm on $W^{p,q}(0,T;L^r(\\Omega))$.\\\\\n\\indent Throughout the paper, we assume that $\\Omega \\subset \\R^{n}$ is an open, bounded, and sufficiently smooth set, where $n \\in \\{1, 2, 3\\}$. When writing solution spaces for $\\psi$, we use the following notational convention:\n\\begin{equation} \\label{sobolev_withtraces}\n\\begin{aligned}\n{H_\\diamondsuit^2(\\Omega)}=&\\,H_0^1(\\Omega)\\cap H^2(\\Omega),\\\\\n{H_\\diamondsuit^3(\\Omega)}=&\\, \\left\\{\\psi \\in H^3(\\Omega)\\,:\\, \\mbox{tr}_{\\partial\\Omega} \\psi = 0, \\ \\mbox{tr}_{\\partial\\Omega} \\Delta \\psi = 0\\right\\}.\n\\end{aligned}\n\\end{equation}\nIn the analysis, we will rely on the continuous embeddings \n$H^1(\\Omega)\\hookrightarrow L^4(\\Omega)$ and $H^2(\\Omega) \\hookrightarrow L^\\infty(\\Omega)$:\n\\begin{equation}\\label{embeddigs}\n\\begin{aligned}\n&\\|v\\|_{L^4(\\Omega)}\\leq C_{H^1, L^4} \\|\\nabla v\\|_{L^2(\\Omega)}, \\quad && v \\in H^1_0(\\Omega)\\\\\n&\\|v\\|_{L^\\infty(\\Omega)}\\leq C_{H^2, L^\\infty} \\|\\Delta v\\|_{L^2(\\Omega)}, \\quad && v \\in {H_\\diamondsuit^2(\\Omega)}.\n\\end{aligned}\n\\end{equation}\n\\indent We often write $x \\lesssim y$ instead of $x \\leq C y$. In such cases, $C>0$ represents a generic constant that may depend on the medium parameters and the final time $T$, but does not depend on the order of differentiation $\\alpha$. \\\\\n\\indent Throughout the paper we make the following assumptions on the (constant) medium parameters:\n\\begin{equation}\n\\tau >0, \\quad c>0, \\quad \\delta>0, \\quad k,\\, \\tilde{k},\\, \\tilde{\\ell}\\, \\in \\R. \n\\end{equation}\n\\subsection*{Coercivity estimates}\nWhen performing energy analysis, we will rely on the following two coercivity estimates.\n\\begin{itemize}\n\t\\item\n\t\\cite[Lemma 1]{Alikhanov:11}: For any absolutely continuous function $w$,\n\t\\begin{equation}\\label{eqn:Alikhanov_1}\n\t{w(t)}\\textup{D}_t^{\\gamma}w(t)\\geq \\tfrac12(\\textup{D}_t^{\\gamma} w^2)(t).\n\t\\end{equation}\n\\end{itemize}\n\\begin{itemize}\t\n\t\\item\n\t\\cite[Lemma 2.3]{Eggermont1987}; see also \\cite[Theorem 1]{VoegeliNedaiaslSauter2016}: For any $w\\in H^{-(1-\\alpha)\/2}(0,t)$,\n\t\\begin{equation}\\label{coercivityI}\n\t\\int_0^t \\langle \\textup{I}^{1-\\alpha} w(s), w(s) \\rangle \\, \\textup{d} s \\geq \\cos ( \\tfrac{\\pi(1-\\alpha)}{2} ) \\| w \\|_{H^{-(1-\\alpha)\/2}(0,t)}^2, \n\t\\end{equation}\t \n\twhere $\\textup{I}^{\\gamma}$ denotes the Abel integral operator:\n\t\\[\n\t\\textup{I}^{\\gamma}w (t)= \\frac{1}{\\Gamma(\\gamma)}\\int_0^t (t-s)^{\\gamma-1} w(s)\\, \\textup{d} s , \\quad \\gamma \\in (0,1)\n\t\\]\nand the negative norm is defined by\n\\[\n\\| w \\|_{H^{-\\gamma}(0,t)}^2 = \\int_{\\mathbb{R}} (1+\\omega^2)^{-\\gamma}|\\hat{w}(\\omega)|^2\\, \\textup{d} \\omega, \\quad \\gamma >0,\n\\]\nwith $\\hat{w}$ being the Fourier transform of the extension by zero of $w$ to all of $\\mathbb{R}$.\n\\end{itemize}\n\\subsection*{The Kato--Ponce inequality} The following product rule estimate holds:\n\\begin{equation}\\label{prodruleest}\n\\|fg\\|_{W^{\\rho,r}(0,T)}\\lesssim \\|f \\|_{W^{\\rho,p_1}(0,T)} \\|g\\|_{L^{q_1}(0,T)}\n+ \\| f \\|_{L^{p_2}(0,T)} \\|g\\|_{W^{\\rho,q_2}(0,T)}\n\\end{equation}\nfor $0\\leq\\rho\\leq\\overline{\\rho}<1$, \n$1 1\/2$. Assume that $f \\in H^{\\alpha-1\/2}(0,T;L^2(\\Omega))$, $\\sigma \\in X^\\sigma_\\textup{fMGT I}$, and \\[(\\psi_0, \\psi_1, \\psi_2) \\in ({H_\\diamondsuit^2(\\Omega)}, {H_\\diamondsuit^2(\\Omega)}, H_0^1(\\Omega)).\\] There exists $\\varrho>0$, independent of $\\alpha$, \n\tsuch that if \\[\\|\\sigma\\|_{X^\\sigma_\\textup{fMGT I}} \\leq \\varrho,\\] then there is a unique $\\psi \\in X_\\textup{fMGT I}$, which satisfies the \\textup{fMGT I} equation in the $L^2(0,T; L^2(\\Omega))$ sense with $(\\psi, \\psi_t, \\psi_{tt})\\vert_{t=0}=(\\psi_0, \\psi_1, \\psi_2)$. Furthermore, this solution fulfills the following estimate:\n\t\\begin{equation} \\label{energy_est_I}\n\t\\begin{aligned}\n\t& \\begin{multlined}[t]\\|\\Dt^{2+\\alpha}\\psi\\|_{L^2_t(L^2)}^2\n\t+ \\nLtwo{\\nabla \\Dt^{1+\\alpha}\\psi(t)}^2\n\t+\\nLtwo{\\Delta \\Dt^{\\alpha}\\psi(t)}^2\\\\\n\t+ C(\\alpha)\\Bigl(\\|\\nabla\\psi_{tt}\\|_{{H_t^{-(1-\\alpha)\/2}}(L^2)}^2\n\t+ \\|\\Dt^{3\/2}\\Delta \\psi\\|_{L^2_t(L^2)}^2\\Bigr) \\end{multlined}\n\t\\\\\n\t\\lesssim&\\, \\|f\\|_{H^{\\alpha-1\/2}(L^2)}^2+\\nLtwo{\\Delta \\psi_0}^2+\\nLtwo{\\Delta \\psi_1}^2+\\nLtwo{\\nabla \\psi_2}^2,\n\t\\end{aligned}\n\t\\end{equation}\n\twhere $C(\\alpha)\\to0$ as $\\alpha\\to1^-$. \n\\end{proposition}\n\\begin{proof}\n\tThe proof follows by discretizing the problem with respect to the spatial variable, using smooth eigenfunctions of the Dirichlet-Laplacian as the basis.\\\\\n\t\n\t\\noindent (I) \\emph{Existence of an approximate solution.} For $n \\in \\N$ fixed, we first prove that the semi-discrete problem has a unique solution. We employ the same notation as in the proof of Proposition~\\ref{Prop:fMGT_III_lower}; that is,\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\psi^n(x,t) =\\sum_{i=1}^n \\xi^n_i(t)\\phi_i(x),\\quad \\psi_{j}^n(x)= \\sum_{i=1}^n \\xi^n_{j, i} \\phi_i(x), \\quad j\\in\\{0,1,2\\}.\n\t\\end{aligned}\n\t\\end{equation}\n\tUsing the mass matrices $M$ and $M_\\sigma=M_\\sigma(t)$, the stiffness matrix $K$, and the source vector $\\boldsymbol{f}$ defined in \\eqref{matrices}, the next step is to rewrite the discretized problem as a system of integral Volterra equations. To this end, let \n\t\\[\\boldsymbol{\\mu}=\\Dt^{2+\\alpha}\\boldsymbol{\\xi} \\qquad \\text{and} \\qquad p^\\gamma(t)=\\frac{1}{\\Gamma(\\gamma+1)}t^\\gamma.\\]\n\tWe can rely on the following identities:\n\t\\[{\\textup{I}}^\\gamma w=p^{\\gamma-1}*w,\\qquad {\\textup{I}}^\\gamma p^i= p^{\\gamma+i}, \\qquad {\\textup{I}}^\\gamma {\\textup{I}}^s = {\\textup{I}}^{\\gamma+s}\\] to rewrite the vector solution and its derivatives as\n\t\\[\n\t\\begin{aligned}\n\t\\boldsymbol{\\xi}_{tt}&= {\\textup{I}}^{1}\\boldsymbol{\\xi}_{ttt}+ p^0 \\boldsymbol{\\xi}_2 \n\t= {\\textup{I}}^\\alpha \\boldsymbol{\\mu}+ p^0 \\boldsymbol{\\xi}_2\n\t= p^{\\alpha-1}*\\boldsymbol{\\mu} + p^0 \\boldsymbol{\\xi}_2,\\\\\n\t\\boldsymbol{\\xi}_t&= {\\textup{I}}^{1} \\boldsymbol{\\xi}_{tt}+ p^0 \\boldsymbol{\\xi}_1 \n\t= \\textup{I}^{1+\\alpha}\\boldsymbol{\\mu}+ p^1 \\boldsymbol{\\xi}_2+ p^0 \\boldsymbol{\\xi}_1\n\t= p^{\\alpha}*\\boldsymbol{\\mu} + p^1 \\boldsymbol{\\xi}_2+ p^0 \\boldsymbol{\\xi}_1,\\\\\n\t\\boldsymbol{\\xi}&= {\\textup{I}}^{1} \\boldsymbol{\\xi}_{t}+ p^0 \\boldsymbol{\\xi}_0 \n\t= \\textup{I}^{2+\\alpha}\\boldsymbol{\\mu}+ p^2 \\boldsymbol{\\xi}_2+ p^1 \\boldsymbol{\\xi}_1+ p^0 \\boldsymbol{\\xi}_0\n\t= p^{\\alpha+1}*\\boldsymbol{\\mu} + p^2 \\boldsymbol{\\xi}_2+ p^1 \\boldsymbol{\\xi}_1+ p^0 \\boldsymbol{\\xi}_0.\n\t\\end{aligned}\n\t\\]\n\tFurthermore, we can rewrite the fractional derivatives as\n\t\\[\n\t\\begin{aligned}\n\t\\Dt^{2-\\alpha}\\boldsymbol{\\xi}&= {\\textup{I}}^\\alpha\\boldsymbol{\\xi}_{tt} \n\t= {\\textup{I}}^{2\\alpha} \\boldsymbol{\\mu}+ {\\textup{I}}^\\alpha p^0 \\boldsymbol{\\xi}_2\n\t= p^{2\\alpha-1}*\\boldsymbol{\\mu} + p^\\alpha \\boldsymbol{\\xi}_2,\\\\\n\t\\Dt^{\\alpha}\\boldsymbol{\\xi}&= \\textup{I}^{1-\\alpha}\\boldsymbol{\\xi}_t\n\t=I^{2}\\boldsymbol{\\mu}+ \\textup{I}^{1-\\alpha} p^1 \\boldsymbol{\\xi}_2+ \\textup{I}^{1-\\alpha} p^0 \\boldsymbol{\\xi}_1\n\t= p^{1}*\\boldsymbol{\\mu} +p^{2-\\alpha} \\boldsymbol{\\xi}_2+ p^{1-\\alpha} \\boldsymbol{\\xi}_1.\\\\\n\t\\end{aligned}\n\t\\]\n\tTherefore, the semi-discrete problem can be equivalently rewritten as a system of Volterra integral equations:\n\t\\begin{equation} \\label{Volterra_system_I}\n\t\\begin{aligned}\n\t&\\begin{multlined}[t]\\tau^\\alpha M \\boldsymbol{\\mu} \n\t+ M_\\sigma(t)\\Bigl(p^{\\alpha-1}*\\boldsymbol{\\mu} + p^0 \\boldsymbol{\\xi}_2\\Bigr)\\\\\n\t+c^2 K \\Bigl(p^{\\alpha+1}*\\boldsymbol{\\mu} + p^2 \\boldsymbol{\\xi}_2+ p^1 \\boldsymbol{\\xi}_1+ p^0 \\boldsymbol{\\xi}_0\\Bigr)\\\\\n\t+\\tau^\\alpha c^2 K \\Bigl(p^{1}*\\boldsymbol{\\mu} +p^{2-\\alpha} \\boldsymbol{\\xi}_2+ p^{1-\\alpha} \\boldsymbol{\\xi}_1\\Bigr)\n\t+\\delta K (p^{2\\alpha-1}*\\boldsymbol{\\mu} + p^\\alpha \\boldsymbol{\\xi}_2)=f.\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tThus, unique solvability of this system in $L^2(0,T)$ follows from~\\cite[Theorem 4.2, p. 241 in \\S 9]{GLS90}. Then from\t\\[ \\left \\{\n\t\\begin{aligned}\n\t&\\Dt^{\\alpha}\\boldsymbol{\\xi}_{tt} =\\boldsymbol{\\mu} \\in L^2(0,T), \\quad \\alpha \\in (\\tfrac12, 1)\\\\\n\t& \\boldsymbol{\\xi}_{tt}(0)=\\boldsymbol{\\xi}_{2}\n\t\\end{aligned} \\right.\n\t\\]\n\twe have a unique $\\boldsymbol{\\xi}_{tt} \\in H^\\alpha(0,T)$; cf.~\\cite[\\S 3.3]{kubica2020time}. Combined with the initial conditions $(\\boldsymbol{\\xi}_0, \\boldsymbol{\\xi}_1)$, this yields a unique $\\boldsymbol{\\xi} \\in H^{2+\\alpha}(0,T)$ and further implies the existence of a unique $\\psi^n \\in H^{2+\\alpha}(0,T; V_n)$. \\\\\n\t\n\t\\noindent (II) \\emph{A priori energy analysis.} We next focus on deriving a uniform energy estimate for $\\psi^n$. We will make use of estimate \\eqref{coercivityI} to treat the fractional terms; that is,\n\t\\begin{equation}\n\t\\int_0^t \\langle \\textup{I}^\\rho w(s), w(s) \\rangle \\, \\textup{d} s \\geq \\cos ( \\tfrac{\\pi\\rho}{2} ) \\| w \\|_{H^{-\\rho\/2}(0,t)}^2 \n\t\\end{equation} \n\tfor $\\rho\\in(0,1)$, as well as the identity $\\int_0^t \\langle w_t(s), w(s) \\rangle \\, \\textup{d} s = \\frac12|w|^2\\,\\big\\vert_0^t$. Thus, the rule of thumb is that for a coercivity estimate on $\\int_0^t \\Dt^r(s) w\\, \\Dt^\\rho w(s)\\, \\textup{d} s $ to yield a non-negative lower bound (up to initial data), the difference $|r-\\rho|$ between the fractional orders must not exceed one. We will consider the multiplier \\[-\\Delta \\Dt^{1+\\alpha}\\psi^n(t)= \\sum_{i=1}^n \\Dt^{1+\\alpha} \\xi^n_i(t) \\,\\lambda_i \\phi_i(x) \\in V_n,\\] for which this rule applies and yields non-negative contributions on the left-hand side for the terms containing $\\Dt^{2+\\alpha}\\psi^n$, $\\psi_{tt}^n$, $-\\Dt^\\alpha\\Delta \\psi^n$, and $-\\Dt^{2-\\alpha} \\Delta \\psi^n$. Multiplying the semi-discrete equation with $-\\Delta \\Dt^{1+\\alpha} \\psi^n$ and integrating over space and $(0,t)$ at first leads to\n\t\\begin{equation}\n\t\\begin{aligned}\n\t&\\, \\begin{multlined}[t]-\\tau^\\alpha \\int_0^t \\prodLtwo{ \\Dt^{2+\\alpha} \\psi^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s -\\int_0^t \\prodLtwo{ \\psi_{tt}^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s \\\\\n\t+ \\tau^\\alpha c^2\\int_0^t \\prodLtwo{{\\textup{D}}_t^\\alpha \\Delta \\psi^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s +\\delta \\int_0^t \\prodLtwo{\\Dt^{2-\\alpha} \\Delta \\psi^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s \n\t\\end{multlined} \\\\\n\t=&\\, \\begin{multlined}[t] -\\int_0^t (f-\\sigma \\psi_{tt}^n+c^2 \\Delta \\psi^n, \\Delta \\Dt^{1+\\alpha} \\psi^n)\\, \\textup{d} s .\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tWe next exchange the order of differentiation in the first and third term on the left as follows:\n\t\\begin{equation}\\label{exchange_derivative}\n\t\\Dt^{2+\\alpha}\\psi^n = \\Dt \\Dt^{1+\\alpha}\\psi^n-p^{-\\alpha}\\psi^n_{tt}(0)\\,, \\qquad\n\t\\Dt^{1+\\alpha}\\psi^n = \\Dt \\Dt^\\alpha\\psi^n-p^{-\\alpha}\\psi^n_{t}(0)\\,, \n\t\\end{equation}\n\tand integrate by parts to obtain \n\t\\[\n\t\\begin{aligned}\n\t&\\int_0^t \\prodLtwo{ \\Dt^\\alpha\\Delta\\psi^n(s)}{p^{-\\alpha}(s)\\Delta \\psi_t^n(0)}\\, \\textup{d} s \\\\\n\t=&\\, \\begin{multlined}[t]-\\int_0^t \\prodLtwo{ \\Dt^{1+\\alpha}\\Delta\\psi^n(s)+p^{-\\alpha}(s)\\Delta \\psi_t^n(0)}{p^{1-\\alpha}\\Delta \\psi_t^n(0)}\\, \\textup{d} s \\\\\n\t+p^{1-\\alpha}(t)\\prodLtwo{ \\Dt^\\alpha\\Delta\\psi^n(t)}{\\Delta \\psi_t^n(0)} \\end{multlined}\\\\\n\t=&\\, -\\int_0^t \\prodLtwo{ \\Dt^{1+\\alpha}\\Delta\\psi^n(s)}{p^{1-\\alpha}(s)\\Delta \\psi_t^n(0)}\\, \\textup{d} s + h_0(t),\n\t\\end{aligned}\n\t\\]\n\twhere we have introduced\n\t\\[h_0(t)=\\frac{t^{2-2\\alpha}}{2\\Gamma(2-2\\alpha)^2} \\|\\Delta \\psi_t^n(0)\\|_{L^2}^2\n\t+p^{1-\\alpha}(t)\\prodLtwo{ \\Dt^\\alpha\\Delta\\psi^n(t)}{\\Delta \\psi_t^n(0)}.\\]\n\tThus, we arrive at the following identity:\n\t\\begin{equation}\n\t\\begin{aligned}\n\t&\\, \\begin{multlined}[t]-\\tau^\\alpha \\int_0^t \\prodLtwo{ \\Dt \\Dt^{1+\\alpha}\\psi^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s -\\int_0^t \\prodLtwo{ \\psi_{tt}^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s \\\\\n\t+ \\tau^\\alpha c^2\\int_0^t \\prodLtwo{{\\textup{D}}_t^\\alpha \\Delta \\psi^n}{\\Delta \\Dt \\Dt^\\alpha\\psi^n}\\, \\textup{d} s +\\delta \\int_0^t \\prodLtwo{\\Dt^{2-\\alpha} \\Delta \\psi^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s \n\t\\end{multlined} \\\\\n\t=&\\, \\begin{multlined}[t] -\\int_0^t \\prodLtwo{f-\\sigma \\psi_{tt}^n+c^2 \\Delta \\psi^n+\\tau^\\alpha p^{-\\alpha}\\psi_{tt}^n(0)-\\tau^\\alpha c^2 p^{1-\\alpha}\\Delta \\psi_t^n(0)}\n\t{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s \\\\\n\t\\hspace*{-5cm}+\\tau^\\alpha c^2 h_0(t).\n\t\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tWe note that \\[\\Dt^{1+\\alpha}\\psi^n=\\psi_{tt}^n=0 \\text{ on } \\partial \\Omega\\] with our choice of the basis functions. Additionally using $\\Dt^{1+\\alpha}\\psi^n=\\textup{I}^{1-\\alpha}\\psi^n_{tt}$ in the second term on the left and integrating by parts in space and time yields \n\t\\begin{equation}\\label{enid_rem}\n\t\\begin{aligned}\n&\\begin{multlined}[t] \\,\\frac{\\tau^\\alpha}{2}\\nLtwo{\\nabla \\Dt^{1+\\alpha}\\psi^n(s)}^2 \\big\\vert_0^t+\\int_0^t\\prodLtwo{\\nabla\\psi_{tt}^n(s)}{\\textup{I}^{1-\\alpha} \\nabla\\psi_{tt}^n(s)}\\, \\textup{d} s \n\t\\\\\t+\\frac{\\tau^\\alpha c^2}{2}\\nLtwo{\\Delta \\Dt^{\\alpha}\\psi^n(s)}^2 \\Big \\vert_0^t+\\delta d \\end{multlined} \\\\\n\t=&\\, \n\t-\\int_0^t\\prodLtwo{\\tilde{f}(s)}{\\Delta \\Dt^{1+\\alpha}\\psi^n(s)}\\, \\textup{d} s \n\t+\\tau^\\alpha c^2 h_0(t), \n\t\\end{aligned}\n\t\\end{equation}\n\twhere we have introduced the short-hand notation\n\t\\begin{equation}\\label{ftilde}\n\t\\tilde{f}=f+c^2\\Delta\\psi^n-\\sigma\\psi_{tt}^n +\\tau^{\\alpha}\\left(p^{-\\alpha}\\psi_{tt}^n(0)-c^2p^{1-\\alpha}\\Delta\\psi_t^n(0)\\right)\n\t\\end{equation}\n\tand\n\t\\[\n\td=\\int_0^t \\prodLtwo{\\Dt^{2-\\alpha} \\Delta \\psi^n}{\\Delta \\Dt^{1+\\alpha} \\psi^n}\\, \\textup{d} s .\n\t\\]\n\tBy the identity\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\Dt^{2-\\alpha}\\psi^n=&\\, {\\textup{I}}^\\alpha\\psi_{tt}=\\textup{I}^{2\\alpha-1}\\textup{I}^{1-\\alpha}\\psi_{tt}^n=\\textup{I}^{2\\alpha-1}\\Dt^{1+\\alpha}\\psi^n,\n\t\\end{aligned}\n\t\\end{equation}\n\twe know that\n\t\\begin{align} \\label{damping_fMGT_I}\n\td = \t\\int_0^t\\prodLtwo{\\textup{I}^{2\\alpha-1} \\Dt^{1+\\alpha}\\Delta\\psi^n(s)}{\\Dt^{1+\\alpha}\\Delta\\psi^n(s)}\\, \\textup{d} s . \n\t\\end{align} \n\tSince $\\alpha>1\/2$, this term can be estimated from below using the fact that $\\textup{I}^\\gamma:H^{-\\gamma}(0,t)\\to L^2(0,T)$ is an isomorphism for $\\gamma \\in [0, 1\/2)$; see~\\cite[Theorem 1]{gorenflo1999operator}. Therefore,\n\t\\[\n\t\\begin{aligned}\n\t\\frac{d}{\\cos(\\pi(\\alpha-1\/2))} \\geq&\\, \n\t\\|\\Dt^{1+\\alpha}\\Delta \\psi^n\\|_{H_t^{1\/2-\\alpha}(L^2)}^2\\\\\n\t\\sim&\\, \n\t\\|\\textup{I}^{\\alpha-1\/2}\\Dt^{1+\\alpha}\\Delta \\psi^n\\|_{L^2_t(L^2)}^2\\sim \n\t\\|\\Dt^{3\/2}\\Delta \\psi^n\\|_{L_t^2(L^2)}^2. \n\t\\end{aligned}\n\t\\]\n\tWe estimate the $\\tilde{f}$ term on the right-hand side of \\eqref{enid_rem} with a view on the possibility of bounding it by $d$ as follows: \n\t\\[\n\t\\begin{aligned}\n\t\\left|\\int_0^t\\prodLtwo{\\tilde{f}(s)}{\\Dt^{1+\\alpha}\\Delta \\psi^n(s)}\\, \\textup{d} s \\right|\n\t\\leq \n\t\\frac{1}{2\\epsilon}\\|\\tilde{f}\\|_{H^{\\alpha-1\/2}_t(L^2)}^2+\\frac{\\epsilon}{2}\\|\\Dt^{1+\\alpha}\\Delta \\psi^n\\|_{H^{1\/2-\\alpha}_t(L^2)}^2.\n\t\\end{aligned}\n\t\\]\n\tIt remains to estimate the terms within $\\tilde{f}$ in this norm; that is, to bound\n\t\\begin{equation}\n\t\\|f+c^2\\Delta\\psi^n-\\sigma\\psi_{tt}^n +\\tau^{\\alpha}\\left(p^{-\\alpha}\\psi_{tt}^n(0)-c^2p^{1-\\alpha}\\Delta\\psi_t^n(0)\\right)\\|_{H^{\\alpha-1\/2}_t(L^2)}.\n\t\\end{equation}\n\tFor the $c^2$ term, it is readily checked that the respective contribution of $c^2\\Delta \\psi^n$ to the above norm of $\\tilde{f}$ (cf. \\eqref{ftilde}) can be bounded by means of $c^2 \\|\\Dt^\\alpha\\Delta\\psi^n\\|_{L^2(L^2)}$ as follows:\n\t\\[\n\t\\begin{aligned}\n\t\\|\\Delta \\psi^n \\|_{H^{\\alpha-1\/2}_t(L^2)}\n\t\\lesssim&\\, \\|\\Dt^{\\alpha-1\/2}\\Delta \\psi^n \\|_{L^2_t(L^2)} + \\nLtwo{\\Delta \\psi^n_0}\\\\\n\t=&\\, \\|\\mathfrak{g}_{1\/2}*\\Dt^\\alpha\\Delta \\psi^n \\|_{L^2_t(L^2)} + \\nLtwo{\\Delta \\psi^n_0}\\\\\n\t\\leq&\\, \\|\\mathfrak{g}_{1\/2}\\|_{L^1(0,T)} \\|\\Dt^\\alpha\\Delta \\psi^n \\|_{L^2_t(L^2)} + \\nLtwo{\\Delta \\psi^n_0}\\,,\n\t\\end{aligned}\n\t\\]\n\tand therefore tackled by the second term on the left-hand side of \\eqref{enid_rem} together with Gronwall's inequality.\\\\\n\t\\indent By the Kato--Ponce inequality \\eqref{prodruleest} with \n\t\\[\\rho=\\alpha-1\/2,\\qquad (p_1, q_1)=\\left(\\frac{2}{2 \\alpha -1}, \\frac{1}{1-\\alpha}\\right), \\qquad (p_2, q_2)=(2, \\infty),\\]\n\twe obtain \n\t\\begin{equation}\\label{estsigmapsitt_I}\n\t\\begin{aligned}\n\t&\\|\\sigma\\psi_{tt}^n\\|_{H_t^{\\alpha-1\/2}(L^2)}\n\t\\\\\n\t\\lesssim&\\,\n\t\\|\\sigma\\|_{W_t^{\\alpha-1\/2,\\frac{2}{2\\alpha-1}}(L^\\infty)}\\|\\psi_{tt}^n\\|_{L_t^{\\frac{1}{1-\\alpha}}(L^2)}\n\t+\\|\\sigma\\|_{L_t^{2}(L^\\infty)}\\|\\psi_{tt}^n\\|_{W_t^{\\alpha-1\/2, \\infty}(L^2)}.\n\t\\end{aligned}\n\t\\end{equation}\n\tBy the Sobolev embedding $L^2(0,T)\\hookrightarrow W^{\\alpha-1\/2,\\frac{2}{2 \\alpha-1}}(0,T)$, we have\n\t\\[\n\t\\|\\sigma\\|_{W_t^{\\alpha-1\/2, \\frac{2}{2 \\alpha-1}}(L^\\infty)}\n\t\\leq C_{H^2,L^\\infty}^\\Omega\\|\\sigma\\|_{X^\\sigma_\\textup{fMGT I}}\\,, \\quad\n\t\\|\\sigma\\|_{L_t^2(L^\\infty)}\n\t\\leq C_{H^2,L^\\infty}^\\Omega\\|\\sigma\\|_{X^\\sigma_\\textup{fMGT I}}\\,.\n\t\\]\n\tTo further estimate the norms of $\\psi_{tt}^n$ in \\eqref{estsigmapsitt_I}, we will use the leading time derivative term $\\Dt^{2+\\alpha}\\psi^n$ as well as its representation via the PDE. That is, we rely on the following Sobolev embeddings:\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\|\\psi_{tt}^n\\|_{L_t^{\\frac{1}{1-\\alpha}}(L^2)}\\lesssim&\\,\n\t\\|\\psi_{tt}^n\\|_{H^{\\alpha-1\/2}_t(L^2)}\\lesssim\n\t\\|\\Dt^{2+\\alpha}\\psi^n\\|_{H^{-1\/2}_t(L^2)}+\\|\\psi_{tt}^n(0)\\|_{L^2}\n\t\\end{aligned}\n\t\\end{equation}\n\tand\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\|\\psi_{tt}^n\\|_{W_t^{\\alpha-1\/2, \\infty}(L^2)}\\lesssim&\\, \\|\\psi_{tt}^n\\|_{H^\\alpha_t(L^2)}\\lesssim \\|\\Dt^{2+\\alpha}\\psi^n\\|_{L^2_t(L^2)}+\\|\\psi_{tt}^n(0)\\|_{L^2},\n\t\\end{aligned}\n\t\\end{equation}\n\twhere\n\t$\\psi_{tt}^n$ satisfies the fractional ODE \n\t\\[\n\t\\tau^\\alpha\\Dt^\\alpha \\psi_{tt}^n + \\psi_{tt}^n = -r\\mbox{ with }r=\\sigma \\psi_{tt}^n - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi^n - \\delta \\Dt^{2-\\alpha} \\Delta \\psi^n -f\n\t\\]\n\tand therefore \n\t\\begin{equation}\\label{fracODE}\n\t\\Dt^\\alpha \\psi_{tt}^n = \\tau^{-\\alpha}\\Bigl(-E_{\\alpha,1}(-(\\tfrac{t}{\\tau})^\\alpha)\\psi_{tt}^n(0)+\\int_0^t E_{\\alpha,\\alpha}(-(\\tfrac{t-s}{\\tau})^\\alpha) r(s)\\, \\textup{d} s -r(t)\\Bigr);\n\t\\end{equation} \n\tsee, e.g.,~\\cite[\\S 3]{kubica2020time}. Thus, we have\n\t\\begin{equation}\\label{Dt2plusalphapn}\n\t\\begin{aligned}\n\t&\\|\\Dt^{2+\\alpha}\\psi^n\\|_{H^{-1\/2}_t(L^2)}\\lesssim\n\t\\|\\Dt^{2+\\alpha}\\psi^n\\|_{L^2_t(L^2)}\\lesssim \\|r\\|_{L^2_t(L^2)}+\\|\\psi_{tt}^n(0)\\|_{L^2}\\\\\n\t\\leq&\\,\\begin{multlined}[t]\\|\\sigma \\psi_{tt}^n\\|_{L^2_t(L^2)} + c^2\\|\\Delta\\psi\\|_{L^2_t(L^2)} +\\tau^\\alpha c^2\\|\\Dt^\\alpha\\Delta \\psi^n\\|_{L^2_t(L^2)}\\\\\n\t+ \\delta \\|\\Dt^{2-\\alpha} \\Delta \\psi^n\\|_{L^2_t(L^2)} +\\|f\\|_{L^2(L^2)}+\\|\\psi_{tt}^n(0)\\|_{L^2}.\\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\tIn here, the terms with factors $c^2$, $\\tau^\\alpha c^2$, and $\\delta$ can be controlled -- in a (generalized) Gronwall inequality fashion -- by left-hand side terms in \\eqref{enid_rem}; to see this for the latter, consider \n\t\\[\\|\\Dt^{2-\\alpha} \\Delta \\psi^n\\|_{L^2_t(L^2)}=\\|\\textup{I}^{\\alpha-1\/2}\\Dt^{3\/2} \\Delta \\psi^n\\|_{L^2_t(L^2)}\\lesssim \\mathfrak{g}_{3\/2-\\alpha}*d.\\]\n\tThus, from \\eqref{estsigmapsitt_I} to \\eqref{Dt2plusalphapn}, we have obtained an estimate of the form\n\t\\[\n\t\\begin{aligned}\n\t\\|\\sigma\\psi_{tt}^n\\|_{H_t^{\\alpha-1\/2}(L^2)}\\lesssim&\\, \\|\\sigma\\|_{X^\\sigma_\\textup{fMGT I}} \\left(\\|\\Dt^{2+\\alpha}\\psi^n\\|_{L^2_t(L^2)}+\\|\\psi_{tt}^n(0)\\|_{L^2}\\right)\\\\\n\t\\lesssim&\\, \\|\\sigma\\|_{X^\\sigma_\\textup{fMGT I}} \\left(\\|\\sigma\\psi_{tt}^n\\|_{H_t^{\\alpha-1\/2}(L^2)} \n+ \\textup{rhs} \\right),\n\t\\end{aligned}\n\t\\]\nwhere \n\\[\\textup{rhs}:=\n c^2\\|\\Delta\\psi^n\\|_{L^2_t(L^2)} +\\tau^\\alpha c^2\\|\\Dt^\\alpha\\Delta \\psi^n\\|_{L^2_t(L^2)}\n\t+ \\mathfrak{g}_{3\/2-\\alpha}*d +\\|f\\|_{L^2(L^2)} +\\|\\psi_{tt}^n(0)\\|_{L^2}.\n\\]\n\tThus, provided $\\|\\sigma\\|_{X^\\sigma_\\textup{fMGT I}}$ is sufficiently small (where the bound can be chosen independent of $\\alpha\\in[\\alpha_0,1)$ for $\\alpha_0>1\/2$),\n\tthe term $\\|\\sigma\\psi_{tt}^n\\|_{H_t^{\\alpha-1\/2}(L^2)}$ is bounded by \na multiple of \\textup{rhs}. By combining this with \\eqref{enid_rem}, \\eqref{Dt2plusalphapn}, and Gronwall's inequality in its generalized version, see, e.g., \\cite[Lemma 7.2]{kubica2020time}, we therefore obtain the following estimate:\n\t\\begin{equation} \\label{discrete_est_I}\n\t\\begin{aligned}\n\t& \\begin{multlined}[t]\\|\\Dt^{2+\\alpha}\\psi^n\\|_{L^2_t(L^2)}^2\n\t+ \\nLtwo{\\nabla \\Dt^{1+\\alpha}\\psi^n(t)}^2\n\t+\\nLtwo{\\Delta \\Dt^{\\alpha}\\psi^n(t)}^2\\\\\n\t+ C(\\alpha)(\\|\\nabla\\psi_{tt}^n\\|_{{H_t^{-(1-\\alpha)\/2}}(L^2)}^2\n\t+ \\|\\Dt^{3\/2}\\Delta \\psi^n\\|_{L^2_t(L^2)}^2) \\end{multlined}\n\t\\\\\n\t\\lesssim&\\, \\|f\\|_{H^{\\alpha-1\/2}(L^2)}^2+\\nLtwo{\\Delta \\psi_0}^2+\\nLtwo{\\Delta \\psi_1}^2+\\nLtwo{\\nabla \\psi_2}^2,\n\t\\end{aligned}\n\t\\end{equation}\n\twhere we have also relied on the uniform boundedness of the approximate data. \n{Here the constant $C(\\alpha)$ tends to zero as $\\alpha\\to1^-$, since it contains the factor $\\cos ( \\tfrac{\\pi(1-\\alpha)}{2} )$ from the coercivity estimate \\eqref{coercivityI}.\n}\n\\\\\n\t\n\t\\noindent (III) \\emph{Passing to the limit.} Thanks to the uniform bound \\eqref{discrete_est_I}, there exists a subsequence, which we do not relabel, such that\n\t\\begin{equation} \\label{weak_limits_I}\n\t\\begin{alignedat}{4} \n\t\\Dt^{2+\\alpha} \\psi^n &\\relbar\\joinrel\\rightharpoonup \\Dt^{2+\\alpha} \\psi &&\\text{ weakly} &&\\text{ in } &&L^2(0,T; L^2(\\Omega)), \\\\\n\t{\\textup{D}}_t^\\alpha \\Delta \\psi^n &\\relbar\\joinrel\\rightharpoonup {\\textup{D}}_t^\\alpha \\Delta \\psi &&\\text{ weakly-$\\star$} &&\\text{ in } &&L^\\infty(0,T; L^2(\\Omega)), \\\\\n\t\\Dt^{1+\\alpha}\\nabla \\psi^n &\\relbar\\joinrel\\rightharpoonup \\Dt^{1 +\\alpha} \\nabla \\psi &&\\text{ weakly-$\\star$} &&\\text{ in } &&L^\\infty(0,T; L^2(\\Omega)), \\\\\n\t\\Dt^{3\/2} \\Delta \\psi^n &\\relbar\\joinrel\\rightharpoonup \\Dt^{3\/2}\\Delta\\psi &&\\text{ weakly} &&\\text{ in } &&L^2(0,T; L^2(\\Omega)).\n\t\\end{alignedat} \n\t\\end{equation} \n\tFurthermore,\n\t\\begin{equation} \\label{weak_limits_I_1}\n\t\\begin{alignedat}{4} \n\t(1+\\sigma)\\psi_{tt}^n &\\relbar\\joinrel\\rightharpoonup (1+\\sigma)\\psi_{tt} &&\\text{ weakly} &&\\text{ in } &&L^2(0,T; L^2(\\Omega)),\\\\\n\t\\Delta \\psi^n &\\relbar\\joinrel\\rightharpoonup \\Delta \\psi &&\\text{ weakly} &&\\text{ in } &&L^2(0,T; L^2(\\Omega)), \\\\\n\t{\\textup{D}}_t^{2-\\alpha} \\Delta \\psi^n &\\relbar\\joinrel\\rightharpoonup {\\textup{D}}_t^{2-\\alpha} \\Delta \\psi &&\\text{ weakly} &&\\text{ in } &&L^2(0,T; L^2(\\Omega)). \n\t\\end{alignedat} \n\t\\end{equation}\n\tThus, we can pass to the limit in the usual way in the semi-discrete problem. Further, weak\/weak-$\\star$ lower semi-continuity of norms implies that the solution we constructed satisfies \\eqref{energy_est_I} a.e. in time.\\\\\n\t\n\t\n\t\\noindent (IV) \\emph{Attainment of the initial conditions.} Similarly to step (IV) in the proof of Proposition~\\ref{Prop:fMGT_III_lower}, we show that $\\psi$ attains its initial conditions by, on one hand concluding from \\eqref{weak_limits_I} that\n\t\\[\n\t\\begin{aligned}\n\t&\\psi^n (0) \\relbar\\joinrel\\rightharpoonup \\psi(0) \\text{ weakly} \\text{ in } {H_\\diamondsuit^2(\\Omega)},\\\\ \n\t&\\psi_t^n (0) \\relbar\\joinrel\\rightharpoonup \\psi_t(0) \\text{ weakly} \\text{ in } H_0^1(\\Omega),\\\\ \n\t&\\psi_{tt}^n (0) \\relbar\\joinrel\\rightharpoonup \\psi_{tt}(0) \\text{ weakly} \\text{ in } L^ 2(\\Omega),\n\t\\end{aligned}\t\\]\n\tand, on the other hand, $\\psi^n(0) \\rightarrow \\psi_0$ in ${H_\\diamondsuit^2(\\Omega)}$, $\\psi_t^n(0) \\rightarrow \\psi_1$ in $H_0^1(\\Omega)$, $\\psi_{tt}^n(0) \\rightarrow \\psi_1$ in $L^2(\\Omega)$ based on our choice of the approximate data. Thus the initial data are attained in an ${H_\\diamondsuit^2(\\Omega)}\\times H_0^1(\\Omega)\\times L^2(\\Omega)$ sense.\\\\\n\t\n\t\n\t\\noindent (IV) \\emph{Uniqueness.} The fact that the obtained solution is unique follows by testing the homogeneous problem\n\t\\[\n\t\\tau^\\alpha \\Dt^{2+\\alpha}\\psi + (1+\\sigma) \\psi_{tt} - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi - \\delta \\Dt^{2-\\alpha} \\Delta \\psi = 0\n\t\\] (with zero initial data) with $\\Dt^{1+\\alpha} \\psi_t$. Analogously to above, but replacing $\\Delta\\to\\nabla$ and $\\nabla\\to\\mbox{id}$, we obtain\n\t\\[\n\t\\begin{aligned}\n\t& \\begin{multlined}[t]\\frac{\\tau^\\alpha}{2}\\nLtwo{\\Dt^{1+\\alpha}\\psi(t)}^2\n\t+ \\frac{\\tau^\\alpha c^2}{2}\\nLtwo{\\nabla\\Dt^{\\alpha}\\psi(t)}^2+C(\\alpha)(\\|\\psi_{tt}\\|_{H_t^{-(1-\\alpha)}(L^2)}^2\n\t+ \\|\\Dt^{3\/2} \\nabla \\psi\\|_{L^2_t(L^2)}^2)\\end{multlined}\n\t\\\\\n\t\\leq&\\, \\left |\\int_0^t\\prodLtwo{c^2\\Delta\\psi-\\sigma\\psi_{tt}}{\\Dt^{1+\\alpha}\\psi}\\, \\textup{d} s \\right| \\\\\n\t\\leq&\\, \\frac{1}{2\\epsilon}\\Bigl(\\|c^2\\nabla\\psi\\|_{H_t^{\\alpha-1\/2}(L^2)}+\\|\\sigma\\psi_{tt}\\|_{H_t^{\\alpha-1\/2}(H^{-1})}\\Bigr)^2+\\frac{\\epsilon}{2} \\|\\Dt^{3\/2}\\nabla\\psi\\|_{L^2_t(L^2)}^2,\n\t\\end{aligned}\n\t\\]\n\twhere \n\t\\[\n\t\\begin{aligned}\n\t\\|c^2\\nabla\\psi\\|_{H^{\\alpha-1\/2}(L^2)}&\\lesssim \\|\\nabla\\Dt^{\\alpha}\\psi\\|_{L^2(L^2)}.\n\t\\end{aligned}\n\t\\]\n\tFurther, on account of the following estimate:\n\t\\begin{equation} \\label{ab_H-1}\n\t\\begin{aligned}\n\t\\|ab\\|_{H^{-1}(\\Omega)}\n\t=& \\, \\|a\\|_{H^{-1}(\\Omega)}\\sup_{v\\in H_0^1(\\Omega)\\setminus\\{0\\}} \\|v\\|_{H_0^1(\\Omega)}^{-1} \\nLtwo{v \\nabla b + b\\nabla v}\\\\\n\t\\leq&\\, \\|a\\|_{H^{-1}(\\Omega)}(C_{H^1,L^6}\\|\\nabla b\\|_{L^3}+\\|b\\|_{L^\\infty}),\n\t\\end{aligned}\n\t\\end{equation}\n\twe have, similarly to \\eqref{estsigmapsitt_I},\n\t\\begin{equation}\\label{sigmaptt}\n\t\\begin{aligned}\n\t\\|\\sigma\\psi_{tt}\\|_{H_t^{\\alpha-1\/2}(H^{-1})}\n\n\t\\lesssim\\,\n\t\\|\\sigma\\|_{X^\\sigma_{\\textup{fMGT I}}}\\|\\psi_{tt}\\|_{L_t^{\\frac{1}{1-\\alpha}}(H^{-1})}\n\t+\\|\\sigma\\|_{X^\\sigma_{\\textup{fMGT I}}}\\|\\psi_{tt}\\|_{W_t^{\\alpha-1\/2, \\infty}(H^{-1})}.\n\t\\end{aligned}\n\t\\end{equation}\n\tAgain using \\eqref{fracODE} with $\\psi$ in place of $\\psi^n$ yields\n\t\\begin{equation}\n\t\\begin{aligned}\n\t&\\|\\psi_{tt}\\|_{L_t^{\\frac{1}{1-\\alpha}}(H^{-1})}\t+\\|\\psi_{tt}\\|_{W_t^{\\alpha-1\/2, \\infty}(H^{-1})} \\lesssim\\, \\|\\Dt^{2+\\alpha} \\psi\\|_{L_t^2(H^{-1})}\\\\\n\t\\lesssim&\\, \\|\\sigma \\psi_{tt} - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi - \\delta \\Dt^{2-\\alpha} \\Delta \\psi\\|_{L_t^2(H^{-1})},\n\t\\end{aligned}\n\t\\end{equation}\n\twhere $\\|\\sigma \\psi_{tt}\\|_{L_t^2(H^{-1})}\\lesssim \\|\\sigma \\psi_{tt}\\|_{H_t^{\\alpha-1\/2}(H^{-1})}$. Therefore, these terms can be absorbed for small enough $\\sigma$ by the left-hand side or handled by Gronwall's inequality to conclude that $\\psi=0$.\n\\end{proof}\nWe next prove a well-posedness result for the corresponding nonlinear problem. To guarantee that the coefficient $\\sigma$ is small enough in the fixed-point iteration, we impose a smallness condition on the data.\n\\begin{theorem}[Local well-posedness of the fJMGT--W I equation] \\label{Thm:fJMGT_W_I} Let $\\alpha \\in [\\alpha_0,1)$ for some $\\alpha_0 > 1\/2$ and $T>0$. Further, assume that $ f \\in H^{\\alpha-1\/2}(0,{T};L^2(\\Omega))$. There exists $\\varrho=\\varrho(\\alpha, T)>0$, such that if\n\t\\begin{equation}\n\t\\|f\\|^2_{H^{\\alpha-1\/2}(L^2)}+\\|\\psi_0\\|_{H^2}^2+\\|\\psi_1\\|_{H^2}^2+\\|\\psi_2\\|_{H^1}^2 \\leq \\varrho^2,\n\t\\end{equation}\n\tthen the initial boundary-value problem\n\t\\begin{equation}\n\t\\left \\{\n\t\\begin{aligned}\n\t\\tau^\\alpha \\Dt^{2+\\alpha}\\psi + &(1+2k\\psi_t) \\psi_{tt} - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi - \\delta \\Dt^{2-\\alpha} \\Delta \\psi = f\\hspace*{-2mm}&&\\text{in }\\Omega\\times(0,T), \\\\[1mm]\n\t&\\psi=\\,0&&\\text{on }\\partial\\Omega\\times(0,T),\\\\[1mm]\n\t&(\\psi, \\psi_t, \\psi_{tt})=\\,(\\psi_0, \\psi_1, \\psi_2)&&\\mbox{in }\\Omega\\times \\{0\\},\n\t\\end{aligned} \\right.\n\t\\end{equation}\n\thas a unique solution $\\psi \\in X_{\\textup{fMGT I}}$, which satisfies\t\n\t\\begin{equation} \\label{energy_est2_fJMGT_W_I}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] \\|\\psi\\|^2_{X_{\\textup{fMGT I}}}\n\t\\end{multlined}\n\t\\lesssim\\,\\begin{multlined}[t] \t\\|f\\|^2_{H^{\\alpha-1\/2}(L^2)}+\\|\\psi_0\\|_{H^2}^2+\\|\\psi_1\\|_{H^2}^2+\\|\\psi_2\\|_{H^1}^2. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tThe proof follows by setting up a fixed-point mapping $\\mathcal{T}:w \\mapsto \\psi$, which associates\n\t\\[\n\tw\\in B_R:=\\{w \\in X_{\\textup{fMGT I}}\\, : \\|w\\|_{ X_{\\textup{fMGT I}}}\\leq R, \\ w(0)=\\psi_0, \\, w_t(0)=\\psi_1, \\, w_{tt}(0)=\\psi_2 \\} \n\t\\]\n\twith the solution $\\psi$ of the linearized problem \\eqref{linearized_fMGTI} with $\\sigma=2k w_t$. We recall that\n\t\\[\n\tX^\\sigma_\\textup{fMGT I} = L^2(0,T;(W^{1,3}\\cap L^\\infty)(\\Omega)),\n\t\\]\n\tand so\n\t\\[\n\t\\|\\sigma\\|_{X^\\sigma_{\\textup{fMGT I}}}= 2|k|\\|w_t\\|_{X^\\sigma_{\\textup{fMGT I}}} \\leq C \\|w\\|_{ X_{\\textup{fMGT I}}} \\lesssim R.\n\t\\]\n\tThus, $\\sigma$ can be made small enough by decreasing $R>0$. The self-mapping is thus an immediate consequence of the energy estimate \\eqref{energy_est_I}, provided we choose $\\varrho$ small enough, so that\n\t\\[\n\tC(R, T) (\\|f\\|^2_{H^{\\alpha-1\/2}(L^2)}+\\|\\psi_0\\|_{H^2}^2+\\|\\psi_1\\|_{H^2}^2+\\|\\psi_2\\|_{H^1}^2) \\leq C(R, T) \\varrho^2 \\leq R^2.\n\t\\]\n\t\\indent We prove strict contractivity of this mapping next. Let $w^{(1)}$, $w^{(2)} \\in B_R$. Denote $\\psi^{(1)}=\\mathcal{T} w^{(1)}$ and $\\psi^{(2)}=\\mathcal{T} w^{(2)}$. Contractivity of $\\mathcal{T}$ follows by considering the difference equation for $\\overline{\\psi}= \\psi^{(1)}-\\psi^{(2)}$:\n\t\\begin{equation} \\label{fJMGT_W_I_diff_contract}\n\t\\begin{aligned}\n\t&\\tau^\\alpha \\Dt^{2+\\alpha}\\overline{\\psi} + (1+2kw_t^{(1)})\\overline{\\psi}_{tt} - c^2\\Delta\\overline{\\psi} -\\tau^\\alpha c^2\\Delta \\Dt^\\alpha\\overline{\\psi} - \\delta \\Dt^{2-\\alpha} \\Delta \\overline{\\psi}_t+2k\\psi^{(2)}_{tt}\\overline{w}_t=0,\n\t\\end{aligned}\n\t\\end{equation}\n\twhich is supplemented by zero initial conditions. Similarly to the proof of uniqueness in Proposition~\\ref{Prop:fMGT_I}, testing with $\\overline{\\psi}_{tt}$ yields\n\t\\[\n\t\\begin{aligned}\n\t&\\,\\begin{multlined}[t] \\frac{\\tau^\\alpha}{2}\\nLtwo{\\Dt^{1+\\alpha}\\overline{\\psi}(t)}^2\n\t+ \\frac{\\tau^\\alpha c^2}{2}\\nLtwo{\\nabla\\Dt^{\\alpha}\\overline{\\psi}(t)}^2\\\\\\hspace*{2cm}+C(\\alpha)(\\|\\psi_{tt}\\|_{H_t^{-(1-\\alpha)}(L^2)}^2\n\t+ \\|\\Dt^{3\/2} \\nabla \\overline{\\psi}\\|_{L^2_t(L^2)}^2) \\end{multlined}\n\t\\\\\n\t\\leq&\\, \\left|\\int_0^t\\prodLtwo{-2k\\psi^{(2)}_{tt}\\overline{w}_t+c^2\\Delta\\overline{\\psi}-2k w_t^{(1)}\\overline{\\psi}_{tt}}{\\Dt^{1+\\alpha}\\overline{\\psi}}\\, \\textup{d} s \\right| \\\\\n\t\\leq&\\, \\begin{multlined}[t]\\frac{\\epsilon}{2} \\|\\Dt^{1+\\alpha}\\overline{\\psi}\\|_{L_t^\\infty(L^2)}^2 \n\t+\\frac{1}{2\\epsilon} \\|-2k\\psi^{(2)}_{tt}\\overline{w}_{t}\\|_{L_t^1(L^2)}^2\n\t+\\frac{\\epsilon}{2} \\|\\Dt^{3\/2}\\nabla\\overline{\\psi}\\|_{L^2_t(L^2)}^2 \\\\\n\t+\\frac{1}{2\\epsilon}\\Bigl(\\|c^2\\nabla\\overline{\\psi}\\|_{H_t^{\\alpha-1\/2}(L^2)}+\\|2k w_t^{(1)}\\overline{\\psi}_{tt}\\|_{H_t^{\\alpha-1\/2}(H^{-1})}\\Bigr)^2.\\end{multlined}\n\t\\end{aligned}\n\t\\]\n\tWe can then rely on the following bound: \n\t\\[\n\t\\begin{aligned}\n\t\\|c^2\\nabla \\overline{\\psi}\\|_{H^{\\alpha-1\/2}(L^2)}&\\lesssim \\|\\nabla\\Dt^{\\alpha}\\overline{\\psi}\\|_{L^2(L^2)}\n\t\\end{aligned}\n\t\\]\n\tand, by \\eqref{estsigmapsitt_I}--\\eqref{Dt2plusalphapn} with $2k w_t^{(1)}$ in place of $\\sigma$, we have\n\t\\[\n\t\\begin{aligned}\n\t&\\|2k w_t^{(1)}\\psi_{tt}\\|_{H^{\\alpha-1\/2}(H^{-1})}\\lesssim \\| w_t^{(1)}\\|_{X^\\sigma_{\\textup{fMGT I}}} \\|\\Dt^{2+\\alpha} \\psi\\|_{L^2(H^{-1})}\\\\\n\t\\lesssim&\\, \\| w_t^{(1)}\\|_{X^\\sigma_{\\textup{fMGT I}}}\\|2k w_t^{(1)} \\psi_{tt}- c^2\\Delta\\overline{\\psi} -\\tau^\\alpha c^2\\Delta \\Dt^\\alpha\\overline{\\psi} - \\delta \\Dt^{2-\\alpha} \\Delta \\overline{\\psi}_t\\|_{L^2(H^{-1})}.\n\t\\end{aligned}\n\t\\]\n\tThus for $\\| w_t^{(1)}\\|_{X_\\sigma} \\lesssim R$ small enough, similarly to the proof of Proposition~\\ref{Prop:fMGT_I}, we obtain\n\t\\begin{equation}\n\t\\begin{aligned}\n\t& \\frac{\\tau^\\alpha}{2}\\nLtwo{\\Dt^{1+\\alpha}\\overline{\\psi}(t)}^2\n\t+ \\frac{\\tau^\\alpha c^2}{2}\\nLtwo{\\nabla\\Dt^{\\alpha}\\overline{\\psi}(t)}^2+\\|\\overline{\\psi}_{tt}\\|_{H_t^{-(1-\\alpha)}(L^2)}^2\n\t+ \\|\\Dt^{3\/2} \\nabla \\overline{\\psi}\\|_{L^2_t(L^2)}^2\n\t\\\\\n\t\\lesssim& \\, R^2\\|2k \\psi_{tt}^{(2)}\\overline{w}_t\\|_{L^2(H^{-1})}^2+\\|2k \\psi_{tt}^{(2)}\\overline{w}_{t}\\|_{L^1(L^2)}.\n\t\\end{aligned}\n\t\\end{equation}\n\tFurthermore,\n\t\\begin{equation}\n\t\\begin{aligned}\n\tR^2\\|2k \\psi_{tt}^{(2)}\\overline{w}_t\\|_{L^2(H^{-1})}^2+\\|2k \\psi_{tt}^{(2)}\\overline{w}_{t}\\|_{L^1(L^2)} \n\t\\lesssim&\\, R^2 \\|\\psi_{tt}^{(2)}\\|^2_{L^\\infty L^4}\\|\\overline{w}_{t}\\|^2_{L^2(L^4)} \\\\\n\t\\lesssim&\\, R^2 \\varrho^2 \\|\\overline{w}_{t}\\|^2_{L^2(L^4)}.\n\t\\end{aligned}\n\t\\end{equation}\n\tWe can further bound the last term as follows:\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\|\\nabla \\overline{w}_t\\|^2_{L^2(L^2)}=\\|\\textup{I}^{\\gamma} \\Dt^{1+\\gamma}\\nabla \\overline{w}\\|^2_{L^2(L^2)} \\leq \\|\\mathfrak{g}_{1-\\gamma}\\|_{L^1(0,T)} \\|\\Dt^{1+\\gamma}\\nabla \\overline{w}\\|^2_{L^2(L^2)},\n\t\\end{aligned}\n\t\\end{equation}\n\tchoosing $\\gamma=1\/2$. Thus, by decreasing $\\varrho>0$, we can guarantee that $\\mathcal{T}$ is strictly contractive in the following norm:\n\t\\[\n\t|||\\psi|||= \\nLinfLtwo{\\Dt^{1+\\alpha}\\overline{\\psi}}^2+ \\nLinfLtwo{\\nabla\\Dt^{\\alpha}\\overline{\\psi}}^2+\\|\\overline{\\psi}_{tt}\\|_{H^{-(1-\\alpha)}(L^2)}^2+ \\|\\nabla \\overline{\\psi}\\|_{L^2(L^2)}^2.\n\t\\]\n\tThe rest of the arguments follow as in Theorem~\\ref{Thm:fJMGT_W_III} and complete the proof.\t\n\\end{proof}\n\\subsection{Analysis of the fJMGT--W equation} To formulate the corresponding result for the fMGT-W equation\n\\[\t\n\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+2k\\psi_t)\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta \\Delta\\psi_{t}=f \n\\]\nwe again need smallness of the coefficient $\\sigma$ in a suitable norm. To this end, let\n\\[\nX^\\sigma_\\textup{fMGT} = H^{\\alpha\/2}(0,T;(W^{1,3}\\cap L^\\infty)(\\Omega))\n\\]\nfor $\\alpha \\in (1\/2, 1)$, {and denote the corresponding norm by $\\|\\cdot\\|_{X^\\sigma_\\textup{fMGT}}$}. We also introduce the solution space by\n\\begin{equation}\n\\begin{aligned}\n\\quad X_{\\textup{fMGT}}=\\, \\left\\{ \\psi\\in H^{2+\\alpha}(0,T; L^2(\\Omega)):\\right.&\\, \\Dt^{1+\\alpha\/2} \\psi \\in L^\\infty(0,T; {H_\\diamondsuit^2(\\Omega)}), \\\\& \\left. \\Dt^{1+\\alpha} \\in L^\\infty(0,T; H_0^1(\\Omega)) \\right\\},\n\\end{aligned}\n\\end{equation}\n{equipped with the norm $\\|\\cdot\\|_{X_{\\textup{fMGT}}}$}. Note that with this choice, again \n\\[\n\\|2k \\psi_t\\|_{X^\\sigma_\\textup{fMGT}} \\lesssim \\|\\psi\\|_{X_{\\textup{fMGT}}}\n\\] \nholds but the energy term $\\|\\Dt^{1+\\alpha\/2}\\Delta \\psi\\|_{L^2(L^2)}$ needed for this purpose comes with an $\\alpha$-dependent coefficient in \\eqref{est_fMGT}.\nFor this reason, while still being able to show well-posedness also of the nonlinear \\textup{fJMGT--W} equation for each $\\alpha\\in(0,1)$, we will not obtain a uniform bound quantifying smallness of the initial data. That is, we will not be able to show that for fixed small enough $(\\psi_0, \\psi_1, \\psi_2)$, there exists a family of solutions to the nonlinear problem. Hence, concerning limits as $\\alpha\\to1-$, we will restrict ourselves to the linear \\textup{fMGT} equation:\n\\[\n\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+\\sigma(x,t))\\psi_{tt}-c^2 \\Delta \\psi -\\tau^\\alpha c^2 {\\textup{D}}_t^\\alpha \\Delta \\psi- \\delta \\Delta\\psi_{t}=f .\n\\]\nWe next prove the well-posedness of the linear time-fractional problem. {Note that under the same regularity conditions on the initial and boundary data, the fMGT equation allows us to prove slightly better regularity of the solution as compared to fMGT I; cf. Proposition~\\ref{Prop:fMGT_I}.}\n\\begin{proposition}[Well-posedness of the fMGT equation] \\label{Prop:fMGT}\n\tLet $\\alpha \\in [\\alpha_0,1)$ for some $\\alpha_0 > 1\/2$. Assume that $f \\in H^{\\alpha-1\/2}(0,T;L^2(\\Omega))$, $\\sigma \\in X^\\sigma_\\textup{fMGT I}$, and \\[(\\psi_0, \\psi_1, \\psi_2) \\in ({H_\\diamondsuit^2(\\Omega)}, {H_\\diamondsuit^2(\\Omega)}, H_0^1(\\Omega)).\\] Then there exists $\\varrho=\\varrho(\\alpha)>0$, such that if $\\|\\sigma\\|_{X^\\sigma_\\textup{fMGT}} \\leq \\varrho$, there is a unique $\\psi \\in X_\\textup{fMGT}$, which satisfies the problem in the $L^2(0,T; L^2(\\Omega))$ sense with $(\\psi, \\psi_t, \\psi_{tt})\\vert_{t=0}=(\\psi_0, \\psi_1, \\psi_2)$. Furthermore, this solution fulfills the following estimate:\n\t\\begin{equation} \\label{est_fMGT}\n\t\\begin{aligned}\n\t& \\begin{multlined}[t]\\|\\Dt^{2+\\alpha}\\psi\\|_{L^2(L^2)}^2\n\t+ \\nLtwo{\\nabla \\Dt^{1+\\alpha}\\psi(t)}^2\n\t+\\nLtwo{\\Delta \\Dt^{\\alpha}\\psi(t)}^2\\\\\n\t+ C(\\alpha)\\Bigl(\\|\\nabla\\psi_{tt}\\|_{{H^{-(1-\\alpha)\/2}}(L^2)}^2\n\t+ \\|\\Dt^{1+\\alpha\/2}\\Delta \\psi\\|_{L^2(L^2)}^2\\Bigr) \\end{multlined}\\\\\n\t\\lesssim&\\, \\|f\\|_{H^{\\alpha-1\/2}(L^2)}^2+\\nLtwo{\\Delta \\psi_0}^2+\\nLtwo{\\Delta \\psi_1}^2+\\nLtwo{\\nabla \\psi_2}^2.\n\t\\end{aligned}\n\t\\end{equation}\n\twhere $C(\\alpha)\\to0$ as $\\alpha\\to1^-$. \n\\end{proposition}\n\\begin{proof}\n{The proof follows similarly to the proof of Proposition~\\ref{Prop:fMGT_I} with the main changes contained in the energy analysis, on which we focus here.} Note that now the semi-discrete problem can be equivalently rewritten as a system of Volterra integral equations:\n\t\\[\n\t\\begin{aligned}\n\t&\\begin{multlined}[t]\\tau^\\alpha M \\boldsymbol{\\mu} \n\t+ M_\\sigma(t)\\Bigl(p^{\\alpha-1}*\\boldsymbol{\\mu} + p^0 \\boldsymbol{\\xi}_2\\Bigr)\n\t+c^2 K \\Bigl(p^{\\alpha+1}*\\boldsymbol{\\mu} + p^2 \\boldsymbol{\\xi}_2+ p^1 \\boldsymbol{\\xi}_1+ p^0 \\boldsymbol{\\xi}_0\\Bigr)\\\\\n\t+\\tau^\\alpha c^2 \\Bigl(p^{1}*\\boldsymbol{\\mu} +p^{2-\\alpha} \\boldsymbol{\\xi}_2+ p^{1-\\alpha} \\boldsymbol{\\xi}_1\\Bigr)\n\t+\\delta K (\tp^{\\alpha}*\\boldsymbol{\\mu} + p^1 \\boldsymbol{\\xi}_2+ p^0 \\boldsymbol{\\xi}_1) =f\\end{multlined}\n\t\\end{aligned}\n\t\\]\n\tin place of \\eqref{Volterra_system_I}; the existence of an approximate solutions follows by the same arguments. We present the energy analysis of the semi-discrete problem here, but omit the superscript $n$ below for simplicity. Multiplying the semi-discrete equation with $-\\Delta \\Dt^{1+\\alpha} \\psi$, yields the energy identity \\eqref{enid_rem}, where now\n\t\\begin{equation} \\label{damping_fMGT}\n\t\\begin{aligned} \n\td =&\\, \\int_0^t \\prodLtwo{\\Dt \\Delta \\psi}{\\Delta \\Dt^{1+\\alpha} \\psi}\\, \\textup{d} s \\\\\n\t=&\\, d_0+\t\\int_0^t\\prodLtwo{{\\textup{I}}^\\alpha \\Dt^{1+\\alpha}\\Delta\\psi(s)}{\\Dt^{1+\\alpha}\\Delta\\psi(s)}\\, \\textup{d} s ,\n\t\\end{aligned}\n\t\\end{equation}\n\twith $d_0=\\prodLtwo{\\Delta\\psi_t(0)}{\\Dt^{\\alpha} \\Delta \\psi(t)}-p^{1-\\alpha}(t)\\|\\Delta\\psi_t(0)\\|_{L^2}^2$ instead of \\eqref{damping_fMGT_I}. Here, we have used the identities\n\t\\[\\Dt\\psi={\\textup{I}}^1\\psi_{tt}+\\psi_t(0)={\\textup{I}}^\\alpha\\Dt^{1+\\alpha}\\psi+\\psi_t(0)\\]\n\tand \\[\n\t\\begin{aligned}\n\t\\int_0^t\\prodLtwo{\\Delta\\psi_t(0)}{\\Dt^{1+\\alpha} \\Delta \\psi(s)}\\, \\textup{d} s \n\t=&\\,\\prodLtwo{\\Delta\\psi_t(0)}{\\int_0^t ((\\Dt \\Dt^{\\alpha} \\Delta \\psi)(s)-p^{-\\alpha}(s)\\Delta\\psi_t(0))\\, \\textup{d} s }\\\\\n\t=&\\, \\prodLtwo{\\Delta\\psi_t(0)}{\\Dt^{\\alpha} \\Delta \\psi(t)}-p^{1-\\alpha}(t)\\|\\Delta\\psi_t(0)\\|_{L^2}^2.\n\t\\end{aligned}\\]\n\tThe damping term can now be estimated from below as follows\n\t\\[\n\t\\frac{d+d_0}{\\cos(\\pi\\gamma\/2)} \\geq \n\t\\|\\Dt^{1+\\alpha}\\Delta \\psi\\|_{H_t^{-\\alpha\/2}(L^2)}^2\\sim \\|\\Dt^{1+\\alpha\/2}\\Delta \\psi\\|_{L_t^2(L^2)}^2. \n\t\\]\n\tFurthermore, we have\n\t\\[\n\t\\begin{aligned}\n\t\\left|\\int_0^t\\prodLtwo{\\tilde{f}(s)}{\\Dt^{1+\\alpha}\\Delta \\psi(s)}\\, \\textup{d} s \\right|\n\t\\leq \n\t\\frac{1}{2\\epsilon}\\|\\tilde{f}\\|_{H^{\\alpha\/2}(L^2)}^2+\\frac{\\epsilon}{2}\\|\\Dt^{1+\\alpha\/2}\\Delta \\psi\\|_{L^2(L^2)}^2\n\t\\end{aligned}\n\t\\]\n\twith\n\t\\begin{equation}\n\t\\tilde{f}=f+c^2\\Delta\\psi^n-\\sigma\\psi_{tt}^n +\\tau^{\\alpha}\\left(p^{-\\alpha}\\psi_{tt}^n(0)-c^2p^{1-\\alpha}\\Delta\\psi_t^n(0)\\right)\n\t\\end{equation}\n\tas before. \tBy the Kato--Ponce inequality \\eqref{prodruleest} with $\\rho=\\alpha\/2$, we then have\n\t\\begin{equation}\\label{estsigmapsitt}\n\t\\begin{aligned}\n\t&\\|\\sigma\\psi_{tt}\\|_{H_t^{\\alpha\/2}(L^2)}\n\t\\\\\n\t\\lesssim&\\,\n\t\\|\\sigma\\|_{W_t^{\\alpha\/2,p_1}(L^\\infty)}\\|\\psi_{tt}\\|_{L_t^{q_1}(L^2)}\n\t+\\|\\sigma\\|_{L_t^{p_2}(L^\\infty)}\\|\\psi_{tt}\\|_{W_t^{\\alpha\/2,q_2}(L^2)}\n\t\\end{aligned}\n\t\\end{equation}\n\tfor $\\frac{1}{p_1}+\\frac{1}{q_1}=\\frac{1}{p_2}+\\frac{1}{q_2}=\\frac12$. \tSimilarly to before, to further estimate the norms of $\\psi_{tt}$ in \\eqref{estsigmapsitt} we will use the leading time derivative term $\\Dt^{2+\\alpha}\\psi$ and its representation via the fractional ODE\n\t\\[\n\t\\tau^\\alpha\\Dt^\\alpha \\psi_{tt} + \\psi_{tt} = -r, \\qquad r=\\sigma \\psi_{tt} - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi - \\delta \\Dt \\Delta \\psi -f.\n\t\\]\n\tTherefore,\n\t\\[\n\t\\begin{aligned}\n\t&\\|\\psi_{tt}\\|_{L^{q_1}(0,t;L^2)}+\\|\\psi_{tt}\\|_{W^{\\alpha\/2, q_2}(0,t;L^2)}\\lesssim \\|\\Dt^{2+\\alpha}\\psi\\|_{L^2(L^2)}\\\\\n\t\\lesssim&\\, \\|\\sigma \\psi_{tt} - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi - \\delta \\Dt \\Delta \\psi -f\n\t\\|_{L^2(L^2)}+\\|\\psi_{tt}(0)\\|_{L^2},\n\t\\end{aligned}\n\t\\]\n\twhere again we take care of the highest order term by using the damping term $d$. Thus, we require\n\t\\begin{equation}\\label{cond_qomega}\n\t2-\\frac{1}{q_1}\\leq 2+\\alpha-\\frac12, \\quad\t2+\\alpha\/2-\\frac{1}{q_2}\\leq 2+\\alpha-\\frac12.\n\n\t\\end{equation}\n\tAdditionally, we aim at choosing the available parameters $p_i$ (yielding $q_i=2p_i\/(p_i-2)$), such that (having in mind that $\\sigma = 2k \\psi_t$ in the fixed point argument later on)\n\t\\[\n\t\\|\\Delta\\psi_t(s)\\|_{W^{\\alpha\/2, p_1}_t(L^2)}\\lesssimd, \\quad \n\t\\|\\Delta\\psi_t(s)\\|_{L^{p_2}_t(L^2)}\\lesssimd\\,,\n\t\\]\n\twhich leads to \n\t\\begin{equation}\\label{cond_pomega}\n\t1+\\frac{\\alpha}{2}-\\frac{1}{p_1}\\leq 1+\\frac{\\alpha}{2}-\\frac12 \\mbox{ and }\n\t1-\\frac{1}{p_2}\\leq 1+\\frac{\\alpha}{2}-\\frac12.\n\t\\end{equation}\n\tIt is readily checked that all conditions in \\eqref{cond_qomega}, \\eqref{cond_pomega} can be satisfied with the choice \n\t\\begin{equation}\\label{p12omega12}\n\t(p_1, q_1)=(2, \\infty)\\,, \\qquad (p_2, q_2)=\\left(\\frac{2}{1-\\alpha}, \\frac{2}{\\alpha}\\right).\n\t\\end{equation}\n\tThus analogously to the proof of Proposition~\\ref{Prop:fMGT_I}, we arrive at \\eqref{est_fMGT}.\n\\end{proof}\n\\begin{theorem}[Local well-posedness of the fJMGT--W equation] \\label{Thm:fJMGT_W}\n\tLet $\\alpha \\in [\\alpha_0,1)$ for some $\\alpha_0 > 1\/2$ and $T>0$. Further, assume that $f \\in H^{\\alpha-1\/2}(0,{T}; L^2(\\Omega))$. Then there exists \n\t$\\varrho=\\varrho(\\alpha,T)>0$\n\tsuch that if\n\t\\begin{equation}\n\t\\|f\\|^2_{H^{\\alpha-1\/2}(L^2)}+\\|\\psi_0\\|_{H^2}^2+\\|\\psi_1\\|_{H^2}^2+\\|\\psi_2\\|_{H^1}^2 \\leq \\varrho ^2,\n\t\\end{equation}\n\tthen the initial boundary-value problem\n\t\\begin{equation}\n\t\\left \\{\n\t\\begin{aligned}\n\t\\tau^\\alpha \\Dt^{2+\\alpha}\\psi + &(1+2k\\psi_t) \\psi_{tt} - c^2\\Delta\\psi -\\tau^\\alpha c^2\\Dt^\\alpha\\Delta \\psi - \\delta \\Delta \\psi_t = f\\hspace*{-2mm}&&\\text{in }\\Omega\\times(0,T), \\\\[1mm]\n\t&\\psi=\\,0&&\\text{on }\\partial\\Omega\\times(0,T),\\\\[1mm]\n\t&(\\psi, \\psi_t, \\psi_{tt})=\\,(\\psi_0, \\psi_1, \\psi_2)&&\\mbox{in }\\Omega\\times \\{0\\},\n\t\\end{aligned} \\right.\n\t\\end{equation}\n\thas a unique solution $\\psi \\in X_{\\textup{fMGT}}$, which satisfies\t\n\t\\begin{equation} \\label{energy_est2_fJMGT_W}\n\t\\begin{aligned}\n\t\\begin{multlined}[t] \\|\\psi\\|^2_{X_{\\textup{fMGT}}}\n\t\\end{multlined}\n\t\\lesssim\\,\\begin{multlined}[t] \t\\|f\\|^2_{H^{\\alpha-1\/2}(L^2)}+\\|\\psi_0\\|_{H^2}^2+\\|\\psi_1\\|_{H^2}^2+\\|\\psi_2\\|_{H^1}^2. \\end{multlined}\n\t\\end{aligned}\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\tThe proof follows in an analogous manner to the proof of Theorem~\\ref{Thm:fJMGT_W_I}, combined with the results of Proposition~\\ref{Prop:fMGT}. We therefore omit the details here.\n\\end{proof}\n\\begin{remark}[On the analysis of the fMGT II equation with $\\boldsymbol{\\sigma \\neq 0}$]\\label{Remark:fMGT_II}\n\tWe note that the \\textup{fMGT II} equation {\n\t\\[\n\t\\tau^\\alpha {\\textup{D}}_t^\\alpha \\psi_{tt}+(1+\\sigma(x,t))\\psi_{tt}-c^2 \\Delta \\psi -(\\tau^\\alpha c^2 +\\delta){\\textup{D}}_t^\\alpha \\Delta \\psi=f\n\t\\]}\n\t does not seem to be tractable this way with $\\sigma\\not=0$. In particular, we would have \n\t\\[\n\td =\\nLtwo{\\Delta\\Dt^\\alpha\\psi}^2 \\Big \\vert_0^t \n\t\\]\n\tin place of \\eqref{damping_fMGT_I} and \\eqref{damping_fMGT}. Thus, the damping term $\\deltad$ is obviously too weak. \\\\\n\t\\indent Note that the multiplier $-\\Delta\\psi_{tt}$ that we have successfully used for the \\textup{fMGT III} equation in Section~\\ref{Sec:Analysis_fJMGT_W_III} does to work out either since then the $\\delta$ term cannot be proven to be nonnegative due to the fact that the difference $2-\\alpha$ of the differentiation orders is larger than one. We provide an analysis of the \\textup{fMGT II} equation with $\\sigma=0$ in Section~\\ref{Sec:LinearAnalysis} by rewriting it as a second-order wave equation with memory.\n\\end{remark}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nThe studies of transition amplitudes from the ground to excited nucleon states off the proton ($\\gamma_vpN^*$ \nelectrocouplings) offer insight into the $N^*$ structure and allow the exploration of the non-perturbative strong \ninteraction mechanisms that are responsible for the resonance formation as relativistic bound systems of \nquarks and gluons~\\cite{Bu12,Az13,Cr14}. The data on $\\gamma_vpN^*$ electrocouplings represent the only \nsource of information on different manifestations of the non-perturbative strong interaction in the generation \nof excited nucleon states of different quantum numbers. \n\nThe CLAS detector at Jefferson Lab is a unique large-acceptance instrument designed for the comprehensive \nexploration of exclusive meson electroproduction. It offers excellent opportunities for the study of \nelectroexcitation of nucleon resonances in detail and with precision. The CLAS detector has provided the \ndominant portion of all data on meson electroproduction in the resonance excitation region. The studies of \ntransition helicity amplitudes from the proton ground state to its excited states represent a key aspect of\nthe $N^*$ program with CLAS~\\cite{Mo11,Bu12,Mo14,Bu14}.\n\nMeson-electroproduction data off nucleons in the $N^*$ region obtained with CLAS open up an opportunity to \ndetermine the $Q^2$-evolution of the $\\gamma_vNN^*$ electrocouplings in a combined analysis of various \nmeson-electroproduction channels for the first time. A variety of measurements of $\\pi^+n$ and $\\pi^0p$ \nsingle-pion electroproduction off the proton, including polarization measurements, have been performed with CLAS \nin the range of $Q^2$ from 0.16 to 6~GeV$^2$ and in the area of invariant masses of the final hadrons \n$W < 2.0$~GeV~\\cite{Joo,Joo2,Joo3,Egiyan,Ungaro,Smith,Park,Biselli,Park15}. Exclusive $\\eta p$ electroproduction \noff the proton was studied with CLAS for $W < 2.3$~GeV and $Q^2$ from 0.2 to 3.1~GeV$^2$~\\cite{Den07}. Furthermore, \ndifferential cross section and polarization asymmetries in exclusive $KY$ electroproduction channels were \nobtained for $W$ from threshold to 2.6~GeV and for $Q^2 < 5.4$~GeV$^2$~\\cite{Car12,Car08,Car07,Gab14,Car09,Car03}. \nThese experiments were complemented by the measurements of nine independent $\\pi^+\\pi^-p$ electroproduction \ncross sections off the proton. The data on charged double-pion electroproduction covered the area of $W < 1.6$~GeV \nat photon virtualities from 0.25 to 0.55~GeV$^2$~\\cite{Fe09}. They are also available from earlier measurements \nwith CLAS for $W$ from 1.40~GeV to 2.10~GeV and 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$~\\cite{Ri03}.\n\nThe electroexcitation amplitudes for the low-lying resonances $\\Delta(1232)3\/2^+$, $N(1440)1\/2^+$, $N(1520)3\/2^-$,\nand $N(1535)1\/2^-$ were determined over a wide range of $Q^2$ in a comprehensive analysis of JLab-CLAS data \non differential cross sections, longitudinally polarized beam asymmetries, and longitudinal target and \nbeam-target asymmetries~\\cite{Az09}. Recently $\\gamma_vNN^*$ electrocouplings of several higher-lying nucleon \nresonances: $N(1675)5\/2^-$, $N(1680)5\/2^+$, and $N(1710)1\/2^+$ have become available for the first time for \n1.5~GeV$^2$ $< Q^2 <$ 4.5~GeV$^2$ from analysis of exclusive $\\pi^+n$ electroproduction off the proton~\\cite{Park15}. \nElectrocouplings for the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances for $Q^2 < 0.6$~GeV$^2$ have been \ndetermined from the data on exclusive $\\pi^+\\pi^-p$ electroproduction off the proton~\\cite{Mo12} together with the \npreliminary results on the electrocouplings of several resonances in the mass range from 1.6~GeV to 1.75~GeV \navailable for the first time from this exclusive channel at 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$~\\cite{Az13,Mo14}. \nThe CLAS results on the $\\gamma_vpN^*$ electrocouplings~\\cite{Az09,Bu12,Mo12,Mo14,Park15} have had a stimulating \nimpact on the theory of the excited nucleon state structure, in particular, on the QCD-based approaches.\n\nThe light cone sum rule (LCSR) approach~\\cite{Br09,Br15} for the first time provided access to the quark \ndistribution amplitudes (DA) inside the $N(1535)1\/2^-$ resonance from analysis of the CLAS results on the\n$\\gamma_vpN^*$ electrocouplings of this state~\\cite{Az09}. Confronting the quark DA's of excited nucleon states \ndetermined from the experimental results on the $\\gamma_vpN^*$ electrocouplings to the LQCD expectations, makes \nit possible to explore the emergence of the resonance structure starting from the QCD Lagrangian. The moments \nof the $N(1535)1\/2^-$ quark DA's derived from the CLAS data are consistent with the LQCD expectations~\\cite{Br14}. \n\nThe Dyson-Schwinger Equations of QCD (DSEQCD) provide a conceptually different avenue for relating the \n$\\gamma_vpN^*$ electrocouplings to the fundamental QCD Lagrangian~\\cite{Cr15,Cr15b,Cr15a,Eich12}. The DSEQCD \napproach allows for the evaluation of the contribution of the three bound dressed quarks, the so-called quark \ncore, to the structure of excited nucleon states starting from the QCD Lagrangian. A successful description of \nthe nucleon elastic form factors and the CLAS results on the $N \\to \\Delta$, $N \\to N(1440)1\/2^+$ transition \nelectromagnetic form factors~\\cite{Az09,Mo12,Bu12,Mo14} at photon virtualities $Q^2 > 2.0$~GeV$^2$ has been \nrecently achieved within the DSEQCD framework~\\cite{Cr13,Cr15,Cr15a}. However, at smaller photon virtualities \n$Q^2 < 1.0$~GeV$^2$, the DSEQCD approach failed to describe the CLAS results~\\cite{Az09,Mo12,Mo14} on the \n$\\Delta(1232)3\/2^+$ and $N(1440)1\/2^+$ $\\gamma_vpN^*$ electrocouplings~\\cite{Cr15,Cr15a}. \n\nFurthermore, most quark models~\\cite{San15,Met12,Ram14,Ram10} that take into account the contributions from \nquark degrees of freedom only, have substantial shortcomings in describing resonance electrocouplings at \n$Q^2 < 1.0$~GeV$^2$ even if they provide a reasonable description of the experimental results at higher photon \nvirtualities. These are the indications for the contributions of degrees of freedom other than dressed quarks \nto the structure of excited nucleon states, contributions that become more relevant at small photon virtualities.\n\nA successful description of the CLAS results on the $N(1440)1\/2^+$ $\\gamma_vpN^*$ electrocouplings \n\\cite{Az09,Mo12,Mo14} has been recently achieved at small photon virtualities up to 0.5~GeV$^2$ within the \nframework of effective field theory employing pions, $\\rho$ mesons, the nucleon, and the Roper $N(1440)1\/2^+$\nresonance as the effective degrees of freedom~\\cite{Tia14}. This success emphasizes the importance of \nmeson-baryon degrees of freedom for the structure of excited nucleon states at small photon virtualities. \nFurthermore, a general unitarity requirement imposes meson-baryon contributions to both resonance electromagnetic \nexcitations and hadronic decay amplitudes. Studies of meson-baryon dressing contributions to the $\\gamma_vpN^*$ \nelectrocouplings from the global analysis of the $N\\pi$ photo-, electro-, and hadroproduction data carried out \nby Argonne-Osaka Collaboration~\\cite{Lee10,Lee091,Lee08} within the framework of a coupled channel approach, \nconclusively demonstrated the contributions from both meson-baryon and quark degrees of freedom to the structure \nof nucleon resonances.\n\nSome quark models that have been developed~\\cite{Az15,Az12,Ob14,Si14,Si09} take into account the contribution from \nboth meson-baryon and quark degrees of freedom to the structure of excited nucleon states. Implementation of \nmeson-baryon degrees of freedom allowed for a considerably improved description of the CLAS results on the\n$N(1440)1\/2^+$ and $N(1520)3\/2^-$ $\\gamma_vpN^*$ electrocouplings at photon virtualities $Q^2 < 1.0$~GeV$^2$,\nwhile simultaneously retaining a good description of these results for $Q^2 > 2.0$~GeV$^2$.\n\nPhysics analyses of the CLAS results~\\cite{Az09,Mo12,Mo14} on the $\\gamma_vpN^*$ electrocouplings revealed the \nstructure of excited nucleon states at photon virtualities $Q^2 < 5.0$~GeV$^2$ as a complex interplay between \nmeson-baryon and quark degrees of freedom. The relative contributions from the meson-baryon cloud and the quark \ncore are strongly dependent on the quantum numbers of the excited nucleons. Analyses of the $A_{1\/2}$ \nelectrocouplings of the $N(1520)3\/2^-$ resonance~\\cite{Lee08,Sa12} demonstrated that this amplitude is dominated \nby quark core contributions in the entire range of $Q^2 < 5.0$~GeV$^2$ measured by CLAS. However, the recent \nanalysis~\\cite{Az15} of the first CLAS results~\\cite{Park15} on the $N(1675)5\/2^-$ $\\gamma_vpN^*$ electrocouplings \nsuggested a dominance of the meson-baryon cloud. The experimental results on the $\\gamma_vpN^*$ electrocouplings \nfor all prominent resonances obtained in a wide range of photon virtualities are of particular importance in order to \nexplore the contributions from different degrees of freedom to the resonance structure. \n\nAnalyses of different exclusive channels are essential for a reliable extraction of the resonance parameters over \nthe full spectrum of excited nucleon states. Currently the separation of the resonant and non-resonant parts of the \nelectroproduction amplitudes can be done only within phenomenological reaction models. Therefore, the credibility of \nany resonance parameters extracted from the meson electroproduction data fit within the framework of any particular \nreaction model should be further examined. Non-resonant mechanisms in various meson-electroproduction channels are \ncompletely different, while the $\\gamma_vNN^*$ electrocouplings are the same. Consistent results for the \n$\\gamma_vpN^*$ electrocouplings of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances that were determined from \nindependent analyses of the major meson electroproduction channels, $\\pi^+n$, $\\pi^0p$, and $\\pi^+\\pi^-p$, \ndemonstrate that the extractions of these fundamental quantities are reliable as these different electroproduction \nchannels have quite different backgrounds~\\cite{Mo12}. Furthermore, this consistency also strongly suggests that the \nreaction models~\\cite{Az09,Mo09,Mo12} developed for the description of these channels will provide a reliable \nevaluation of the $\\gamma_vNN^*$ electrocouplings for analyzing either single- or charged double-pion \nelectroproduction data. These models then make it possible to determine the electrocouplings for almost all \nwell-established resonances that decay preferentially to the $N\\pi$ and\/or $N\\pi\\pi$ final states. The information \non the $\\gamma_vNN^*$ electrocouplings available from the exclusive charged double-pion electroproduction off the\nproton is still rather limited and will be extended by the results of this paper.\n\nIn this paper we present the results on the electrocouplings of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and \n$\\Delta(1620)1\/2^-$ resonances at the photon virtualities 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$, obtained from the \nanalysis of the CLAS data on $\\pi^+\\pi^-p$ electroproduction off the proton~\\cite{Ri03}. The analysis was carried \nout employing the JM reaction model~\\cite{Mo09,Mo12}, which has been further developed to provide a framework for \nthe determination of the $\\gamma_vpN^*$ electrocouplings from a combined fit of unpolarized differential cross \nsections in a broader kinematic area of $W$ and $Q^2$ in comparison with that covered in our previous studies\n\\cite{Mo09,Mo12}. This paper extends the available information on the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ \nelectrocouplings from the charged double-pion exclusive electroproduction off the proton and provides the first \nresults on the electrocouplings and the hadronic decay widths of the $\\Delta(1620)1\/2^-$ resonance to the \n$\\pi \\Delta$ and $\\rho N$ final states. \n\nThe paper is organized as follows. In Section~\\ref{genjm} we describe the JM reaction model employed for the \nextraction of the resonance parameters and the fitted experimental data. A special fitting procedure that allowed us \nto account for the experimental data and the reaction model uncertainties is presented in Section~\\ref{fit}. The \nresults on the $\\gamma_vpN^*$ electrocouplings and the hadronic decays of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and \n$\\Delta(1620)1\/2^-$ resonances to the $\\pi \\Delta$ and $\\rho N$ final states extracted from the CLAS data~\\cite{Ri03} \nare presented in Section~\\ref{nstarelectrocoupl}. Insights into the non-perturbative strong interaction mechanisms \noffered by our results and their impact on hadron structure theory are discussed in Section~\\ref{impact} with \nsummary and outlook in Section~\\ref{concl}.\n\n\\section{Analysis Tools for Evaluation of the $\\gamma_vpN^*$ Resonance Electrocouplings and Hadronic Decay Widths}\n\\label{genjm}\n\nThe $\\gamma_vpN^*$ electrocouplings and hadronic decay widths of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and \n$\\Delta(1620)1\/2^-$ resonances to the $\\pi \\Delta$ and $\\rho N$ final states were extracted from the fit of the \nCLAS charged double-pion electroproduction data~\\cite{Ri03} at $W$ from 1.41~GeV to 1.66~GeV in three $Q^2$-bins \ncentered at $Q^2$=0.65~GeV$^2$, 0.95~GeV$^2$, and 1.3~GeV$^2$. The JM meson-baryon model~\\cite{Mo09,Mo12} was \nemployed for the description of the measured observables in the $\\gamma_v p \\to \\pi^+\\pi^-p$ exclusive channel. \nThis model was successfully used in our previous extraction of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonance \nelectrocouplings at smaller $Q^2 < 0.6$~GeV$^2$~\\cite{Mo12} from the CLAS charged double-pion electroproduction \ndata~\\cite{Fe09} at $W < 1.57$~GeV. In our current analysis of the CLAS $\\pi^+\\pi^-p$ electroproduction data\n\\cite{Ri03}, the JM model was further developed in order to provide a data description in a wider area of $W$ \nfrom 1.40~GeV to 1.82~GeV and at photon virtualities $Q^2$ from 0.5~GeV$^2$ to 1.5~GeV$^2$. In this section we \ndescribe the differential cross sections we fit for the resonance parameter extraction. We also present the basic \nfeatures of the JM model relevant for the extraction of the $\\gamma_vpN^*$ electrocouplings from the data\n\\cite{Ri03}, focusing on the model updates needed to achieve a good description of the measured differential cross \nsections. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{pictures\/2pi_theta_phi.eps}\n\\includegraphics[width=8cm]{pictures\/2pi_alpha_pim.eps}\n\\caption{Kinematic variables for the description of $e p \\to e' p' \\pi^+ \\pi^-$ in the CM frame of \nthe final-state hadrons corresponding to the explicit assignment presented in Section~\\ref{kinxsect}. Panel (a) \nshows the $\\pi^-$ spherical angles $\\theta_{\\pi^-}$ and $\\varphi_{\\pi^-}$. Plane C represents the electron \nscattering plane. Plane A is defined by the 3-momenta of the initial state proton and the final state $\\pi^-$. \nPanel (b) shows the angle $\\alpha_{[\\pi^-p][\\pi^+p']}$ between the two defined hadronic planes A and B or the \nplane B rotation angle around the axis aligned along the 3-momentum of the final $\\pi^-$. Plane B is defined by \nthe 3-momenta of the final state $\\pi^+$ and $p'$. The unit vectors $\\overline{\\gamma}$ and $\\overline{\\beta}$ are \nnormal to the $\\pi^-$ three-momentum in the planes A and B, respectively.} \n\\label{kinematic}\n\\end{center}\n\\end{figure}\n\n\\subsection{Differential Cross Sections and Kinematic Variables}\n\\label{kinxsect}\n\nAt a given invariant mass $W$ and photon virtuality $Q^2$, the $\\gamma_v p \\to \\pi^+\\pi^-p$ reaction\ncan be fully described as a five-fold differential cross section $d^5\\sigma\/d^5\\tau$, where $d^5\\tau$ is \nthe differential of the five independent variables in the center-of-mass (CM) frame of the final\n$\\pi^+ \\pi^- p$ state. There are many possible choices~\\cite{Byc} of the five independent variables. After \ndefining $M_{\\pi^+p}$, $M_{\\pi^-p}$, and $M_{\\pi^+\\pi^-}$ as invariant mass variables of the three possible \ntwo-particle pairs in the $\\pi^+\\pi^-p$ system, we adopt here the following assignment for the computation of the\nfive-fold differential cross section:\\\\\n \\\\\n$d^5\\tau=dM_{\\pi^+p}dM_{\\pi^+\\pi^-}d\\Omega_{\\pi^-}d\\alpha_{[\\pi^-p][\\pi^+p']}$, where $\\Omega_{\\pi^-}$ \n($\\theta_{\\pi^-}$, $\\varphi_{\\pi^-}$) are the final state $\\pi^-$ spherical angles with respect to the \ndirection of the virtual photon with the $\\varphi_{\\pi^-}$ defined as the angle between the hadronic plane A and \nthe electron scattering plane C, see Fig~\\ref{kinematic} (a), and $\\alpha_{[\\pi^-p][\\pi^+p']}$ is the rotation \nangle of the plane B defined by the momenta of the final state $\\pi^+p'$ around the axis defined by the final \nstate $\\pi^-$ momentum, see Fig.~\\ref{kinematic} (b).\\\\ \n \\\\\nAll frame-dependent variables are defined in the final hadron CM frame. The relations between the momenta of the \nfinal-state hadrons and the aforementioned five variables can be found in Ref.~\\cite{Fe09}.\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=11.5cm]{pictures\/1diff065151description.eps}\n\\includegraphics[width=11.5cm]{pictures\/1_diff_095161description.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) Description of the CLAS $ep \\to e'p'\\pi^+\\pi^-$ data~\\cite{Ri03} within the \nframework of the JM model~\\cite{Mo09,Mo12} after implementation of the phases for the $2\\pi$ direct production \nmechanisms discussed in Section~\\ref{pipipmech} at $W$ = 1.51~GeV, $Q^2$=0.65~GeV$^2$ (top) and at $W$ = 1.61~GeV, \n$Q^2$=0.95~GeV$^2$ (bottom). Full model results are shown by the black thick solid lines together with the \ncontributions from the isobar channels $\\pi^-\\Delta^{++}$ (thin red lines), $\\pi^+ \\Delta^0$ (blue dash-dotted \nlines), $\\pi^+ D^0_{13}(1520)$ (black dotted lines), and the $2\\pi$ direct production mechanisms (magenta \ndash-dotted lines). The contributions from other mechanisms described in Section~\\ref{pipipmech} are comparable \nwith the data uncertainties and are not shown in the plot.} \n\\label{isochan}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=14.0cm]{pictures\/jmdiag.eps}\n\\vspace{-0.1cm}\n\\caption{The $ep\\to e'p' \\pi^+\\pi^-$ electroproduction mechanisms incorporated into the JM model\n\\cite{Mo09,Mo12}: a) full amplitude; b) $\\pi^- \\Delta^{++}$ and $\\pi^-\\Delta^{++}(1600)\\frac{3}{2}^+$ isobar \nchannels; c) $\\pi^+ \\Delta^0$, $\\pi^+ N^0(1520)\\frac{3}{2}^-$, and $\\pi^+ N^0(1685)\\frac{5}{2}^+$ isobar \nchannels; d) $\\rho p$ meson-baryon channel; e) the 2$\\pi$ direct electroproduction mechanisms.}\n\\label{jmmech}\n\\end{center}\n\\end{figure*}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|} \\hline\n & 0.5-0.8 \\\\\n & 0.65 central value \\\\ \\cline{2-2}\n & 0.8-1.1 \\\\\n$Q^2$ Intervals, GeV$^2$ & 0.95 central value \\\\ \\cline{2-2}\n & 1.1-1.5 \\\\\n & 1.3 central value \\\\ \\hline\n$W$ Intervals, GeV & 1.41-1.71 \\\\\ncovered in each $Q^2$ bin & 13 bins \\\\ \\hline\n\\end{tabular}\n\\caption{Kinematic area covered in the fit of the CLAS $\\pi^+\\pi^-p$ electroproduction cross \nsections~\\cite{Ri03} for the extraction of the resonance parameters.}\n\\label{wq2bins} \n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nOne-fold differential & Interval & Number of \\\\\ncross section & Covered & Bins \\\\ \\hline\n$\\frac{d\\sigma}{dM_{\\pi^+p}}$ ($\\mu$b\/GeV) & $M_{\\pi^+p_{min}}$-$M_{\\pi^+p_{max}}$ & 10 \\\\\n$\\frac{d\\sigma}{dM_{\\pi^+\\pi^-}}$ ($\\mu$b\/GeV) & $M_{\\pi^+\\pi^-_{min}}$-$M_{\\pi^+\\pi^-_{max}}$ & 10 \\\\\n$\\frac{d\\sigma}{dM_{\\pi^-p}}$ ($\\mu$b\/GeV) & $M_{\\pi^-p_{min}}$-$M_{\\pi^-p_{max}}$ & 10 \\\\\n$\\frac{d\\sigma}{d(-cos(\\theta_{\\pi^-}))}$ ($\\mu$b\/rad) & 0-180$^\\circ$ & 10 \\\\\n$\\frac{d\\sigma}{d(-cos(\\theta_{\\pi^+}))}$ ($\\mu$b\/rad) & 0-180$^\\circ$ & 10 \\\\\n$\\frac{d\\sigma}{d(-cos(\\theta_{p'}))}$ ($\\mu$b\/rad) & 0-180$^\\circ$ & 10 \\\\\n$d\\sigma\/d\\alpha_{[\\pi^-p][\\pi^+p']}$ ($\\mu$b\/rad) & 0-360$^\\circ$ & 5\\\\\n$d\\sigma\/d\\alpha_{[\\pi^+p][\\pi^-p']}$ ($\\mu$b\/rad) & 0-360$^\\circ$ & 5\\\\\n$d\\sigma\/d\\alpha_{[\\pi^+\\pi^-][p p']}$ ($\\mu$b\/rad)& 0-360$^\\circ$ & 5\\\\ \\hline\n\\end{tabular}\n\\caption{List of the fit one-fold differential cross sections measured with CLAS~\\cite{Ri03} and the\nbinning over the kinematic variables. $M_{min_{i,j}}=M_{i}+M_{j}$, $M_{max_{i,j}}=W-M_{k}$, where \n$M_{i,j}$ and $M_k$ are the invariant masses of the final state hadron pair $i,j$, and the mass of the \nthird final state hadron $k$, respectively.}\n\\label{1diffbins}\n\\end{center}\n\\end{table}\n\nThe $\\pi^+\\pi^-p$ electroproduction data have been collected in the bins of a seven-dimensional space. As \nmentioned above, five variables are needed to fully describe the final hadron kinematics, while to describe \nthe initial state kinematics, two others variables, $W$ and $Q^2$, are required. The huge number of seven \ndimensional bins over the reaction phase space ($\\approx$ 500,000 bins) does not allow us to use the\ncorrelated multi-fold differential cross sections in the analysis of the data, where the statistics decrease \ndrastically with increasing $Q^2$. More than half of the seven-dimensional phase-space bins of the final state\nhadrons are not populated due to statistical limitations. This is a serious obstacle for any analysis method \nthat employs information on the behavior of multi-fold differential cross sections. We therefore use the \nfollowing one-fold differential cross sections in each bin of $W$ and $Q^2$ covered by the data:\n\\begin{itemize}\n\\item invariant mass distributions for the three pairs of the final state particles \n$d\\sigma\/dM_{\\pi^+\\pi^-} $, $d\\sigma\/dM_{\\pi^+ p}$, and \n$d\\sigma\/dM_{\\pi^- p}$;\n\\item angular distributions for the spherical angles of the three final state particles \n$d\\sigma\/d(-\\cos\\theta_{\\pi^-})$, $d\\sigma\/d(-\\cos\\theta_{\\pi^+})$, and $d\\sigma\/d(-\\cos\\theta_{p'})$ \nin the CM frame;\n\\item angular distributions for the three $\\alpha$-angles determined in the CM frame: \n$d\\sigma\/d\\alpha_{[\\pi^-p][\\pi^+p']}$, $d\\sigma\/d\\alpha_{[\\pi^+p][\\pi^-p']}$, and\n$d\\sigma\/d\\alpha_{[\\pi^+\\pi^-][p p']}$. The $d\\sigma\/d\\alpha_{[\\pi^+p][\\pi^-p']}$ and\n$d\\sigma\/d\\alpha_{[\\pi^+\\pi^-][p p']}$ differential cross sections are defined analogously to \n$d\\sigma\/d\\alpha_{[\\pi^-p][\\pi^+p']}$ describe above. More details on these observables can be found in \nRefs.~\\cite{Fe09,Mo12}. \n\\end{itemize} \nThe one-fold differential cross sections were obtained by integrating the five-fold differential cross \nsections over the other four relevant kinematic variables of $d^5\\tau$. However, the angular \ndistributions for the spherical angles of the final state $\\pi^+$ and $p$, as well as for the rotation angles \naround the axes along the momenta of these final state hadrons, cannot be obtained with the aforementioned \n$d^5\\tau$, since this differential does not depend on these variables. Two other sets of $d^5\\tau^{'}$ and \n$d^5\\tau^{''}$ differentials are required, which contain $d\\Omega_{\\pi^+}d\\alpha_{[\\pi^+p][\\pi^-p']}$ and \n$d\\Omega_{p'}d\\alpha_{[pp'][\\pi^+\\pi^-]}$, respectively, as described in Refs.~\\cite{Fe09,Mo09}. The five-fold \ndifferential cross sections evaluated over the other two $d^5\\tau^{'}$ and $d^5\\tau^{''}$ differentials were \ncomputed from the five-fold differential cross section over the $d^5\\tau$ differential detailed above by means \nof cross section interpolation. For each kinematic point in the five-dimensional phase space determined by the \nvariables of the other two $d^5\\tau^{'}$ and $d^5\\tau^{''}$ differentials, the four-momenta of the three \nfinal state hadrons were computed, and from these values the five variables of the $d^5\\tau$ were determined. \nThe $d^5\\sigma\/d^5\\tau$ cross sections were interpolated into this five-dimensional kinematic point.\nAll details related to the evaluation of the nine one-fold differential cross sections from the CLAS data \non $\\pi^+\\pi^-p$ electroproduction off the proton can be found in Ref.~\\cite{Fe09}. \n\nAn example of the data analyzed in two particular bins of $W$ and $Q^2$ is shown in Fig.~\\ref{isochan}. The \nkinematic area covered in our analysis and the data binning are summarized in Tables~\\ref{wq2bins}\nand \\ref{1diffbins}.\n\n\\subsection{The Reaction Model for Extraction of the Resonance Parameters}\n\\label{pipipmech}\n\nA phenomenological analysis of the CLAS $\\pi^+\\pi^- p$ electroproduction data~\\cite{Ri03} was carried out for \n$W < 1.82$~GeV and at photon virtualities 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$. This work allows us to establish \nall essential mechanisms that contribute to the measured cross sections. The peaks in the invariant mass \ndistributions provide evidence for the presence of the channels arising from \n$\\gamma_v p \\to Meson+Baryon \\to \\pi^+\\pi^- p$ having an unstable baryon or meson in the intermediate state. \nPronounced dependencies in the angular distributions further allow us to establish the relevant $t$-, $u$-, and \n$s$-channel exchanges. The mechanisms without pronounced kinematic dependencies are identified through \nexamination of various differential cross sections, with their presence emerging from the correlation patterns. \nThe phenomenological reaction model JM~\\cite{Mo09,Mo12,Ri00,Az05} was developed with the primary objective to \ndetermine the $\\gamma_vpN^*$ electrocouplings and the corresponding $\\pi \\Delta$ and $\\rho N$ partial hadronic \ndecay widths from fitting all measured observables in the $\\pi^+\\pi^- p$ electroproduction channel. The \nrelationships between the $\\pi^+\\pi^-p$ electroproduction cross sections and the three-body production \namplitudes employed in the JM model are given in Appendix D of Ref.~\\cite{Mo09}. \n\nThe amplitudes of the $\\gamma_v p \\to \\pi^+\\pi^- p$ reaction are described in the JM model as a superposition \nof the $\\pi^-\\Delta^{++}$, $\\pi^+\\Delta^0$, $\\rho p$, $\\pi^+ D_{13}^0(1520)$, $\\pi^+ F_{15}^0(1685)$, and\n$\\pi^- \\Delta^{++}(1600)$ sub-channels with subsequent decays of the unstable hadrons to the final $\\pi^+\\pi^-p$ \nstate, and additional direct 2$\\pi$ production mechanisms, where the final $\\pi^+\\pi^- p$ state comes about \nwithout going through the intermediate process of forming unstable hadron states. The mechanisms incorporated \ninto the JM model are shown in Fig.~\\ref{jmmech}. \n\nThe JM model incorporates contributions from all well-established $N^*$ states with listed in Table~\\ref{nstlist} \nconsidering the resonant contributions only to $\\pi \\Delta$ and $\\rho p$ sub-channels. We also have included the \n$3\/2^+(1720)$ candidate state, whose existence is suggested in the analysis~\\cite{Ri03} of the CLAS \n$\\pi^+\\pi^- p$ electroproduction data. In the versions of the JM model beginning in 2012~\\cite{Mo12}, the \nresonant amplitudes are described by a unitarized Breit-Wigner ansatz as proposed in Ref.~\\cite{Ait72}; the\nmodel was modified to make it fully consistent with a relativistic Breit-Wigner parameterization of each individual \n$N^*$ state contributions in the JM model ~\\cite{Ri00} that also accounts for the energy-dependent resonance hadronic \ndecay widths. A unitarized Breit-Wigner ansatz accounts for the transition between the same and different resonances \nin the dressed resonant propagator, which makes the resonant amplitudes consistent with restrictions imposed by a \ngeneral unitarity condition~\\cite{Ait78,Lee08}. Quantum number conservation in the strong interaction allows \nfor transitions between the pairs of $N^*$ states $N(1520)3\/2^- \\leftrightarrow N(1700)3\/2^-$, \n$N(1535)1\/2^- \\leftrightarrow N(1650)1\/2^-$, and $3\/2^+(1720) \\leftrightarrow N(1720)3\/2^+$ incorporated into the \nJM model. We found that the use of the unitarized Breit-Wigner ansatz has a minor influence on the $\\gamma_vNN^*$ \nelectrocouplings, but it may substantially affect the $N^*$ hadronic decay widths determined from fits to the CLAS \ndata. \n\nThe non-resonant contributions to the $\\pi \\Delta$ sub-channels incorporate a minimal set of current-conserving \nBorn terms~\\cite{Mo09,Ri00}. They consist of $t$-channel pion exchange, $s$-channel nucleon exchange, $u$-channel \n$\\Delta$ exchange, and contact terms. Non-resonant Born terms were reggeized and current conservation was \npreserved as proposed in Refs.~\\cite{Gu97hy,Gu97by}. The initial- and final-state interactions in $\\pi \\Delta$ \nelectroproduction are treated in an absorptive approximation, with the absorptive coefficients estimated from the \ndata from $\\pi N$ scattering~\\cite{Ri00}. Non-resonant contributions to the $\\pi \\Delta$ sub-channels further \ninclude additional contact terms that have different Lorentz-invariant structures with respect to the contact \nterms in the sets of Born terms. These extra contact terms effectively account for non-resonant processes in the \n$\\pi \\Delta$ sub-channels beyond the Born terms, as well as for the final-state interaction effects that are \nbeyond those taken into account by the absorptive approximation. Parameterizations of the extra contact terms in \nthe $\\pi \\Delta$ sub-channels are given in Ref.~\\cite{Mo09}. A phenomenological treatment of the initial and \nfinal state interactions~\\cite{Ri00} along with the extra-contact-terms~\\cite{Mo09} in the $\\pi \\Delta$ sub-channels\ndetermined from fits to the data are important in order to account for the constraints imposed by unitarity on the \nnon-resonant amplitudes of these sub-channels.\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=7.0cm]{pictures\/alphap065156.eps}\n\\includegraphics[width=7.0cm]{pictures\/alphap065156ph.eps}\\\\\n\\includegraphics[width=7.0cm]{pictures\/alphap095154.eps}\n\\includegraphics[width=7.0cm]{pictures\/alphap095154ph.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) Manifestation of the direct 2$\\pi$ electroproduction mechanism relative phases in the \nCLAS data~\\cite{Ri03} on the angular distributions over the angle $\\alpha_{[\\pi^+\\pi^-][p p']}$. The JM model \nresults with the relative phases equal to zero are shown in the left column, while the computed cross sections with \nphases based on fits to the CLAS data~\\cite{Ri03} are shown in the right column. The sample plots shown are for\n$W$=1.56~GeV, $Q^2$=0.65~GeV$^2$ (top row) and $W$=1.54~GeV, $Q^2$=0.95~GeV$^2$ (bottom row). The curves for\nthe different contributing meson-baryon channels are the same as those shown in Fig.~\\ref{isochan}.}\n\\label{pidirphase}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=5.5cm]{pictures\/wdep141181065.eps}\n\\includegraphics[width=5.5cm]{pictures\/wdep141181095.eps}\n\\includegraphics[width=5.5cm]{pictures\/wdep141181130.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) Description of the fully integrated $\\pi^+\\pi^-p$ electroproduction cross sections \nachieved within the framework of the updated JM model discussed in Section~\\ref{pipipmech} together with the \ncross sections for the various contributing mechanisms: full cross section (black solid), $\\pi^-\\Delta^{++}$ \n(red thin solid), $\\rho p$ (green thin solid), $\\pi^+\\Delta^0$ (blue thin dashed), $\\pi^+ N^0(1520)3\/2^-$ (black \ndotted), direct 2$\\pi$ mechanisms (magenta thin dot-dashed), and $\\pi^+ N^0(1685)5\/2^+$ (red thin dashed). The data \nfits were carried out at $W < 1.71$~GeV.} \n\\label{integsec}\n\\end{center}\n\\end{figure*}\n\nThe contributions from the $\\rho p$ meson-baryon channel are quite small in the range of $W < 1.71$~GeV where \nthe resonance parameters presented in this paper are determined. However, reliable accounting of this channel \nis important for ascertaining the electrocouplings and the corresponding hadronic parameters of the resonances \nin the aforementioned range of $W$. Non-resonant amplitudes in the $\\rho p$ sub-channel are described within the \nframework of a diffractive approximation, which also takes into account the effects caused by $\\rho$-line \nshrinkage~\\cite{Bu07}. The latter effects play a significant role in near-threshold and sub-threshold \n$\\rho$-meson production for $W < 1.71$~GeV. The previous analyses of the CLAS data ~\\cite{Fe09,Ri03} have \nrevealed the presence of the $\\rho p$ sub-channel contributions for $W > 1.5$~GeV.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|} \\hline\n$N^*$ states & Mass,\\ & Total & BF & BF & $N^*$ electro- \\\\\nincorporated & GeV & decay & ${\\pi \\Delta}$, \\% & ${\\rho p}$, \\% & coupling \\\\\ninto the & & width & & & variation \\\\\ndata fit & & $\\Gamma_{tot}$, & & & in the fit \\\\\n & & GeV & & & \\\\ \\hline\n$N(1440)1\/2^+$ & var & var & var & var & \\cite{Az051} var \\\\\n$N(1520)3\/2^-$ & var & var & var & var & \\cite{Az051} var \\\\\n$N(1535)1\/2^-$ & var & var & var & var & \\cite{Az09} fix \\\\\n$\\Delta(1620)1\/2^-$ & var & var &var & var & \\cite{Az05,Mo05nstar} var\\\\\n$N(1650)1\/2^-$ & var & var & var & var & \\cite{Bu03} var \\\\\n$N(1680)5\/2^+$ & 1.68 & 0.12 & 12 & 5.5 & \\cite{Az05,Mo05nstar} var \\\\\n$N(1700)3\/2^-$ & 1.74 & 0.19 & 53 & 45 & \\cite{Az05,Mo05nstar} fix \\\\\n$\\Delta(1700)3\/2^-$ & 1.70 & 0.26 & 89 & 2 & \\cite{Az05,Mo05nstar} fix \\\\\n$3\/2^+(1720)$ & 1.72 & 0.09 & 55 & 7 & \\cite{Mo14} fix \\\\\n$N(1720)3\/2^+$ & 1.73 & 0.11 & 47 & 36 & \\cite{Mo14} fix \\\\ \\hline\n\\end{tabular}\n\\caption{List of resonances invoked in the $\\pi^+\\pi^-p$ fit and their parameters: total decay widths \n$\\Gamma_{tot}$ and branching fractions (BF) to the $\\pi \\Delta$ and $\\rho N$ final states. The quoted values \nfor the hadronic parameters are taken from earlier fits~\\cite{Az05,Mo14} to the CLAS $\\pi^+\\pi^-p$ data\n\\cite{Ri03}. The quantities labeled as $var$ correspond to the variable parameters fit to the CLAS \n$\\pi^+\\pi^-p$ data~\\cite{Ri03} within the framework of the current JM model version. Start values for \nthe resonance electrocouplings are taken from the references listed in the last column and extrapolated to \nthe $Q^2$ area covered by the CLAS experiment~\\cite{Ri03}. $3\/2^+(1720)$ represents the candidate $N^*$ state \nwith the signal reported in a previous analysis of CLAS data~\\cite{Ri03}.}\n\\label{nstlist} \n\\end{center}\n\\end{table}\n\nThe $\\pi^+ N^0(1520)3\/2^-$, $\\pi^+ N^0(1685)5\/2^+$, and $\\pi^- \\Delta(1600)3\/2^+$ sub-channels are described \nin the JM model by non-resonant contributions only. The amplitudes of the $\\pi^+ N^0(1520)3\/2^-$ sub-channel \nwere derived from the non-resonant Born terms in the $\\pi \\Delta$ sub-channels by implementing an additional \n$\\gamma_5$-matrix that accounts for the opposite parities of $\\Delta(1232)3\/2^+$ and $ N(1520)3\/2^-$~\\cite{Mo05nstar}. \nThe magnitudes of the $\\pi^+ N^0(1520)3\/2^-$ production amplitudes were independently fit to the data for each \nbin in $W$ and $Q^2$. The contributions from the $\\pi^+ N^0(1520)3\/2^-$ sub-channel should be taken into account \nfor $W > 1.5$~GeV.\n\nThe $\\pi^+ N^0(1685)5\/2^+$ and $\\pi^- \\Delta^{++}(1600)3\/2^+$ sub-channel contributions are seen in the data\n\\cite{Ri03} at $W > 1.6$~GeV. These contributions are almost negligible at smaller $W$. The effective contact \nterms were employed in the JM model for parameterization of these sub-channel amplitudes~\\cite{Mo05nstar}. The\nmagnitudes of the $\\pi^+ N^0(1685)5\/2^+$ and $\\pi^- \\Delta^{++}(1600)3\/2^+$ sub-channel amplitudes were fit to \nthe data for each bin in $W$ and $Q^2$.\n\nIn general, unitarity requires the presence of so-called $2\\pi$ direct production mechanisms in the $\\pi^+\\pi^- p$ \nelectroproduction amplitudes, where the final $\\pi^+\\pi^- p$ state is created without going through the intermediate \nstep of forming unstable hadron states~\\cite{Ait78}. These $2\\pi$ direct production processes \nare beyond the aforementioned contributions from the two-body sub-channels. 2$\\pi$ direct production amplitudes \nwere established for the first time in the analysis of the CLAS $\\pi^+\\pi^-p$ electroproduction data~\\cite{Mo09}. \nThey are described in the JM model by a sequence of two exchanges in the $t$ and\/or $u$ channels by unspecified \nparticles that may come from two Regge trajectories. The amplitudes of the $2\\pi$ direct production mechanisms are \nparameterized by a Lorentz-invariant contraction between spin-tensors of the initial and final-state particles, \nwhile two exponential propagators describe the above-mentioned exchanges by unspecified particles. The magnitudes \nof these amplitudes are fit to the data for each bin in $W$ and $Q^2$. The contributions from the $2\\pi$ direct \nproduction mechanisms are maximal and substantial ($\\approx$ 30\\%) for $W < 1.5$~GeV and they decrease with \nincreasing $W$, contributing less than 10\\% for $W > 1.7$~GeV. However, even in this kinematic regime, $2\\pi$ direct \n production mechanisms can be seen in the $\\pi^+\\pi^-p$ electroproduction cross sections due to an \ninterference of the amplitudes with the two-body sub-channels. Explicit expressions for the above-mentioned \n2$\\pi$ direct production amplitudes can be found in Appendices A-C of Ref.~\\cite{Mo09}. We are planning to \nexplore the possibility to replace this phenomenological ansatz by the $B_{5}$ Veneziano model that was employed \nsuccessfully in the studies of charged double-kaon photoproduction \\cite{Jpac}. \n\nThe studies of the final state hadron angular distributions over $\\alpha_{i}~(i=[\\pi^-p][\\pi^+p'],[\\pi^+p][\\pi^-p'], \n[\\pi{+}\\pi^-][p p']$) conclusively demonstrated the need to implement the relative phases for all $2\\pi$ direct \nproduction mechanisms included in the JM model. Figure~\\ref{pidirphase} shows the comparison of the measured data\n\\cite{Ri03} to the differential cross sections $d\\sigma\/d\\alpha_{[\\pi^+\\pi^-][p p']}$ computed within the \nframework of the JM model for values of the relative phases of the $2\\pi$ direct production mechanisms fit to \nthe data and for values of these phases equal to zero. The computed cross sections, assuming zero phases for all \n$2\\pi$ direct production amplitudes, underestimate the measured $d\\sigma\/d\\alpha_{[\\pi^+\\pi^-][p p']}$ cross \nsections at $\\alpha_{[\\pi^+\\pi^-][p p']}$ around 180$^\\circ$ (left panel in Fig.~\\ref{pidirphase}). This is a \nconsequence of destructive interference of these contributions with the amplitudes of other relevant processes at \n$Q^2$=0.95 GeV$^2$ and insufficient constructive interference at $Q^2$=0.65 GeV$^2$. Fits to the data phases of \n$2\\pi$ direct production mechanisms change the interference pattern and allow us to \nimprove the description of the $d\\sigma\/d\\alpha_{[\\pi^+\\pi^-][p p']}$ angular distributions at $W > 1.48$~GeV in \nall three $Q^2$-bins under study, while retaining the same or even better quality of description of the other \neight one-fold differential cross sections. Examples of the achieved improvements implementing the relative \nphases of the $2\\pi$ direct production mechanisms are shown in the right column of Fig.~\\ref{pidirphase}. \n\n\nThe JM model provides a reasonable description of the nine $\\pi^+\\pi^- p$ one-fold differential cross sections for\n$W < 1.8$~GeV and $Q^2 < 1.5$~GeV$^2$. As a typical example, the nine one-fold differential cross \nsections and their corresponding descriptions for $W = 1.51$~GeV and $Q^2$ = 0.65~GeV$^2$ and for $W = 1.61$~GeV \nand $Q^2$ = 0.95~GeV$^2$ are shown in Fig.~\\ref{isochan}, together with the contributions from each of the\nindividual mechanisms incorporated into the JM model. Any contributing mechanism will be manifested by\nsubstantially different shapes in the cross sections for the observables, all of which are highly correlated \nthrough the underlying reaction dynamics. The simultaneous description of all the nine one-fold differential \ncross sections allows for identifying the essential mechanisms contributing to the $\\pi^+\\pi^-p$ \nelectroproduction off the proton. \n\nDescriptions of the fully integrated $\\pi^+\\pi^-p$ electroproduction cross sections are shown in \nFig.~\\ref{integsec} together with the contributions from the meson-baryon mechanisms of the JM model. The \nmajor part of the $\\pi^+\\pi^-p$ electroproduction off the proton at $W < 1.6$~GeV is due to contributions \nfrom the two $\\pi \\Delta$ isobar channels, $\\pi^- \\Delta^{++}$ and $\\pi^+ \\Delta^0$. The $\\Delta^{++}$(1232) \nresonance is clearly seen in all $\\pi^+ p$ mass distributions for $W > 1.4$~GeV, while contributions from the \n$\\pi^+ \\Delta^0$ isobar channel are needed to better describe the data in the low mass regions of the $\\pi^- p$\n mass distributions. The strength of the $\\pi^- \\Delta^{++}$ isobar channel observed in the data~\\cite{Ri03,Fe09} \nis approximately nine times larger than that of $\\pi^+ \\Delta^0$~\\cite{Mo09} due to isospin invariance. The CLAS \ndata~\\cite{Ri03} demonstrated sub-leading but still important contributions from the $\\pi^+D_{13}^0(1520)$ \nmeson-baryon channel. This contribution is needed for a description of the $\\pi^+$ CM-angular \ndistributions at forward angles and allows us to better describe the $\\pi^-p$ invariant mass distributions as $W$\nincreases (see Fig.~\\ref{isochan}). The contributions from $2\\pi$ direct production mechanisms shown in \nFig.~\\ref{integsec} were obtained with the phases of these mechanisms derived from the CLAS data~\\cite{Ri03}. \nThese contributions are substantial, up to 30\\% at $W < 1.5$~GeV. They decrease sharply as $W$ increases. Direct \n$2\\pi$ production mechanisms become minor at $W > 1.7$~GeV, but they still should be taken into account because \nof their interference with larger amplitudes of other contributing mechanisms. $2\\pi$ direct production mechanisms \nare kinematically allowed in the entire range of $W$, while meson-baryon channels with final mesons\/baryons \nheavier than the pion\/nucleon can be open at $W$ larger than the respective threshold values. This may explain the \nbiggest contributions from $2\\pi$ direct production mechanisms at small $W < 1.5$~GeV. The $\\pi^+\\pi^-p$ final \nstate interaction with all open meson-baryon channels may be a plausible explanation for the sharp decrease of \nthese mechanism contributions at $W > 1.5$~GeV, see Fig~\\ref{integsec}. A quantitative description of this \npronounced effect in the behavior of the $2\\pi$ direct production mechanisms represents a challenging task for the \nglobal multi-channel analysis of exclusive meson electroproduction within the framework of the coupled channel \napproaches under development by the Argonne-Osaka group~\\cite{Lee10,Lee13a}.\n\nAccounting for the restrictions imposed by unitarity on the $\\pi^+\\pi^-p$ electroproduction amplitudes \nrepresents an important requirement for reliable extraction of the resonance parameters. However, a rigorous\nimplementation of unitarity for this three-body exclusive channel is still far from the reach of reaction theory \nand is outside the scope of this paper. To our knowledge, none of the available models is capable of providing \nfully unitarized amplitudes to fit the $\\pi^+\\pi^-p$ data to determine the electroproduction amplitudes. A very \npromising step in this direction was made by the Argonne-Osaka group~\\cite{Lee13,Kam09}. Nevertheless, their \napproach is still under development. In this paper we employ a strategy that allows us to account phenomenologically \nfor unitarity constraints in extracting the resonance parameters. As was mentioned above, we incorporated several \nfeatures in the JM model in order to account for the unitarity restrictions on the resonant\/non-resonant \n$\\pi^+\\pi^-p$ electroproduction amplitudes: a) the unitarized Breit-Wigner ansatz for the resonant amplitudes, \nb) the phenomenological treatment of the initial and final state interactions and the inclusion of the \nextra contact terms for the non-resonant amplitudes of the $\\pi \\Delta$ sub-channels, and c) direct 2$\\pi$ \nproduction mechanisms. A good description of the nine one-fold differential cross sections in the entire \nkinematic area of $W$ and $Q^2$ analyzed in this paper strongly suggests a reliable parameterization of the \nsquared $\\pi^+\\pi^-p$ electroproduction amplitudes achieved for the CLAS data~\\cite{Ri03} fit within the framework \nof the JM model updated as was described in earlier. The $\\pi^+\\pi^-p$ electroproduction amplitudes determined\nfrom a fit to the data account for the restrictions imposed by unitarity on their magnitudes at the real energy \naxis because the measured differential cross sections should be consistent with the unitarity constraints. The \nresonant contributions to the differential cross sections were obtained from these amplitudes switching off the \nnon-resonant parts. In Section~\\ref{fit} we will discuss in detail the extraction of the resonant contributions to \nthe differential cross sections. The resonant parameters were extracted from the resonant contributions to the\ndifferential cross sections employing the unitarized Breit-Wigner ansatz for the resonant amplitudes. Therefore, \nthe unitarity constraints on the resonant amplitudes were fully taken into account. The resonant parameters were \nobtained at the real energy axis at the resonant point $W=M_{N^*}$. The reliability of the resonance parameters \nobtained in this way is determined by credible isolation of the resonant contributions to the differential cross \nsections, which will be discussed in Section~\\ref{fit}. \n\n\\section{The CLAS Data Fit}\n\\label{fit}\n\nThe resonance parameters obtained in our work were determined in the simultaneous fit to the CLAS $\\pi^+\\pi^-p$ \nelectroproduction differential cross sections ~\\cite{Ri03} in the three $Q^2$-bins listed in Table~\\ref{wq2bins}.\nThe $W$-area included in the fit is limited to $W < 1.71$~GeV. Currently the resonance content for the structure \nin the $W$-dependence of the fully integrated $\\pi^+\\pi^-p$ cross sections at $W \\approx 1.7$~GeV~\\cite{Ri03} is \nstill under study~\\cite{Mo15}. For this reason the resonance parameters for the states located at $W$ above \n1.64~GeV are outside of this paper scope. \n\nIn order to provide a realistic evaluation of the resonance parameters, we abandoned the traditional least-squares \nfit, since the parameters extracted in such a fit correspond to a single presumed global minimum, while the \nexperimental data description achieved with other local minima may be equally good within the data uncertainties. \nFurthermore, the traditional evaluation of the fit-parameter uncertainties, based on the error propagation matrix, \ncannot be used for the same reason.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|} \\hline\n & Ranges covered \\\\\n & in variations of the \\\\ \nVariable \\, parameters & start parameters, \\\\\n\t\t & \\% from their values \\\\ \\hline\nMagnitude of the additional & \\\\\ncontact term amplitude in the & 40.0 \\\\\n$\\pi^- \\Delta^{++}$ sub-channel & \\\\ \\hline\nMagnitude of the additional & \\\\\ncontact term amplitude in the & 45.0 \\\\\n$\\pi^+ \\Delta^0$ sub-channel & \\\\ \\hline\nMagnitude of the & \\\\\n$\\pi^+ N^0(1520)3\/2^-$ amplitude& 45.0 \\\\ \\hline\nMagnitudes of the six & \\\\\n2$\\pi$ direct production & 20.0-30.0 \\\\\namplitudes & \\\\ \\hline\n\\end{tabular}\n\\caption{Variable parameters of the non-resonant mechanisms incorporated into the JM model~\\cite{Mo09,Mo12}. \nThe ranges in the table correspond to the 3$\\sigma$ areas around the start values of the parameters.}\n\\label{bckpar} \n\\end{center}\n\\end{table}\n\nThe special data fit procedure described in Ref.~\\cite{Mo12} was employed for extraction of the resonance \nparameters. It allows us to obtain not only the best fit, but also to establish bands of the computed cross \nsections that are compatible with the data within their uncertainties. In the fit we simultaneously vary the \nresonant and non-resonant parameters of the JM model given in Tables~\\ref{nstlist} and~\\ref{bckpar}, respectively. \nMore details on the non-resonant mechanisms listed in Table~\\ref{bckpar} can be found in Refs.~\\cite{Mo09,Mo12}. \nThese non-resonant mechanisms have an essential influence on the data description at $W < 1.71$~GeV. The values \nof the aforementioned non-resonant\/resonant parameters were evaluated under simultaneous variation of:\n\\begin{itemize}\n\\item the magnitudes of additional contact-term amplitudes in the $\\pi^- \\Delta^{++}$ and $\\pi^+ \\Delta^0$ isobar \nchannels (2 parameters per $Q^2$-bin);\n\\item the magnitudes of the $\\pi^+ N^0(1520)3\/2^-$ isobar channel (1 parameter per $Q^2$-bin);\n\\item the magnitudes of all direct 2$\\pi$ production amplitudes (9 parameters per $Q^2$-bin including the phases \ndescribed in the Section~\\ref{pipipmech});\n\\item and the variable resonant parameters listed in Table~\\ref{nstlist}. The CLAS $\\pi^+\\pi^-p$ data~\\cite{Ri03} at \n$W < 1.71$~GeV are mostly sensitive to the variable electrocouplings of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, \n$\\Delta(1620)1\/2^-$, and $N(1650)1\/2^-$ states (9 resonance electrocouplings per $Q^2$-bin), as well as the $\\pi \\Delta$ \nand $\\rho p$ hadronic decay widths of these four resonances and of the $N(1535)1\/2^-$ state (12 parameters that \nremain the same in the entire kinematic area covered by the fit). \n\\end{itemize}\n\nAll of the aforementioned parameters are sampled around their start values, employing unrestricted normal \ndistributions. In this way we mostly explore the range of $\\approx$ 3$\\sigma$ around the start parameter values. \nThe $W$-dependencies of the magnitudes of the amplitudes of all non-resonant contributions are determined by \nadjusting their values to reproduce the measured nine one-fold differential charged double-pion electroproduction \ncross sections~\\cite{Ri03}. We apply multiplicative factors to the magnitudes of all non-resonant amplitudes. \nThey remain the same in the entire $W$-interval covered by the fit within any $Q^2$-bin, but they depend on the\nphoton virtuality $Q^2$ and are fit to the data in each $Q^2$-bin independently. The multiplicative factors are \nvaried around unity employing normal distributions with the $\\sigma$ values in the ranges listed in \nTable~\\ref{bckpar}. In this way we retain the smooth $W$-dependencies of the non-resonant contributions\nestablished in the adjustment to the data and explore the possibility to improve the data description in the\nsimultaneous variation of the resonant and non-resonant parameters. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nResonances &$Q^2_{cent.}$=0.65 &$Q^2_{cent.}$=0.95 &$Q^2_{cent.}$=1.30 \\\\\n & GeV$^2$ & GeV$^2$ & GeV$^2$ \\\\\n\\hline\n$N(1440)1\/2^+$ & 40 & 30 & 40 \\\\\n$N(1520)3\/2^-$ & 20 & 20 & 30 \\\\\n$\\Delta(1620)1\/2^-$ & 40 & 40 & 40 \\\\\n$N(1650)1\/2^-$ & 40 & 40 & 50 \\\\ \\hline\n\\end{tabular}\n\\caption{$\\sigma$ parameters employed in the variation of the resonance electrocouplings in \\% of their start \nvalues. The $\\sigma$ parameters listed for the $\\Delta(1620)1\/2^-$ were applied as a variation of the \n$S_{1\/2}(Q^2)$ electrocouplings only. The variation of the $A_{1\/2}(Q^2)$ electrocouplings of this state is \ndescribed in Section~\\ref{nstarelectrocoupl}.}\n\\label{varelcoupl} \n\\end{center}\n\\end{table}\n\nIn this fit we also vary the $\\gamma_vpN^*$ electrocouplings and the $\\pi \\Delta$ and $\\rho p$ hadronic \npartial decay widths of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and $\\Delta(1620)1\/2^-$ resonances around their \nstart values. The start values of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ electrocouplings were determined by \ninterpolating the results from the analyses~\\cite{Az09,Mo12} of the CLAS data on $N\\pi$ and $\\pi^+\\pi^-p$ \nelectroproduction off the proton in the range 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$. The electrocouplings of the \n$N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances were varied employing normal distributions with the $\\sigma$ \nparameters listed in Table~\\ref{varelcoupl} in terms of \\% of their start values. There were no restrictions on \nthe minimum or maximum trial electrocoupling values, allowing us to mostly explore the area of $\\approx$ 3$\\sigma$ \naround their start values.\n\nThe $\\pi^+\\pi^-p$ electroproduction channel also has some sensitivity to the $N(1535)1\/2^-$ state, which couples \ndominantly to the $N\\pi$ and $N\\eta$ final states. The $N(1535)1\/2^-$ electrocouplings were taken from the CLAS \nanalysis of $N \\pi$ electroproduction~\\cite{Az09} and varied strictly inside the uncertainties reported in that\npaper. \n\nThe start values of the $\\Delta(1620)1\/2^-$ and $N(1650)1\/2^-$ electrocouplings were taken from a preliminary \nanalysis~\\cite{Mo14}. In this study the resonance electrocouplings were adjusted to describe the nine \n$\\pi^+\\pi^-p$ one-fold differential cross sections~\\cite{Ri03} in the $W$-interval from 1.41 to 1.80~GeV and at \n0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$. However, the results~\\cite{Mo14} do not allow us to draw unambiguous \nconclusions regarding the resonant content of the structure at $W$ $\\approx$ 1.7~GeV. Therefore, we are using the \nresults of this analysis as an estimate for the resonance electrocoupling start points to fit the charged \ndouble-pion electroproduction data~\\cite{Ri03} for $W < 1.71$~GeV and 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$.\n\nSince the resonance content of the structure at $W \\approx$ 1.7~GeV is still under study, we present in this paper \nthe fit results for the resonances with masses less than 1.64 GeV. In the extraction of these resonance parameters \nwe also account for the contributions of the tails from several excited proton states in the third resonance region, \n$N(1685)5\/2^+$, $N(1720)3\/2^+$, and $\\Delta(1700)3\/2^-$, with their start electrocoupling values taken \nfrom the analyses of Refs.~\\cite{Mo14,Mo12a} and varied within 15\\% of their parameters. The \n$N(1685)5\/2^+$ state decays mostly to the $N\\pi$ final states. The electrocouplings of this state determined in \nthe analyses of $\\pi^+\\pi^-p$ electroproduction~\\cite{Mo14,Mo12a} are consistent with the results of independent \nanalysis of $N\\pi$ electroproduction~\\cite{Tia11}. This suggests a reasonable evaluation of the aforementioned \nthird resonance region state electrocouplings in the analyses~\\cite{Mo14,Mo12a} of the $\\pi^+\\pi^-p$ electroproduction \ndata as the start values for extraction of the resonance parameters for the states with masses less than 1.65~GeV. \nThe contributions from the tails of the $N(1675)5\/2^-$, $N(1700)1\/2^+$, and $N(1700)3\/2^-$ resonances were found \nto be negligible for $W < 1.64$~GeV.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\n$N^*$ states & Mass, & Total decay width, $\\Gamma_{tot}$, \\\\\n & MeV & MeV \\\\ \\hline\n$N(1440)1\/2^+$ & 1430-1480 & 200-450 \\\\\n$N(1520)3\/2^-$ & 1515-1530 & 100-150 \\\\\n$N(1535)1\/2^-$ & 1510-1560 & 100-200 \\\\\n$\\Delta(1620)1\/2^-$ & 1600-1660 & 130-160 \\\\\n$N(1650)1\/2^-$ & 1640-1670 & 140-190 \\\\ \\hline\n\\end{tabular}\n\\caption{The ranges of the resonance masses and total $N^*$ hadronic decay widths employed to constrain \nthe variation of their partial hadronic decay widths to the $\\pi \\Delta$ and $\\rho N$ final states in the \nfit of the CLAS $\\pi^+\\pi^-p$ electroproduction data~\\cite{Ri03}.}\n\\label{hadrrange} \n\\end{center}\n\\end{table}\n\nThe $\\pi \\Delta$ and $\\rho p$ hadronic decay widths of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and $N(1535)1\/2^-$ \nresonances were varied around their start values taken from previous analyses of the CLAS double-pion \nelectroproduction data~\\cite{Mo12}. For the $\\Delta(1620)1\/2^-$ state, the start values of these parameters \nwere computed as the products of the $N^*$ total decay widths from Ref.~\\cite{Rpp12} and the branching \nfractions for decays to the $\\pi \\Delta$ and $\\rho N$ final states were taken from analyses of $\\pi N \\to \\pi\\pi N$ \nhadroproduction~\\cite{Man92}. The ranges for the variations of the $\\pi \\Delta$ and $\\rho p$ hadronic decay \nwidths were restricted by the total $N^*$ decay widths and their uncertainties shown in Table~\\ref{hadrrange}. \nThe total $N^*$ decay widths were obtained by summing the partial widths over all decay channels. The partial \nhadronic decay widths to all final states other than $\\pi \\Delta$ and $\\rho p$ were computed as the products of \nRPP~\\cite{Rpp12} values of the $N^*$ total decay widths and branching fractions for decays to particular hadronic \nfinal states, which were taken from the analysis of Ref.~\\cite{Man92}. We varied the $\\pi \\Delta$ and $\\rho p$ \nhadronic decay widths of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, $N(1535)1\/2^-$, $\\Delta(1620)1\/2^-$, and \n$N(1650)1\/2^-$ resonances simultaneously with their masses, keeping the hadronic $N^*$ parameters independent \nof $Q^2$. The $\\pi \\Delta$ and $\\rho p$ hadronic decay widths of all other resonances obtained in the analyses of \nRefs.~\\cite{Mo14,Mo12a} and noted in Table~\\ref{nstlist} as ``fix\" were kept unchanged.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n$W$ intervals, & $\\chi^2\/d.p.$ intervals for selected\\\\\n GeV & computed $\\pi^+\\pi^-p$ cross sections \\\\ \\hline\n1.41-1.51 & 2.12-2.30 \\\\\n1.46-1.55 & 2.27-2.60 \\\\\n1.51-1.61 & 2.55-2.85 \\\\\n1.56-1.66 & 2.63-2.72 \\\\ \n1.61-1.71 & 2.49-2.68 \\\\ \\hline\n\\end{tabular}\n\\caption{Quality of the fit of the CLAS data~\\cite{Ri03} on $\\pi^+\\pi^-p$ electroproduction off the proton \nwithin the framework of the updated JM model described in Section~\\ref{pipipmech}. The resonance parameters \nare determined from the data fit at $W$ $<$ 1.71 GeV.}\n\\label{fitqual} \n\\end{center}\n\\end{table}\n\n\n For each trial set of the JM model resonant and non-resonant parameters we computed nine one-fold differential \n$\\pi^+\\pi^-p$ cross sections and the $\\chi^2$ per data point values ($\\chi^2$\/$d.p.$). The $\\chi^2$\/$d.p.$\nvalues were estimated in point-by-point comparisons between the measured and computed one-fold differential \ncross sections in all bins of $W$ from 1.41~GeV to 1.71~GeV and in the three $Q^2$-bins covered by the CLAS \n$\\pi^+\\pi^-p$ data~\\cite{Ri03}. In the fit we selected the computed one-fold differential cross sections closest \nto the data with $\\chi^2\/d.p.$ less than a predetermined maximum value $\\chi^2_{max}\/d.p.$. The values of \n$\\chi^2_{max}\/d.p.$ were obtained by requiring that the computed cross sections with smaller $\\chi^2\/d.p.$ be \nwithin the data uncertainties for the majority of the data points, based on point-by-point comparisons between \nthe measured and the computed cross sections, see examples in Fig.~\\ref{fitsec},~\\ref{fitsec1}. In this fit procedure we \nobtained the $\\chi^2\/d.p.$ intervals within which the computed cross sections described the data equally well \nwithin the data uncertainties.\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=11.5cm]{pictures\/fit_1diff_065151.eps}\n\\includegraphics[width=11.5cm]{pictures\/fit_1diff_095151.eps}\\\\\n\\vspace{-0.1cm}\n\\caption{(Color Online) Examples of fits to the CLAS data~\\cite{Ri03} on the nine one-fold differential \n$\\pi^+\\pi^-p$ electroproduction cross sections in particular bins of $W$ and $Q^2$ within the framework of the \nupdated JM model described in Section~\\ref{pipipmech}. The curves correspond to those fits with $\\chi^2\/d.p.$ \nwithin the intervals listed in Table~\\ref{fitqual}. The resonant and non-resonant contributions determined \nfrom the data fit within the framework of the JM15 model are shown by blue triangles and green squares, \nrespectively.} \n\\label{fitsec}\n\\end{center}\n\\end{figure*}\n\n\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=11.5cm]{pictures\/fit_1diff_095161.eps}\n\\includegraphics[width=11.5cm]{pictures\/fit_1diff_130161.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) The same as in Fig.~\\ref{fitsec}, but in other bins of $W$ and $Q^2$.} \n\\label{fitsec1}\n\\end{center}\n\\end{figure*}\n\n\nWe fit the CLAS data~\\cite{Ri03} consisting of nine one-fold differential cross sections of the\n$ep \\to e'p'\\pi^+\\pi^-$ electroproduction reaction in all bins of $W$ and $Q^2$ in the kinematic regions of \n$W$: 1.41~GeV $< W <$ 1.51~GeV, 1.46~GeV $< W <$ 1.56~GeV, 1.51~GeV $< W <$ 1.61~GeV, 1.56~GeV $< W <$ 1.66~GeV,\n 1.61~GeV $< W <$ 1.71~GeV and 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$ within the framework of the fit procedure \ndescribed above. The five intervals of $W$ listed in Table~\\ref{fitqual} were fit independently. Each of the \naforementioned $W$ intervals contained 375 fit data points. The $\\chi^2\/d.p.$ intervals that correspond to an \nequally good data description within the data uncertainties are shown in Table~\\ref{fitqual}. Their values \ndemonstrate the quality of the CLAS $\\pi^+\\pi^-p$ data description achieved in the fits. Examining the \ndescription of the nine one-fold differential cross sections, we found that the $\\chi^2\/d.p.$ values were \ndetermined mostly by the deviations of only a few experimental data points from the computed fit cross sections. \nThere were no discrepancies in describing the shapes of the differential cross sections, which would manifest \nthemselves systematically in neighboring bins of $W$ and $Q^2$. Typical fit examples for $W$=1.51~GeV and \nneighboring $Q^2$ intervals centered at 0.65~GeV$^2$ and 0.95~GeV$^2$, as well as for $W$=1.61~GeV and $Q^2$ \nintervals centered at 0.95~GeV$^2$ and 1.30~GeV$^2$, are shown in Fig.~\\ref{fitsec},~\\ref{fitsec1}.\n \nSince only statistical data uncertainties were used in the computation of the $\\chi^2\/d.p.$ values listed in \nTable~\\ref{fitqual}, we concluded that a reasonable data description was achieved. The $\\chi^2\/d.p.$ values of \nour fits are comparable with those obtained in the fit of the CLAS $N\\pi$ and $\\pi^+\\pi^-p$ electroproduction \ndata published in Refs.~\\cite{Az09,Park15} and in Ref.~\\cite{Mo12}, respectively.\n\nFor each computed cross section point the resonant\/non-resonant contributions were estimated switching off\nthe non-resonant\/resonant amplitudes, respectively. The determined resonant\/non-resonant contributions to the \nnine one-fold differential cross sections are shown in Fig.~\\ref{fitsec},~\\ref{fitsec1}. The results suggest the unambiguous \nand credible separation between the resonant\/non-resonant contributions achieved fitting the CLAS data~\\cite{Ri03} \nwithin the framework of the JM model. The determined resonant\/non-resonant contributions are located within well \ndefined ranges (see Fig.~\\ref{fitsec},~\\ref{fitsec1}) and show no evidence for separation ambiguities, which would manifest\nthemselves as substantial differences between the particular computed resonant\/non-resonant cross sections and the \nranges determined for the resonant\/non-resonant contributions as shown in Fig.~\\ref{fitsec},~\\ref{fitsec1}. Such features in \nthe behavior of the resonant\/non-resonant contributions remain unseen in the entire area of $W$ and $Q^2$ covered \nby our analysis. Furthermore, the uncertainties of the resonant\/non-resonant contributions are comparable with the \nuncertainties of the measured cross sections, demonstrating again unambiguous resonant\/non-resonant separation of \na good accuracy. The credible isolation of the resonant contributions makes it possible to determine the resonance \nparameters from the resonant contributions employing for their description the amplitudes of the unitarized \nBreit-Wigner ansatz that fully accounts for the unitarity restrictions on the resonant amplitudes. \n\nThe resonance parameters obtained from each of these equally good fits were averaged and their mean values were \ntaken as the resonance parameters extracted from the data. The dispersions in these parameters were taken as \nthe uncertainties. The resonance electrocoupling uncertainties obtained in this manner are shown in \nFigs.~\\ref{p11comp}, \\ref{d13comp}, \\ref{s31comp}. Our fitting procedure allowed us to obtain more realistic \nuncertainties that take into account both statistical uncertainties in the data and systematic uncertainties \nimposed by the use of the JM reaction model. Furthermore, we consistently account for the correlations between \nvariations of the non-resonant and resonant contributions while extracting the resonance parameters. \n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=8.5cm]{pictures\/p11_a12_3wbin.eps}\n\\includegraphics[width=8.5cm]{pictures\/p11_s12_3wbin.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) Electrocouplings of the $N(1440)1\/2^+$ resonance determined from analysis of the \nCLAS $\\pi^+\\pi^-p$ electroproduction data~\\cite{Ri03} carried out independently in three intervals of $W$: \n1.41~GeV $<$ $W$ $<$ 1.51~GeV (black squares), 1.46~GeV $<$ $W$ $<$ 1.56~GeV (red circles), and 1.51~GeV \n$<$ $W$ $<$ 1.61~GeV (blue triangles).} \n\\label{p11comp}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=5.5cm]{pictures\/d13_a12_3wbin.eps}\n\\includegraphics[width=5.5cm]{pictures\/d13_s12_3wbin.eps}\n\\includegraphics[width=5.5cm]{pictures\/d13_a32_3wbin.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) Electrocouplings of the $N(1520)3\/2^-$ resonance determined from analysis of the \nCLAS $\\pi^+\\pi^-p$ electroproduction data~\\cite{Ri03} carried out independently in three intervals of $W$: \n1.41~GeV $<$ $W$ $<$ 1.51~GeV (black squares), 1.46~GeV $<$ $W$ $<$ 1.56~GeV (red circles), and 1.51~GeV \n$<$ $W$ $<$ 1.61~GeV (blue triangles).}\n\\label{d13comp}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=8.5cm]{pictures\/s31_a12_3wbin.eps}\n\\includegraphics[width=8.5cm]{pictures\/s31_s12_3wbin.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) Electrocouplings of the $\\Delta(1620)1\/2^-$ resonance determined from analysis of \nthe CLAS $\\pi^+\\pi^-p$ electroproduction data~\\cite{Ri03} carried out independently in three intervals of \n$W$: 1.51~GeV $<$ $W$ $<$ 1.61~GeV (black squares), 1.56~GeV $<$ $W$ $<$ 1.66~GeV (red circles), and \n1.61~GeV $<$ $W$ $<$ 1.71~GeV (blue triangles).}\n\\label{s31comp}\n\\end{center}\n\\end{figure*} \n\n\\section{Evaluation of the $\\gamma_vpN^*$ Resonance Electrocouplings and Hadronic Decay Widths to the \n$\\pi \\Delta$ and $\\rho N$ Final States}\n\\label{nstarelectrocoupl}\n\nThe procedure described in Section~\\ref{fit} allowed us to determine the $\\gamma_vpN^*$ electrocouplings of \nthe $N(1440)1\/2^+$, $N(1520)3\/2^-$, and $\\Delta(1620)1\/2^-$ resonances and their uncertainties. Our analysis \nextended the information on the electrocouplings of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ states providing the \nfirst results in the range of photon virtualities 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$ from the CLAS data. The \n$\\Delta(1620)1\/2^-$ resonance decays preferentially to the $N\\pi\\pi$ final state, making the charged double-pion \nelectroproduction channel the major source of information on the electrocouplings of this state. Our \nstudies provide the first results on the electrocouplings and hadronic decays of this resonance to the \n$\\pi \\Delta$ and $\\rho p$ final states from analysis of exclusive charged double-pion electroproduction.\n\nA special approach was developed for the evaluation of the $\\Delta(1620)1\/2^-$ electrocouplings. The analysis \nof the CLAS $\\pi^+\\pi^-p$ electroproduction data revealed that the $A_{1\/2}$ electrocoupling of this resonance \nwas much smaller than the $S_{1\/2}$ for 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$~\\cite{Mo14}. The $A_{1\/2}$ variations \ncomputed as a percentage of the start value became too small. For realistic uncertainty estimates we varied \n$A_{1\/2}$ in a much wider range that made its tested absolute values comparable with those for the $S_{1\/2}$ \nelectrocoupling. We fit the CLAS data~\\cite{Ri03} on $\\pi^+\\pi^-p$ electroproduction by varying $A_{1\/2}$, as \ndescribed above, keeping the variation of all other resonant and non-resonant parameters as described in \nSection~\\ref{fit}.\n\nIn order to compare our results on the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ electrocouplings in the $\\pi^+\\pi^-p$ \nelectroproduction channel with their values from the analysis of $N\\pi$ electroproduction, we must use in both \nof the exclusive electroproduction channels common branching fractions for the decays of these resonances to \nthe $N\\pi$ and $N\\pi\\pi$ final states. According to the RPP~\\cite{Rpp12}, the sum of the branching fractions \ninto the $N\\pi$ and $N\\pi\\pi$ final states accounts for almost 100\\% of the total decay widths of the \n$N(1440)1\/2^+$ and $N(1520)3\/2^-$ states. Since the $N\\pi$ exclusive electroproduction channels are most \nsensitive to contributions from the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances, we re-evaluated the branching \nfraction for decay to the $N\\pi\\pi$ final states $BF(N\\pi\\pi)_{corr}$ as:\n\\begin{equation}\n\\label{bnpipi}\nBF(N\\pi\\pi)_{corr}=1-BF(N\\pi).\n\\end{equation} \nFor these resonance decays to the $N\\pi\\pi$ final states it turns out that the estimated branching fractions \n$BF(N\\pi\\pi)_{corr}$ from Eq.(\\ref{bnpipi}) are slightly ($<$10\\%) different with respect to those obtained \nfrom the $\\pi^+\\pi^-p$ fit ($BF(N\\pi\\pi)_{0}$). Therefore, we multiplied the $\\pi \\Delta$ and $\\rho p$ hadronic \ndecay widths of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ from the $\\pi^+\\pi^-p$ fit by the ratio \n$\\frac{BF(N\\pi\\pi)_{corr}}{BF(N\\pi\\pi)_{0}}$. The $N(1440)1\/2^+$ and $N(1520)3\/2^-$ electrocouplings obtained \nin our analysis were then multiplied by the correction factors \n\\begin{equation}\n\\label{bnpipi1}\nC_{hd}=\\sqrt{\\frac{BF(N\\pi\\pi)_{0}}{BF(N\\pi\\pi)_{corr}}}\n\\end{equation} \nin order to keep the resonant parts and the computed differential $\\pi^+\\pi^-p$ cross sections unchanged \nunder the re-scaling of the resonance hadronic decay parameters described above.\n\nThe electrocouplings of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and $\\Delta(1620)1\/2^-$ resonances were determined \nin our analysis for 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$, where there is still no data on observables of other \nexclusive meson electroproduction channels measured with CLAS. We have developed special procedures to test \nthe reliability of the resonance $\\gamma_vpN^*$ electrocouplings and their $\\pi\\Delta$ and $\\rho p$ partial \nhadronic decay widths extracted from the charged double pion electroproduction data only. In order to check the \nreliability of the extracted $\\gamma_vpN^*$ electrocouplings, we carried out the extraction of the resonance \nparameters of all of the aforementioned resonances independently, fitting the CLAS $\\pi^+\\pi^-p$ electroproduction \ndata~\\cite{Ri03} in the five overlapping intervals of $W$ given in Table~\\ref{fitqual} covering in each fit the \nthree $Q^2$-bins centered at 0.65~GeV$^2$, 0.95~GeV$^2$, and 1.30~GeV$^2$. The non-resonant amplitudes in each of \nthe aforementioned $W$-intervals are different, while the resonance parameters should remain the same as they are \ndetermined from the data fit in different $W$-intervals. The $N(1440)1\/2^+$, $N(1520)3\/2^-$, and $\\Delta(1620)1\/2^-$ \nstate electrocouplings extracted in the fit of the $\\pi^+\\pi^-p$ CLAS data~\\cite{Ri03} in the different $W$-intervals \nare shown in Figs.~\\ref{p11comp}, \\ref{d13comp}, and \\ref{s31comp}. \n\nThe values of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ electrocouplings, as well as the $S_{1\/2}$ electrocoupling\nof the $\\Delta(1620)1\/2^-$, obtained from independent analyses of the different $W$-intervals, are consistent \nwithin the uncertainties. The values of the $A_{1\/2}$ electrocoupling of the $\\Delta(1620)1\/2^-$ state from the \nfit of the $W$-interval from 1.51~GeV to 1.61~GeV are different in comparison to the fit results of the two others \n$W$-intervals. We consider the values of the $\\Delta(1620)1\/2^-$ electrocouplings determined in the $W$-interval \nfrom 1.56~GeV to 1.66~GeV as the most reliable, since the others $W$-intervals overlap only over part of the \nresonance line width of the $\\Delta(1620)1\/2^-$. The consistent results on the $\\gamma_vpN^*$ electrocouplings \nfrom the independent analyses of different $W$-intervals strongly support the reliable extraction of these \nfundamental quantities, as well as the capability of the JM model to provide reliable information on the \n$\\gamma_vpN^*$ resonance electrocouplings from analysis of the data on exclusive charged double-pion \nelectroproduction.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nQ$^2$, & $A_{1\/2},$ & $S_{1\/2}$ \\\\\nGeV$^2$ & GeV$^{-1\/2}$*1000 & GeV$^{-1\/2}$*1000 \\\\ \\hline\n$0.65$ & $21.4 \\pm 6.2$ & $25.7 \\pm 5.9$ \\\\\n$0.95$ & $29.6 \\pm 6.5$ & $25.6 \\pm 6.2$ \\\\\n$1.30$ & $42.6 \\pm 9.3$ & $29.4 \\pm 5.5$ \\\\ \\hline\n\\end{tabular}\n\\caption{Electrocouplings of the $N(1440)1\/2^+$ resonance determined from this analysis of $\\pi^+\\pi^-p$ \nelectroproduction off the proton~\\cite{Ri03} at 1.41~GeV $< W <$ 1.51~GeV within the framework of the \nupdated JM model described in Section~\\ref{pipipmech}.}\n\\label{p11el} \n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nQ$^2$, & $A_{1\/2},$ & $S_{1\/2}$, & $A_{3\/2}$, \\\\\nGeV$^2$ & GeV$^{-1\/2}$*1000 & GeV$^{-1\/2}$*1000 & GeV$^{-1\/2}$*1000 \\\\ \\hline\n$0.65$ & $-52.9 \\pm 7.5$ & $-29.4 \\pm 3.1$ & $42.9 \\pm 7.1$ \\\\\n$0.95$ & $-50.8 \\pm 7.9$ & $-27.3 \\pm 7.1$ & $32.7 \\pm 6.2$ \\\\\n$1.30$ & $-39.6 \\pm 6.3$ & $-9.8 \\pm 2.9$ & $25.2 \\pm 3.3$ \\\\ \\hline\n\\end{tabular}\n\\caption{Electrocouplings of the $N(1520)3\/2^-$ resonance determined from this analysis of $\\pi^+\\pi^-p$ \nelectroproduction off the proton~\\cite{Ri03} at 1.46~GeV $< W <$ 1.56~GeV within the framework of the \nupdated JM model described in Section~\\ref{pipipmech}.}\n\\label{d13el} \n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nQ$^2$, & $A_{1\/2},$ & $S_{1\/2}$ \\\\\nGeV$^2$ & GeV$^{-1\/2}$*1000 & GeV$^{-1\/2}$*1000 \\\\ \\hline\n$0.65$ & $15.5 \\pm 10.2$ & $-46.3 \\pm 3.8$ \\\\\n$0.95$ & $12.5 \\pm 5.4$ & $-30.9 \\pm 7.0$ \\\\\n$1.30$ & $5.5 \\pm 4.4$ & $-17.2 \\pm 5.6$ \\\\ \\hline\n\\end{tabular}\n\\caption{Electrocouplings of the $\\Delta(1620)1\/2^-$ resonance determined from this analysis of $\\pi^+\\pi^-p$ \nelectroproduction off the proton~\\cite{Ri03} at 1.56~GeV $< W <$ 1.66~GeV within the framework of the \nupdated JM model described in Section~\\ref{pipipmech}.}\n\\label{s31el} \n\\end{center}\n\\end{table}\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nParameter & Current analysis of the CLAS & Previous analysis \\cite{Mo12} of the CLAS & RPP \\\\\n & $\\pi^+\\pi^-p$ data \\cite{Ri03} at & $\\pi^+\\pi^-p$ data \\cite{Fe09} 0.25 at & \\\\\n & 0.5 GeV$^2$ $< Q^2 < 1.5$~GeV$^2$ & 0.25 ~GeV$^2$ $< Q^2 < 0.6$~GeV$^2$ & \\\\ \\hline\nBreit-Wigner mass, MeV & 1454 $\\pm$ 11 & 1458 $\\pm$ 12 & 1420-1470 ($\\approx$ 1440) \\\\\nBreit-Wigner width, MeV & 352 $\\pm$ 37 & 363 $\\pm$ 39 & 200-450 ($\\approx$ 300) \\\\\n$\\pi \\Delta$ partial decay width, MeV & 120 $\\pm$ 41 & 142 $\\pm$ 48 & \\\\\n$\\pi \\Delta$ BF, & 20\\%-52\\% & 23\\%-58\\% & 20\\%-30\\% \\\\\n$\\rho p$ partial decay width, MeV & 4.9 $\\pm$ 2.2 & 6.2 $\\pm$ 4.1 & \\\\\n$\\rho p$ BF & $<$ $~$2.0\\% & $<$ $~$2.0\\% & $<$ $~$8.0\\% \\\\ \\hline\n\\end{tabular}\n\\caption{Hadronic parameters of the $N(1440)1\/2^+$ resonance determined from the CLAS data~\\cite{Ri03} on\n$\\pi^+\\pi^-p$ electroproduction off the proton within the framework of the updated JM model described in \nSection~\\ref{pipipmech} in comparison with the results of our previous analysis~\\cite{Mo12} and RPP~\\cite{Rpp12}.}\n\\label{hpp11} \n\\end{center}\n\\end{table*}\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nParameter & Current analysis of the CLAS & Previous analysis \\cite{Ri03} of the CLAS & RPP \\\\\n & $\\pi^+\\pi^-p$ data \\cite{Ri03} at & $\\pi^+\\pi^-p$ data at & \\\\\n & 0.5~GeV$^2$ $< Q^2 < 1.5$~GeV$^2$ & 0.25 ~GeV$^2$ $< Q^2 < 0.6$~GeV$^2$ & \\\\ \\hline\nBreit-Wigner mass, MeV & 1522 $\\pm$ 5 & 1521 $\\pm$ 4 & 1515-1525 ($\\approx$ 1520) \\\\\nBreit-Wigner width, MeV & 125 $\\pm$ 4 & 127 $\\pm$ 4 & 100-125 ($\\approx$ 115) \\\\\n$\\pi \\Delta$ partial decay width, MeV & 36 $\\pm$ 5 & 35 $\\pm$ 4 & \\\\\n$\\pi \\Delta$ BF & 25\\%-34\\% & 24\\%-32\\% & 15\\%-25\\% \\\\\n$\\rho p$ partial decay width, MeV & 13 $\\pm$ 6 & 16 $\\pm$ 5 & \\\\\n$\\rho p$ BF & 4.8\\%-16\\% & 8.4\\%-17\\% & 15\\%-25\\% \\\\ \\hline\n\\end{tabular}\n\\caption{Hadronic parameters of the $N(1520)3\/2^-$ resonance determined from the CLAS data~\\cite{Ri03} on\n$\\pi^+\\pi^-p$ electroproduction off the proton within the framework of the updated JM model described in \nSection~\\ref{pipipmech} in comparison with the results of our previous analysis~\\cite{Mo12} and RPP~\\cite{Rpp12}.}\n\\label{hpd13} \n\\end{center}\n\\end{table*}\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nParameter & Current analysis of the CLAS & RPP \\\\\n & $\\pi^+\\pi^-p$ data~\\cite{Ri03} at & \\\\ \n & 0.5~GeV$^2$ $< Q^2 < 1.5$~GeV$^2$ & \\\\ \\hline\nBreit-Wigner mass, MeV & 1631 $\\pm$ 12 & 1600-1660 ($\\approx$ 1630) \\\\\nBreit-Wigner width, MeV & 148 $\\pm$ 10 & 130-150 ($\\approx$ 140) \\\\\n$\\pi \\Delta$ partial decay width, MeV & 66 $\\pm$ 23 & \\\\\n$\\pi \\Delta$ BF, & 27\\%-64\\% & 30\\%-60\\% \\\\\n$\\rho p$ partial decay width, MeV & 70 $\\pm$ 21 & \\\\\n$\\rho p$ BF & 31\\%-63\\% & 7\\%-25\\% \\\\ \\hline\n\\end{tabular}\n\\caption{Hadronic parameters of the $\\Delta(1620)1\/2^-$ resonance determined from the CLAS data~\\cite{Ri03} \non $\\pi^+\\pi^-p$ electroproduction off the proton within the framework of the updated JM model described in \nSection~\\ref{pipipmech} in comparison with RPP~\\cite{Rpp12}.}\n\\label{hps31} \n\\end{center}\n\\end{table*}\n\nThe final results on the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and $\\Delta(1620)1\/2^-$ electrocouplings are listed \nin Tables~\\ref{p11el},~\\ref{d13el}, and \\ref{s31el}. They were determined from the fit of the CLAS data~\\cite{Ri03} \nin the $W$-intervals given in the captions of Tables~\\ref{p11el},~\\ref{d13el}, and \\ref{s31el} covering the three \n$Q^2$-bins centered at 0.65~GeV$^2$, 0.95~GeV$^2$, and 1.30~GeV$^2$. The intervals over $W$ within which the \nresonance electrocouplings were extracted were determined by the requirement that the selected $W$ intervals \noverlap the area of masses below and above the central resonance mass values. The resonance electrocoupling \nuncertainties reflect both the experimental data uncertainties and the systematic uncertainties imposed by \nthe extraction model. \n\nThe $A_{1\/2}$ electrocouplings of the $N(1440)1\/2^+$ state are positive and increase with $Q^2$, supporting \nthe zero crossing observed for this electroexcitation amplitude in our previous analyses of the CLAS $N\\pi$ \nand $N\\pi\\pi$ electroproduction data~\\cite{Az09,Mo12}. The $A_{1\/2}$ electrocouplings of the $N(1520)3\/2^-$ \nstate are negative and increase with photon virtualities, confirming the local minimum at $Q^2 \\approx 0.45$~GeV$^2$ \nobserved in our previous analyses~\\cite{Az09,Mo12}. The electroexcitation of the $\\Delta(1620)1\/2^-$ resonance \nis dominated by longitudinal electrocouplings in the entire area of photon virtualities covered in our analysis, \n0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$.\n\nIn this analysis we also obtained the hadronic decay widths of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and \n$\\Delta(1620)1\/2^-$ resonances to the $\\pi \\Delta$ and $\\rho p$ final states. These parameters were \ndetermined from the $\\pi^+\\pi^-p$ electroproduction data~\\cite{Ri03} under simultaneous variations of the\nresonance masses, $\\gamma_vpN^*$ electrocouplings, and hadronic decay widths to the $\\pi \\Delta$ and $\\rho N$ \nfinal states under the requirement of $Q^2$-independence of the resonance masses and hadronic decay parameters. \n\nThe $N(1440)1\/2^+$ and $N(1520)3\/2^-$ masses, as well as the branching fractions for the decays to the \n$\\pi \\Delta$ and $\\rho N$ final states extracted in the fit of the data~\\cite{Ri03}, are given in \nTables~\\ref{hpp11} and \\ref{hpd13} in comparison with the results of our previous analysis~\\cite{Mo12} of the \nCLAS $\\pi^+\\pi^-p$ electroproduction data~\\cite{Fe09} carried out at smaller $W$ and $Q^2$. The results of our \ncurrent analysis on the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ masses, and their total and partial hadronic decay \nwidths to the $\\pi \\Delta$ and $\\rho p$ final states are consistent. A successful description of the CLAS \n$\\pi^+\\pi^-p$ electroproduction data over different and wide ranges of photon virtualities, 0.25~GeV$^2$ \n$< Q^2 <$ 0.6~GeV$^2$ (previous analysis~\\cite{Mo12}) 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$ (current analysis), \nstrongly support the reliable separation of the resonant and non-resonant contributions achieved within the \nframework of the JM model and the credible extraction of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonance \nparameters. Both the current and previous analyses of the CLAS $\\pi^+\\pi^-p$ electroproduction data suggest \nthat the $\\rho p$ hadronic decay widths of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances are smaller than\nthose reported in the RPP, and that the $\\pi \\Delta$ hadronic decay widths of the $N(1520)3\/2^-$ are larger than \nthose reported in RPP~\\cite{Rpp12}. The successful description of the CLAS $\\pi^+\\pi^-p$ electroproduction data\n\\cite{Fe09,Ri03} in a wide area of $Q^2$ from 0.25~GeV$^2$ to 1.5~GeV$^2$ achieved with $Q^2$-independent \nresonance hadronic parameters, makes the results presented in Tables~\\ref{hpp11} and \\ref{hpd13} reliable. They \noffer new information on the hadronic decays of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances to the $\\pi \\Delta$ \nand $\\rho N$ final states that may be considered as input for the upcoming RPP edition. \n\nFor the first time the hadronic decay parameters of the $\\Delta(1620)1\/2^-$ resonance listed in Table~\\ref{hps31}, \nhave become available from the analysis of the $\\pi^+\\pi^-p$ electroproduction data. The mass, total width, and \nthe branching fractions for decays of the $\\Delta(1620)1\/2^-$ to the $\\pi \\Delta$ final states obtained in our \nanalysis are in good agreement with the RPP results~\\cite{Rpp12}. The current analysis suggests much \nlarger values of the branching fractions for decays of the $\\Delta(1620)1\/2^-$ to the $\\rho p$ final states in \ncomparison with those presented in RPP. A successful description of the CLAS $\\pi^+\\pi^-p$ electroproduction \ndata~\\cite{Ri03} with $Q^2$ independent values of the $\\Delta(1620)1\/2^-$ hadronic decay widths strongly supports \nthe branching fraction values listed in Table~\\ref{hps31}. The large values determined for the branching fraction \nfor decays of the $\\Delta(1620)1\/2^-$ to the $\\rho p$ final states represent an interesting and unexpected \nresult, since the $\\Delta(1620)1\/2^-$ state is located in the sub-threshold area for $\\rho p$ electroproduction \noff the proton. The absence of $\\rho$ peaks in the data on the $\\pi^+\\pi^-$ invariant mass distributions at \n$W \\approx 1.6$~GeV, in combination with large $\\rho p$ hadronic decays of the $\\Delta(1620)1\/2^-$, impose \nrestrictions on the upper limits of the $A_{1\/2}$ electrocouplings for the $\\Delta(1620)1\/2^-$ state, making their \nabsolute values much smaller than those of the $S_{1\/2}$ electrocouplings.\n\n\n\n\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=14.5cm]{pictures\/nstdse.eps}\n\\vspace{-0.1cm}\n\\caption{Description of the resonance electroexcitation amplitudes within the framework of DSEQCD\n\\cite{Cr14,Cr15,Cr15a}: A) the amplitude for the transition $p \\to$ three dressed quarks or the ground state \nwave function $\\psi_p$, B) the amplitude for the transition three dressed quarks $\\to N^*$ or the excited \nnucleon state wave function $\\psi_{N^*}$, C) the amplitude that describes the interaction between the virtual \nphoton and three dressed quarks bound by the non-perturbative strong interaction between pairs of correlated \nquarks (di-quark) and by the dressed quark exchange between the di-quark pair and third quark. The virtual photon \ninteractions with the quark and di-quark currents are shown on the left and top right diagrams, respectively. \nThe di-quark currents incorporate the transitions between di-quarks of the same and different quantum numbers. \nThe full $N \\rightarrow N^*$ transition amplitude can be found in Fig.~C1 of Ref.~\\cite{Cr15}.} \n\\label{diagdse}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=8.5cm]{pictures\/f1roperdse.eps}\n\\includegraphics[width=8.5cm]{pictures\/f2roperdse.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) The $F_1^*$ and $F_2^*$ transition $p \\to N(1440)1\/2^+$ form factors: experimental \nresults from analyses of the CLAS data on $N\\pi$~\\cite{Az09} (red circles) and $\\pi^+\\pi^-p$~\\cite{Mo12} \n(black triangles) electroproduction off the proton and the results of this present work (blue squares). The\ndata are shown in comparison with DSEQCD evaluations~\\cite{Cr15a} start from the QCD Lagrangian (black \ndashed line) and after accounting for the meson-baryon cloud contributions as described in Section~\\ref{impp11d13} \n(blue thick solid line).}\n\\label{p11datdse}\n\\end{center}\n\\end{figure*} \n\n\\section{Impact on Studies of the $N^*$ Structure from the New CLAS Results}\n\\label{impact}\n\nIn this section we discuss the impact of the new CLAS results on the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and\n$\\Delta(1620)1\/2^-$ electrocouplings and their partial hadronic decay widths to the $\\pi \\Delta$ and \n$\\rho N$ final states on the contemporary understanding of the structure of these states. We will also \noutline new possibilities for hadron structure theory to employ these experimental results in order \nto explore how the dynamical properties of three constituent dressed quarks inside the resonance quark core \nemerge from QCD. \n\n\\subsection{$N(1440)1\/2^+$ and $N(1520)3\/2^-$ Resonances} \n\\label{impp11d13} \n\nPrevious studies of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances with the CLAS detector~\\cite{Az09,Mo12} \nhave provided the dominant part of the world-wide information available on their electrocouplings in a wide \nrange of photon virtualities 0.25~GeV$^2$ $< Q^2 <$ 5.0~GeV$^2$. This paper extends the CLAS results on the\n$N(1440)1\/2^+$ and $N(1520)3\/2^-$ $\\gamma_vpN^*$ electrocouplings in the range of photon virtualities from\n0.5~GeV$^2$ to 1.5~GeV$^2$ where there is limited availability of data. Previous studies of $\\pi^+\\pi^-p$ \nelectroproduction~\\cite{Mo12} have allowed us to determine the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ partial \ndecay widths to the $\\pi \\Delta$ and $\\rho p$ final states. Our current studies confirmed the previous \nresults~\\cite{Mo12} on these hadronic decays. Currently the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ states, \ntogether with the $\\Delta(1232)3\/2^+$ and $N(1535)1\/2^-$ resonances~\\cite{Bu12}, represent the most explored \nexcited nucleon states. Detailed information on the electrocouplings of these states that are available for \nthe first time from CLAS, have already provided a profound impact on the contemporary understanding of the \nnucleon resonance structure~\\cite{Bu12,Az13,Mo12}. \n\nRecent progress in the studies of resonance structure achieved within the framework of the Dyson-Schwinger \nEquations of QCD (DSEQCD)~\\cite{Cr14,Cr15,Cr15a} has allowed us for the first time to interpret the \nexperimental results on the nucleon elastic form factors, as well as the magnetic $p \\to \\Delta$ and \n$p \\to N(1440)1\/2^+$ Dirac ($F_1^*$) and Pauli ($F_2^*$) transition from factors start from the QCD \nLagrangian. Currently this approach is capable of evaluating the contributions from the quark core of three \ndressed quarks to the nucleon elastic and $p \\to N^*$ transition form factors. DSEQCD approaches describe \nthe ground and excited nucleons as bound systems of three dressed quarks that represent the complex objects \ngenerated non-perturbatively from an infinite number of QCD quarks and gauge gluons. The dynamical properties \nof dressed quarks, the momentum dependent mass $M(p)$ and form factors, that enter into the quark electromagnetic \ncurrent, are determined start from the QCD Lagrangian employing the towers of gap equations for quarks and \ngluons~\\cite{Cr14}. The ground and excited nucleon state masses and the transition amplitudes, $p \\to$ three \ndressed quarks (the ground state wave function) and three dressed quarks $\\to N^*$ (the excited nucleon state \nwave function), are obtained in a Poincar$\\acute{\\rm e}$ covariant approach employing Faddeev equations for the \nthree dressed quarks. The non-perturbative interactions between the three dressed quarks are reduced to a \nquark-quark interaction that generates pairs of correlated quarks, the so-called dynamical di-quark, and \ndressed quark exchanges between the di-quark pair and third quark shown in the parts labeled ``C'' in \nFig.~\\ref{diagdse}~\\cite{Cr14,Cr15c}. The ground and excited nucleon state masses emerge as poles in the \nenergy dependence of the amplitude with the respective spin-parity that comes from the Faddeev equation \nsolution. The ground\/excited nucleon state wave functions represent the residues of the Faddeev equation \nsolutions at the respective pole positions. The resonance electroexcitation amplitudes, depicted in \nFig.~\\ref{diagdse}, are evaluated as the product of three amplitudes: A) ground state $p \\to$ three dressed \nquarks, B) three dressed quarks $\\to$ resonance $N^*$, and C) interaction between real\/virtual photons and \nthe three dressed quarks. The latter part C is described mostly by real\/virtual photon couplings to the dressed quark \nand di-quark pair currents. All details on the evaluations of resonance electroexcitation amplitudes can be \nfound in Refs.~\\cite{Cr15,Cr15a}.\n\nThe resonance electroexcitation amplitudes shown in Fig.~\\ref{diagdse} should be sensitive to the momentum \ndependence of the dressed quark mass $M(p)$, since it affects all quark propagators and dressed quark currents. \nMoreover, it was shown in Refs.~\\cite{Cr15,Cr15a} that the momentum dependence of the dressed quark mass has a \npronounced influence on the wave functions of the ground and excited nucleon states. DSEQCD studies of \nexperimental results on elastic nucleon form factors~\\cite{Cr13} confirmed these expectations and revealed \nconsiderable sensitivity of the nucleon elastic form factors to the momentum dependence of the dressed quark \nmass function. It was found that the location of the zero crossing for the ratio $\\mu_pG_E\/G_M(Q^2)$ is \ndetermined by the derivative of the dressed quark mass function $M(p)$. \n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=8.5cm]{pictures\/p11a12dseqm.eps}\n\\includegraphics[width=8.5cm]{pictures\/p11s12dseqm.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) The $A_{1\/2}$ and $S_{1\/2}$ $\\gamma_vpN^*$ electrocouplings of the $N(1440)1\/2^+$ \nresonance: experimental results from analyses of the CLAS data on $N\\pi$~\\cite{Az09} (red circles) and \n$\\pi^+\\pi^-p$~\\cite{Mo12} (black triangles) electroproduction off the proton and the results of this \nwork (blue squares). The data are shown in comparison with the DSEQCD evaluations~\\cite{Cr15a} (blue thick \nsolid) and the results from constituent quark models that account for the contributions from both the quark core \nand the meson-baryon cloud:~\\cite{Az12} (thin red solid) and~\\cite{Ob14} (thin red dashed). The calculations of \nthin red line includes pion loops and a parametrization of the running quark mass, and the calculations of \ndashed red line contains $N\\sigma$ contributions and fixed quark mass. The meson-baryon cloud contributions \nobtained from the experimental data (see Section~\\ref{impp11d13}) are shown by the magenta thick dashed lines.}\n\\label{p11a12s12dseqm}\n\\end{center}\n\\end{figure*} \n\nThe need to employ a \nmomentum-dependent dressed quark mass function was conclusively demonstrated in the studies of the \n$N \\to \\Delta$ magnetic transition form factor within the DSEQCD framework~\\cite{Cr14}.\nComputations employing a dressed quark with a momentum-independent mass generated by simplified contact \nquark-quark interactions were able to describe the experimental results only in a very limited range of \nphoton virtualities $Q^2 < 3.0$~GeV$^2$. Instead, the DSEQCD evaluation with running quark mass successfully \nreproduced the experimental data at $Q^2 > 1.0$~GeV$^2$ in the entire range of photon virtualities covered by \nmeasurements reaching up to 8.0~GeV$^2$. \n\nThe recent DSEQCD studies of the $N(1440)1\/2^+$ resonance electroexcitation~\\cite{Cr15a} derive from a realistic \nquark-quark interaction that generates a momentum-dependent dressed quark mass function. The evaluated \ncontributions from the quark core to the Dirac $F_1^*$ and to the Pauli $F_2^*$ $p \\to N(1440)1\/2^+$ transition \nform factors are shown in Fig.~\\ref{p11datdse} by the dashed lines in comparison with the CLAS experimental \nresults published in Refs.~\\cite{Az09,Mo12}, as well as with those obtained in this present work. DSEQCD \nreasonably reproduces the experimental results for $Q^2 > 2.5$~GeV$^2$. However, a pronounced disagreement for \n$Q^2 < 1.0$~GeV$^2$, in particular, for the Pauli $F_2^*$ form factor, suggests significant contributions \nfrom degrees of freedom other than the quark core, presumably the meson-baryon cloud found in the global \nmulti-channel analysis of exclusive meson photo-, electro-, and hadroproduction data~\\cite{Lee08}. These \ncontributions are still beyond the scope of DSEQCD studies~\\cite{Cr15a}. However, we have to account for the \nfraction of the meson-baryon degrees of freedom in the wave functions of the ground and excited nucleon states. \nWe choose to estimate this contribution by multiplying the $p \\to N(1440)1\/2^+$ transition form factors computed \nwithin the DSEQCD approach~\\cite{Cr15a} by a common $Q^2$-independent factor fit to the data for $Q^2 > 3.0$~GeV$^2$, \nwhere the meson-baryon cloud contributions are much smaller than those from the quark core. The fit value of this \nfactor of 0.73 is consistent with the results of a recent advanced light front quark model~\\cite{Az12}, which \nemploys the parameterization of running quark mass function in spirit of DSEQCD~\\cite{Cr14}. The $p \\to N(1440)1\/2^+$ transition form factors obtained in \nthis way are shown in Fig.~\\ref{p11datdse} by the solid blue lines. A good description of the experimental \nresults for $Q^2 > 1.5$~GeV$^2$ is achieved within the entire range of photon virtualities covered by the \nmeasurements.\n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=5.8cm]{pictures\/d13a12qmdat.eps}\n\\includegraphics[width=5.8cm]{pictures\/d13s12qmdat.eps}\n\\includegraphics[width=5.8cm]{pictures\/d13a32qmdat.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) The $A_{1\/2}$, $S_{1\/2}$, and $A_{3\/2}$ $\\gamma_vpN^*$ electrocouplings of the\n$N(1520)3\/2^-$ resonance: experimental results from analyses of the CLAS data on $N\\pi$~\\cite{Az09} (red \ncircles) and $\\pi^+\\pi^-p$~\\cite{Mo12} (black triangles) electroproduction off the proton and the results \nof this present work (blue squares). The data are shown in comparison with the predictions of the \nhypercentral quark model~\\cite{Sa12} (thin black solid). The meson-baryon cloud contributions obtained in a\nglobal multi-channel $N\\pi$ photo-, electro-, and hadroproduction data analysis~\\cite{Lee08} are shown by the\nthick dashed magenta lines.}\n\\label{d13elcoupl}\n\\end{center}\n\\end{figure*} \n\nThe dressed quark mass function used in the DSEQCD computations of the $p \\to N(1440)1\/2^+$ transition form \nfactors~\\cite{Cr15a} is {\\it exactly the same} as that employed in the previous evaluations of the nucleon \nelastic and magnetic $p \\to \\Delta$ transition form factors~\\cite{Cr15,Cr13}. The $\\Delta(1232)3\/2^+$ and \n$N(1440)1\/2^+$ excited nucleon states have a distinctively different structure: spin-flavor flip for the\n$\\Delta(1232)3\/2^+$ and the first radial excitation of three dressed quarks for the $N(1440)1\/2^+$. A successful \ndescription of the elastic and transition form factors to nucleon resonances of distinctively different \nstructure achieved with the same dressed quark mass function strongly underlines:\n\\begin{itemize}\n\\item the relevance of dynamical dressed quarks with the properties predicted by the DSEQCD approach~\\cite{Cr14} \nas constituents of the quark cores for the structure of the ground and excited nucleon states;\n\\item the capability of the DSEQCD approach~\\cite{Cr15,Cr15a} to map out the dressed quark mass function from \nthe experimental results on the $Q^2$-evolution of the nucleon elastic and $p \\to N^*$ transition form \nfactors ($\\gamma_vpN^*$ electrocouplings) from a combined analysis.\n\\end{itemize} \nConsistent results on the momentum dependence of the dressed quark mass function obtained from independent \nanalyses of nucleon elastic and $p \\to N^*$ form factors, i.e. $\\gamma_vpN^*$ electrocouplings for the transition \nto excited nucleons with different quantum numbers, are critical in order to prove the reliable access to this \nfundamental quantity.\n\nDSEQCD analyses~\\cite{Cr15,Cr15a} of the CLAS results on the $p \\to \\Delta$ and $p \\to N(1440)1\/2^+$ transition \nform factors (the latter shown in Fig.~\\ref{p11a12s12dseqm}) have demonstrated the capability of accessing the dressed \nquark mass function from the experimental data for the first time. Studies of the dressed quark mass function \nwill address the most challenging and still open problems of the Standard Model on the nature of the dominant \npart of the hadron mass, quark-gluon confinement, its emergence from QCD, and its relation to dynamical chiral \nsymmetry breaking, which is expected to be the source of more than 98\\% of the hadron mass in universe~\\cite{Cr14}.\n\nRecent advances in the development of the constituent quark models make it possible to extend the $Q^2$-range \nfor a better description of the $\\gamma_vpN^*$ electrocouplings in comparison with DSEQCD approaches, taking \ninto account both contributions from the quark core and the meson-baryon cloud. The two models~\\cite{Az12,Ob14} \ndescribe the structure of the $N(1440)1\/2^+$ resonance as an interplay between the \ncontributions from the inner core of three dressed quarks in the first radial excitation and an external \nmeson-baryon cloud. Both approaches treat the quark core contributions within the light front framework. \nThe first model~\\cite{Az12} employs a phenomenological momentum-dependent dressed quark mass motivated by \nthe DSEQCD results~\\cite{Cr15,Cr15a}, while the second~\\cite{Ob14} employs constituent quarks of \nmomentum-independent mass. The meson baryon cloud is modeled by $\\pi N$ loops in the first approach~\\cite{Az12}, \nwhile the $\\sigma p$ loops are employed in the second approach~\\cite{Ob14}. The CLAS experimental results on the\n$A_{1\/2}$ and $S_{1\/2}$ $\\gamma_vpN^*$ electrocouplings of the $N(1440)1\/2^+$ resonance are shown in \nFig.~\\ref{p11a12s12dseqm} in comparison with the expectations from DSEQCD~\\cite{Cr15a} and from the \naforementioned two advanced constituent quark models~\\cite{Az12,Ob14}. Accounting for the meson-baryon cloud \ncontributions allowed us to considerably improve the description of the experimental data at $Q^2 < 2.0$~GeV$^2$, \nconfirming the relevance of meson-baryon degrees of freedom in the $N(1440)1\/2^+$ structure at these distances \nthat had previously been established in multi-channel analyses of exclusive meson photo-, electro-, and \nhadroproduction experimental data~\\cite{Lee08}.\n\nThe CLAS results on the $\\gamma_vpN^*$ electrocouplings of the $N(1520)3\/2^-$ resonance are shown in \nFig.~\\ref{d13elcoupl}. The currently available models for the description of the structure of this state \naccount for quark core contributions only. The quark core contributions to the $\\gamma_vpN^*$ \nelectrocouplings of most well-established excited nucleon states were explored within the framework of \ntwo conceptually different approaches: a) hypercentral constituent quark model~\\cite{Sa12} and b) \nBethe-Salpeter approach that employs structureless constituent quarks with momentum-independent mass and an\ninstanton quark-quark interaction~\\cite{Met12}. The hypercentral constituent quark model provides a reasonable \ndescription of the experimental results at $Q^2 > 1.0$~GeV$^2$ as shown in Fig.~\\ref{d13elcoupl}. At smaller \nphoton virtualities there are substantial discrepancies between the model~\\cite{Sa12} and the CLAS results. \nA similar observation comes from the comparison of the CLAS results with the Bethe-Salpeter approach\n\\cite{Met12}. Estimates for the contributions from the meson-baryon cloud to the structure of the \n$N(1520)3\/2^-$ resonance were obtained in Ref.~\\cite{Lee08} from a global multi-channel analysis of the \nexperimental data on exclusive pion photo-, electro-, and hadroproduction. The absolute values of the \nmeson-baryon cloud shown in Fig.~\\ref{d13elcoupl} are maximal at small photon virtualities where discrepancies \nbetween the quark model expectations and the experimental data are largest. Hence, the meson-baryon cloud \ncontributions may explain the difference between the CLAS data and the quark model expectations for the \n$\\gamma_vpN^*$ electrocouplings of the $N(1520)3\/2^-$ resonance. The aforementioned studies of the CLAS data \nin Fig.~\\ref{d13elcoupl} suggest that the structure of the $N(1520)3\/2^-$ resonance arises from the contributions \nfrom the inner core of three dressed quarks in the first orbital excitation with $L=1$ and the external \nmeson-baryon cloud. The contributions from the meson-baryon cloud are strongly dependent on the helicity of \nthe $N^*$ electroexcitation amplitudes. They decrease with photon virtuality $Q^2$. \n\n\\begin{figure*}[htp]\n\\begin{center}\n\\includegraphics[width=8.5cm]{pictures\/s31a12qm.eps}\n\\includegraphics[width=8.5cm]{pictures\/s31s12qm.eps}\n\\vspace{-0.1cm}\n\\caption{(Color Online) The first results on the $A_{1\/2}$ and $S_{1\/2}$ $\\gamma_vpN^*$ electrocouplings of \nthe $\\Delta(1620)1\/2^-$ resonance from the CLAS data on $\\pi^+\\pi^-p$ electroproduction off the proton\n\\cite{Ri03} in comparison with a hypercentral constituent quark model~\\cite{Sa12} (thick black solid lines) \nand the Bethe-Salpeter approach~\\cite{Met12} (blue dashed lines). The photocoupling value is taken from the\nRPP~\\cite{Rpp12}.} \n\\label{s31qm}\n\\end{center} \n\\end{figure*}\n\nThe CLAS data on the $\\gamma_vpN^*$ electrocouplings of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances, \ntogether with the results from previous studies~\\cite{Az09} and recent analyses of the $N^*$ electroexcitations \nin the third resonance region~\\cite{Park15,Az15a}, strongly suggest that the structure of nucleon resonances \nfor $Q^2 < 5.0$~GeV$^2$ is determined by a complex interplay between the inner core of three dressed quarks \nbound to the states with the quantum numbers of the nucleon resonance and the external meson-baryon cloud. The \nquark core fully determines the spins and parities of the resonances, while the meson-baryon cloud affects the\nresonance masses, electroexcitation amplitudes, and hadronic decay widths.\n\nAccess to the different components in the resonance structure represents a challenging objective. The credible \nDSEQCD evaluation of the quark core contributions to the electrocouplings of the $N(1440)1\/2^+$ \\cite{Cr15a} \nallows us to estimate the meson-baryon cloud contributions as the difference between the fit of the experimental \nresults and the quark core electroexcitation amplitudes from DSEQCD~\\cite{Cr15a}. The meson-baryon cloud contributions \nto the electrocouplings of the $N(1440)1\/2^+$ state obtained in this way are shown in Fig.~\\ref{p11a12s12dseqm} \nby the thick dashed magenta lines. The meson-baryon cloud contributions to the $A_{1\/2}$ electrocouplings of \nthe $N(1440)1\/2^+$ are maximal for $Q^2 < 1.0$~GeV$^2$. At $Q^2 > 1.0$~GeV$^2$ they rapidly decrease with photon \nvirtualities and become negligible for $Q^2 > 2.0$~GeV$^2$. The meson-baryon cloud contributions to the \n$S_{1\/2}$ electrocouplings of the $N(1440)1\/2^+$ show a rather slow $Q^2$-evolution for \n2.0~GeV$^2$ $< Q^2 <$ 5.0~GeV$^2$. The $S_{1\/2}$ electrocouplings of the $N(1440)1\/2^+$ are proportional to \nthe difference \n\\begin{equation}\n\\label{ropers12}\nS_{1\/2} \\sim F_1^*-\\frac{Q^2}{(M_R-M_N)^2}F_2^* ,\n\\end{equation}\nwhere $M_R$ and $M_N$ are the $N(1440)1\/2^+$ and nucleon masses, respectively. For $Q^2 > 2.0$~GeV$^2$, \nthe contributions from the quark core almost cancel out, making the $S_{1\/2}$ electrocouplings of the\n$N(1440)1\/2^+$ more sensitive to the meson-baryon cloud contributions for $Q^2 > 2.0$~GeV$^2$.\n\nThe analysis of the CLAS data has revealed a substantial dependence of the meson-baryon cloud contributions on the \nquantum numbers of the excited nucleon states and the transition helicity amplitudes. The magnitudes of \nthe meson-baryon dressing amplitudes for the $A_{1\/2}$ electrocouplings of the $N(1520)3\/2^-$ are much smaller \nthan for either the $S_{1\/2}$ or $A_{3\/2}$ electrocouplings (see Fig.~\\ref{d13elcoupl}), as well as with \nthe $A_{1\/2}$ electrocoupling for the $N(1440)1\/2^+$ (see Fig.~\\ref{p11a12s12dseqm}). This makes the $A_{1\/2}$ \nelectrocoupling of the $N(1520)3\/2^-$ attractive for the studies of quark degrees of freedom in the structure \nof the $N(1520)3\/2^-$ resonance. \n\nStudies of the parton content of excited nucleons have been already initiated by the Regensburg University \ntheory group~\\cite{Br09,Br14}. Recent developments in the Light-Cone-Sum-Rule (LCSR) approach allowed us \nfor the first time to determine the partonic structure of the $N(1535)1\/2^-$ resonance~\\cite{Br15} from the \nCLAS experimental results on the electrocouplings of this state for $Q^2 > 2.0$~GeV$^2$~\\cite{Az09}. The analysis \nof the $N(1520)3\/2^-$ electrocouplings within the framework of the LCSR approach were carried out in Ref.~\\cite{Al14}. \nHowever, this approach employs quark distribution amplitudes for the nucleon ground states only. Future LCSR \nevaluations of the $p \\to N(1520)3\/2^-$ electromagnetic transition amplitudes that incorporate the \n$N(1520)3\/2^-$ quark distribution amplitudes are needed in order to explore the partonic structure of the\n$N(1520)3\/2^-$ resonance.\n\n\\subsection{$\\Delta(1620)1\/2^-$ Resonance}\n\\label{imps31}\n\nThe $\\gamma_vpN^*$ electrocouplings and the partial $\\pi \\Delta$ and $\\rho p$ hadronic decay widths (see \nTable~\\ref{hps31}) of the $\\Delta(1620)1\/2^-$ resonance obtained for the first time from CLAS data on $\\pi^+\\pi^-p$ \nelectroproduction off the proton~\\cite{Ri03} have revealed very unusual properties of this state (Fig.~\\ref{s31qm}). \nCurrently it is the only well-established $N^*$ state produced via electroexcitation that is dominated by the \nlongitudinal $S_{1\/2}$ amplitude for 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$. The $\\rho p$ channel opens for the central \n$\\rho$ mass at the threshold of $W$=1.71~GeV. Despite of the much smaller central mass 1.62~GeV, the \n$\\Delta(1620)1\/2^+$ state has a large branching fraction for decays to the $\\rho p$ final state as listed in \nTable~\\ref{hps31}.\n\nThe attempts to describe the electrocouplings of the $\\Delta(1620)1\/2^-$ resonance within the framework of \nthe constituent quark models, accounting for the contributions from only three dressed quarks in the first \norbital excitation that belongs to the [70,1$^-$] $SU(6)$ spin-flavor multiplet, were not successful. As \nshown in Fig~\\ref{s31qm}, the hypercentral constituent quark model~\\cite{Sa12} does allow for a reasonable\ndescription of the longitudinal electrocouplings, but it underestimates the transverse $A_{1\/2}$ \nelectrocouplings. Instead, the above-mentioned Bethe-Salpeter approach~\\cite{Met12} offers a good description \nof the transverse $A_{1\/2}$ electrocoupling of the $\\Delta(1620)1\/2^-$, but underestimates the longitudinal \n$S_{1\/2}$ electrocouplings. The unquenched constituent quark models~\\cite{Sa15a} currently employed in the \nstudies of mesons offer a promising opportunity to explore both the hadron wave functions and the hadronic \ndecays. The extension of these approaches into the baryon sector looks promising in order to understand the nature \nof the $\\Delta(1620)1\/2^-$ resonance from the combined analysis of the electroexcitation amplitudes and the \nhadronic decays of this resonance. \n\nThe large branching fraction for the hadronic decays to the $\\rho p$ final state of the deeply sub-threshold \n$\\Delta(1620)1\/2^-$ state makes it attractive to search for an admixture of exotic configurations such as \n$qqq(q\\bar{q})$ that may facilitate the resonance decays to the $\\rho p$ final state. \n \n\\section{Summary and Outlook}\n\\label{concl}\n\nPhenomenological analysis of CLAS data~\\cite{Ri03} on $\\pi^+\\pi^-p$ electroproduction off the proton at \ninvariant masses of the final hadron system 1.40~GeV $< W <$ 1.82~GeV and photon virtualities $Q^2$ from \n0.5~GeV$^2$ to 1.5~GeV$^2$ was carried out with the primary objective of determining the $\\gamma_vpN^*$ \nresonance electrocouplings and their partial hadronic decay widths to the $\\pi \\Delta$ and $\\rho p$ final states \nfor all prominent $N^*$ states with masses below 1.64~GeV. The JM reaction model~\\cite{Mo09,Mo12} previously \nemployed for the extraction of the resonance parameters from $\\pi^+\\pi^-p$ electroproduction data~\\cite{Fe09} was \nfurther developed for extraction of the resonance parameters in a wider area of $W$ and $Q^2$. In order to \ndescribe the data~\\cite{Ri03} on the final hadron distributions over the $\\alpha_i$ angles for $W > 1.5$~GeV, \nthe phases of the direct double-pion electroproduction amplitudes were implemented and fit to the measured nine \none-fold differential cross sections. The updated JM model provides a good description of all available CLAS \ndata on $\\pi^+\\pi^-p$ electroproduction off the proton at 1.40~GeV $< W <$ 1.82~GeV and $Q^2$ from 0.5~GeV$^2$ \nto 1.5~GeV$^2$. The achieved quality of the data fit~\\cite{Ri03} is comparable to that obtained in reaction \nmodels employed previously for extraction of the resonance electrocouplings from CLAS data on $N\\pi$\n\\cite{Az09,Park15} and $\\pi^+\\pi^-p$~\\cite{Mo12} electroproduction off the proton. The contributions to \ncharged double-pion electroproduction off the proton from all relevant meson-baryon channels and direct double \npion production mechanisms determined from CLAS data within the framework of the updated JM model, shown \nin Fig.~\\ref{integsec}, are of interest for the future modeling of different exclusive meson electroproduction \nchannels that are relevant in the resonance region. These results can also be used in global multi-channel \nanalyses aimed at extraction of the resonance parameters from all available data on exclusive meson \nphoto-, electro-, and hadroproduction. \n\nThe $\\gamma_vpN^*$ electrocouplings of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ resonances were determined from the\nexclusive charged double-pion electroproduction cross sections measured with CLAS at $Q^2$ from 0.5~GeV$^2$ to \n1.5~GeV$^2$. Consistent values of the $N(1440)1\/2^+$ and $N(1520)3\/2^-$ electrocouplings obtained in \nindependent analyses of three $W$-intervals, where the non-resonant contributions are different, strongly \nsupport the reliable extraction of these fundamental quantities. Furthermore, the hadronic decay widths of \nthese resonances to the $\\pi \\Delta$ and $\\rho p$ final states obtained in our analysis are consistent with \nthose previously determined in this exclusive channel at smaller photon virtualities \n$Q^2 < 0.55$~GeV$^2$~\\cite{Mo12}. Successful description of the CLAS $\\pi^+\\pi^- p$ electroproduction data\n\\cite{Fe09,Ri03} in a wide range of photon virtualities from 0.25~GeV$^2$ to 1.5~GeV$^2$ with $Q^2$-independent \nhadronic decay widths of the contributing resonances, supports a reliable separation between the resonant and \nnon-resonant contributions achieved in the updated JM model and confirm reliable extraction of the resonance \nparameters. The $\\Delta(1620)1\/2^-$ resonance decays preferentially to the $N\\pi\\pi$ final state. The \n$\\pi^+\\pi^-p$ exclusive electroproduction off the proton represents the major source of information on the\nelectrocouplings of this resonance. Our studies provide for the first time information on the $\\gamma_vpN^*$ \nelectrocouplings and the $\\pi \\Delta$ and $\\rho p$ partial hadronic decay widths of the $\\Delta(1620)1\/2^-$ \nresonance. \n\nDue to the recent progress in DSEQCD studies of excited nucleon states~\\cite{Cr14,Cr15}, the first evaluations \nof the $p \\to N(1440)1\/2^+$ Dirac $F_1^*$ and Pauli $F_2^*$ transition form factors starting from the QCD \nLagrangian have recently become available~\\cite{Cr15a}. A good description of the CLAS experimental results \nwas obtained at $Q^2 > 2.0$~GeV$^2$ in the DSEQCD approach. In this application the same momentum-dependent \ndressed quark mass function was employed that was also\nused in the previous DSEQCD computations of the nucleon elastic~\\cite{Cr13} and \nmagnetic $p \\to \\Delta$ transition form factors~\\cite{Cr15}. A successful description of \nthe nucleon elastic and electromagnetic transition form factors to excited nucleon states of distinctly \ndifferent structure strongly supports a reliable access to the dressed quark mass function achieved in the \nanalysis~\\cite{Cr15a}. Mapping out the dressed quark mass function from available and future data on \n$p \\to N^*$ transition form factors will address the most challenging and still open problems of the Standard \nModel on the nature of the dominant part of the hadron mass, quark-gluon confinement, and their emergence from \nQCD~\\cite{Az13,Cr14}. These prospects motivate the future studies of the excited nucleon state structure at \nhigh photon virtualities from 5~GeV$^2$ to 12~GeV$^2$ with the CLAS12 detector after the completion of the \nJefferson Lab 12~GeV upgrade~\\cite{Az13,Go12,Ca14,Temple15}. \n\nAnalyses of the experimental results on the $\\gamma_vpN^*$ electrocouplings of the $N(1440)1\/2^+$, \n$N(1520)3\/2^-$, and $\\Delta(1620)1\/2^-$ resonances in the entire range of photon virtualities covered by the \nmeasurements employing the DSEQCD approach~\\cite{Cr15,Cr15a}, advanced quark models~\\cite{Az12,Met12,Ob14}, and \na global multi-channel analysis~\\cite{Lee10,Lee091,Lee08}, have convincingly demonstrated that their structure \nat $Q^2 < 5.0$~GeV$^2$ is determined by a complex interplay between the inner core of three dressed quarks and \nthe external meson-baryon cloud. A successful description of the quark core contributions to the electrocouplings \nof the $N(1440)1\/2^+$ resonance within the framework of DSEQCD~\\cite{Cr15a} makes it possible to outline \nmeson-baryon cloud contributions for this state at the resonant point ($W=M_{N^*}$) from the experimental results \non the $\\gamma_vpN^*$ electrocouplings. We observed pronounced differences for the meson-baryon cloud contributions \nto different electroexcitation amplitudes and their strong dependence on the quantum numbers of the excited nucleon \nstate and photon virtuality. In particular, small contributions from the meson-baryon cloud to the $A_{1\/2}$ \nelectrocouplings of the $N(1520)3\/2^-$ make this resonance attractive for the exploration of its quark components. \nThe studies of resonance electrocouplings over the full spectrum of excited nucleon states of different quantum \nnumbers are critical in order to explore different components in the $N^*$ structure.\n \nAvailable for the first time, $\\Delta(1620)1\/2^-$ resonance electrocouplings and hadronic decay widths to the \n$\\pi \\Delta$ and $\\rho p$ final states have demonstrated a rather peculiar behavior. The $\\Delta(1620)1\/2^-$ \nstate is the only known resonance produced via electroexcitation that is dominated by the longitudinal $S_{1\/2}$ \nelectrocoupling in a wide range of photon virtualities 0.5~GeV$^2$ $< Q^2 <$ 1.5~GeV$^2$. Furthermore, the\n$\\Delta(1620)1\/2^-$ resonance has a large branching fraction (above 30\\%) for the decay into the $\\rho N$ final \nstates. This is a rather unusual feature for decays of a resonance located in the deeply sub-threshold region for \nthe production of the $\\rho p$ final state. Failures in describing the $\\Delta(1620)1\/2^-$ electrocouplings within \nthe framework of quark models that account for the contributions of the quark core only~\\cite{Sa12,Met12}, indicate \nthat the structure of this state can be more complex than that assumed in quark models described by the orbital \nexcitation of three quarks with the total orbital momentum $L=1$. Further experimental data are needed in order to \nestablish the nature of this state. In the near term future, new CLAS results on the $\\Delta(1620)1\/2^-$ \nelectrocouplings are expected at photon virtualities from 0.3~GeV$^2$ to 1.0~GeV$^2$ with a much finer $Q^2$-binning \n\\cite{Fe12}. The results on the $\\Delta(1620)1\/2^-$ electrocouplings from CLAS data on charged double-pion \nelectroproduction off the proton eventually will also be extended in $Q^2$ up to 5.0~GeV$^2$. Further developments in \nhadron structure theory that will allow us to perform a combined analysis of the resonance electrocouplings and \nhadronic decay widths are critical in order to understand the nature of the $\\Delta(1620)1\/2^-$ state. A search for \ncontributions from exotic $qqq(q\\bar{q})$ configurations to the structure of this state that may facilitate the decays \nof the $\\Delta(1620)1\/2^-$ resonance to the $\\rho p$ final state are of particular interest. \n \n\\section{Acknowledgments}\n\nWe would like to acknowledge the outstanding efforts of the staff of the Accelerator and the Physics Divisions \nat Jefferson Lab that made this evaluation of the $N(1440)1\/2^+$, $N(1520)3\/2^-$, and $\\Delta(1620)1\/2^-$ \nelectrocouplings and hadronic decay parameters possible. We are grateful to I.G. Aznauryan, V.M. Braun, \nI.C. Cl\\\"{o}et, M.M. Giannini, T-S.~H. Lee, M.R. Pennington, C.D. Roberts, E. Santopinto, J. Segovia, and \nA.P. Szczepaniak for theoretical support and helpful discussions. This work was supported in part by the U.S. \nDepartment of Energy and the National Science Foundation, the Skobeltsyn Institute of Nuclear Physics and the\nPhysics Department at Moscow State University, Ohio University, and the University of South Carolina. The \nSoutheastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator \nFacility for the United States Department of Energy under contract DE-AC05-84ER40150.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nDenoising Diffusion Models (DDMs) \\cite{sohl2015deep}\\cite{ho2020denoising}\\cite{song2019generative} have emerged as a powerful class of generative models. DDMs learn to reverse a gradual multi-step noising process to match a data distribution. Samples are then produced by a Markov Chain that starts from white noise and progressively denoises it into an image. This class of models has shown excellent capabilities in synthesising high-quality images \\cite{nichol2021improved}\\cite{rombach2022high}, audio \\cite{chen2020wavegrad}, and 3D shapes \\cite{zhou20213d} \\cite{zenglion}, recently outperforming Generative Adversarial Networks (GANs) \\cite{goodfellow2014generative} on image synthesis.\n\nHowever, GANs require a single forward pass to generate samples, while the iterative DDM design requires hundreds or thousands of inference timesteps and, consequently, forward passes through a denoising neural network.\nThe slow sampling process thus represents one of the most significant limitations of DDMs. It is well-known that there is a trade-off between sample quality and speed, measured in the number of timesteps \\cite{song2020denoising}\\cite{nichol2021improved}. However, it is currently unclear how low the number of timesteps can be pushed while retaining high quality for a given data distribution \\cite{chen2020wavegrad}.\nThis issue is the focus of a lot of current research in the field, with recent works proposing acceleration solutions which can be divided into two categories: learning-free sampling and learning-based sampling.\nThe learning-free approach focuses on modifying the sampling process without the need for training \\cite{song2020denoising}\\cite{song2020score}\\cite{jolicoeur2021gotta}\\cite{karras2022elucidating}, while the learning-based approach uses techniques such as truncation \\cite{lyu2022accelerating}\\cite{zheng2022truncated}, knowledge distillation \\cite{salimans2021progressive}\\cite{luhman2021knowledge}, dynamic programming \\cite{watson2021learning} and differentiable sampler search \\cite{watson2022learning} to improve sampling speed. \n\nIn this paper, we propose MMD-DDM, a technique to finetune a pretrained DDM with a large number of timesteps in order to optimize the features of the generated data under the constraint of a reduced number of timesteps. This is done by directly optimizing the weights of the denoising neural network via backpropagation through the sampling chain. The minimization objective is the Maximum Mean Discrepancy (MMD) \\cite{gretton2006kernel} between real and generated samples in a perceptually-relevant feature space. This allows to specialize the model for a fixed and reduced computational budget with respect to the original training; the use of MMD represents a different and, possibly complementary, objective to the original denoising loss. Our proposed approach is extremely fast, requiring only a small number of finetuning iterations. Indeed, the finetuning procedure can be performed in minutes, or at most few hours for more complex datasets, on standard hardware. Moreover, it is agnostic to the sampling procedure, making it appealing even for future models employing new and improved procedures. \n\nExtensive experimental evaluation suggests that the proposed solution significantly outperforms state-of-the-art approaches for fast DDM inference. MMD-DDM is able to substantially reduce the number of timesteps required to reach a target fidelity. We also need to remark that the choice of feature space for the MMD objective may artificially skew results, if the evaluation metric is based on the same feature space being optimized, such as Inception features and the FID score. We discuss the importance of this point for fair evaluation and present results on different metrics and features spaces, such as CLIP features \\cite{radford2021learning}, in order to present a fair assessment of the method.\n\n\n\n\\section{Background and Related Work}\n\\subsection{Denoising Diffusion Models}\nDDMs (\\cite{ho2020denoising}\\cite{sohl2015deep}) leverage the diffusion process to model a specific distribution starting from random noise. They are based on a predefined Markovian forward process, by which data are progressively noised in $T$ steps. $T$ is set to be sufficiently large such that ${\\bm{x}}_T$ is close to white Gaussian noise (in practice, $T \\geq 1000$ is often used). The forward process can be written as:\n\\begin{eqnarray}\n\\label{eqn:fwd}\n q({\\bm{x}}_0,...,{\\bm{x}}_T) &=& q({\\bm{x}}_0)\\prod_{t=1}^T q({\\bm{x}}_t|{\\bm{x}}_{t-1}) \\\\\n q({\\bm{x}}_t|{\\bm{x}}_{t-1}) &=& \\mathcal{N}({\\bm{x}}_t|\\sqrt{1-\\beta_t}{\\bm{x}}_s, \\beta_t {\\bm{I}}) \n\\end{eqnarray}\nwhere $q({\\bm{x}}_0)$ denotes the real data distribution and $\\beta_t$ the variance of the Gaussian noise at timestep $t$.\nThe reverse process traverses the Markov Chain backwards and can be written as:\n\\begin{eqnarray}\n p_\\theta({\\bm{x}}_{0:T}) &=& p({\\bm{x}}_T) \\prod\\nolimits_{t=1}^T p_\\theta({\\bm{x}}_{t-1} \\mid {\\bm{x}}_t) \\\\\n p_\\theta({\\bm{x}}_{t-1} \\mid {\\bm{x}}_{t}) &=& \\mathcal{N}({\\bm{x}}_{t-1} \\mid \\mu_{\\theta}({{\\bm{x}}}_{t}, t), \\sigma_t^2\\bm{I})~. \\label{eq:reverse_process}\n\\end{eqnarray}\nThe parameters of the learned reverse process $p_\\theta$ can be optimized by maximizing an evidence lower bound (ELBO) on the training set.\nUnder a specific parametrization choice \\cite{ho2020denoising}, the training objective can be simplified to that of a noise conditional score network \\cite{vincent2011connection} \\cite{song2019generative}: \n\\begin{align}\n\\min_\\theta \\mathcal{L}(\\theta) = \\mathbb{E}_{x_0, \\epsilon, t} || \\epsilon - \\epsilon_{\\theta}({\\bm{x}}_t, t)||^2_2\n\\end{align}\nwhere ${\\bm{x}}_0 \\sim q_\\text{data}$, $\\epsilon \\sim \\mathcal{N}(0, \\bm{I})$ and $t$ is uniformly sampled from \\{$1,...,T$\\}.\n\n\\subsection{Accelerated Sampling for DDMs}\\label{sec:related}\n\nAccelerated DDM sampling is currently a hot research topic. At a high level, the different approaches can be divided into two categories: learning-free sampling and learning-based sampling \\cite{yang2022diffusion}. The learning-free approaches do not require training and instead focus on modifying the sampling process to make it more efficient. One example is the work of Song et al. \\cite{song2020denoising} (DDIM), in which they define a new family of non-Markovian diffusion processes that maintains the same training objectives as a traditional DDPM. They demonstrate that alternative ELBOs may be built using only a sub-sequence of the original timesteps $\\tau \\in$ \\{$1,...,T$\\}, obtaining faster samplers compatible with a pre-trained DDPM.\nOther works focus on using the Score SDE formulation \\cite{song2020score} of continuous-time DDMs to develop faster sampling methods. For example, Song et al. \\cite{song2020score} propose the use of higher-order solvers such as Runge-Kutta methods, while Jolicoeur et al. \\cite{jolicoeur2021gotta} propose the use of SDE solvers with adaptive timestep sizes. Another approach is to solve the probability flow ODE, which has been shown by Karras et al. \\cite{karras2022elucidating} to provide a good balance between sample quality and sampling speed when using Heun's second-order method. Additionally, customized ODE solvers such as the DPM-solver \\cite{lu2022dpm} and the Diffusion Exponential Integrator sampler \\cite{zhang2022fast} have been developed specifically for DDMs and have been shown to be more efficient than general solvers. These methods provide efficient and effective ways to speed up the sampling process in continuous-time DDMs. \n\nThe other main line of approaches for efficient sampling is the learning-based one. Some of these approaches \\cite{lyu2022accelerating}\\cite{zheng2022truncated} involve truncating the forward and reverse diffusion processes to improve sampling speed, while others \\cite{salimans2021progressive} \\cite{luhman2021knowledge} use knowledge distillation to create a faster model that requires fewer steps. Another approach (GENIE) \\cite{dockhorngenie}, based on truncated Taylor methods, trains an additional model on top of a first-order score network to create a second-order solver that produces better samples with fewer steps. Dynamic programming techniques \\cite{watson2021learning} have also been used to find the optimal discretization scheme for DDMs by selecting the best time steps to maximize the training objective, although the variational lower bound does not correlate well with sample quality, limiting the performance of the method. In a successive work \\cite{watson2022learning}, the sampling procedure was directly optimized using a common perceptual evaluation metric (KID) \\cite{binkowski2018demystifying}, but this required a long training time (30k training iterations).\nIn this work. the authors backpropagate through the sampling chain using reparametrization and gradient rematerialization in order to make the optimization feasible. Our work is closely related to \\cite{watson2022learning}, since we similarly backpropagate through the sampling chain. However, we use the MMD \\cite{gretton2006kernel}\\cite{gretton2012kernel} to finetune the weights of a pretrained DDM without optimizing the sampling strategy. In essence, the proposed method is complementary to \\cite{watson2022learning}: instead of optimizing the sampling procedure, keeping the model fixed, we directly optimize the model leaving the sampling procedure unchanged. This leads to better results with as few as 500 finetuning iterations. We also remark that our approach is decoupled from the sampling strategy and can be used in conjunction with other training-free acceleration methods such as DDIM.\n\n\n\\subsection{MMD in Generative Models}\nThe MMD \\cite{gretton2006kernel}\\cite{gretton2012kernel} is a distance on the space of probability measures. It is a non-parametric approach that does not make any assumptions about the underlying distributions, and can be used to compare a wide range of distributions. Generative models trained by minimizing the MMD were first considered in \\cite{pmlr-v37-li15}\\cite{dziugaite2015training}. These works optimized a generator to minimize the MMD with a fixed kernel, but struggled with the complex distribution of natural images where pixel distances are of little value. Successive works \\cite{li2017mmd} \\cite{binkowski2018demystifying} addressed this problem by adversarially learning the kernel for the MMD loss, reaching results comparable to GANs trained with a Wasserstein critic. In this work we apply the MMD in the context of diffusion models, demonstrating its effectiveness in finetuning a pretrained DDM under a more restrictive timesteps constraint.\n\n\n\n\n\n\\section{Method}\n\n\\subsection{Overview}\nWe propose MMD-DDM, a technique to accelerate inference in DDMs while maintaining high sample quality, based on finetuning a pretrained diffusion model. The finetuning process minimizes an unbiased estimator of the MMD between real and generated samples, evaluated over a perceptually-relevant feature space. We backpropagate through the sampling process with the aid of the reparametrization trick and gradient checkpointing. This is done only for a small subset of the original timesteps and can be combined with existing techniques for timestep selection or acceleration of the sampling process.\nThe reduction in timesteps with respect to the original model degrades the distribution of the generated data. However, the main idea behind the proposed approach is that it is possible to recover part of this degradation by analyzing the generated data in a perceptual feature space and imposing that the reduced DDM produces perceptual features similar to those of real data via MMD minimization. \nBy utilizing this approach, we are thus able to maximize the model performance under a fixed computational budget. It is interesting to notice that older approaches that utilized MMD as sole objective for image generation failed to capture their complex data distribution. On the other hand, our approach avoids that as it leverages the strong baseline provided by the pretrained DDM, albeit degraded by the timesteps constraint.\n\n\n\\subsection{Finetuning with MMD}\nWe are interested in learning a model distribution $p_{\\theta}({\\bm{x}}_0)$ that approximates the real data distribution $q({\\bm{x}}_0)$. Starting from a pretrained diffusion model, we know from previous work (DDIM \\cite{song2020denoising}) that it is possible to sample from $p_{\\theta}^{(\\mathcal{T})}({\\bm{x}}_0)$, i.e., the learned distribution using a subset of the original timesteps $\\mathcal{T} \\subset $ \\{$1,...,T$\\}, accepting a complexity-quality tradeoff.\nThe MMD \\cite{gretton2006kernel} is an integral probability metric that we use to measure the discrepancy between the real data distribution $q({\\bm{x}}_0)$ and the generated data distribution with the given budget of timesteps $p_{\\theta}^{(\\mathcal{T})}({\\bm{x}}_0)$. Mathematically, it is defined as: \n\\begin{align}\n\\label{eq:mmdprimal}\n\\centering\n\\mathrm{MMD}(p_\\theta^{(\\mathcal{T})},q) = \\Vert \\mathbb{E}_{{\\bm{x}} \\sim p_\\theta^{(\\mathcal{T})}} \\varphi({\\bm{x}}) - \\mathbb{E}_{{\\bm{y}} \\sim q}\\varphi({\\bm{y}}) \\Vert\n\\end{align}\nwhere $\\varphi$ represents a function mapping raw images to a perceptually-meaningful feature space. This is needed as MMD would not perform well on the pixel space, since it is well known that images live on a low-dimensional manifold within the high-dimensional pixel space. However, once the images are mapped into an appropriate feature space, MMD is proven to have strong discriminative performances, as proved by the success of the KID \\cite{binkowski2018demystifying} as evaluation metric for perceptual quality. The choice of feature space is critical for the performance of the proposed method and for the fair assessment of methods optimizing quality metrics, which will be presented in Secs. \\ref{sec:feature_spaces} and \\ref{sec:feature}.\n\nIn order to use the MMD as our loss function, given a batch of generated samples $\\{{{\\bm{x}}_i}\\}_{i=1}^N \\sim p_{\\theta}^{(\\mathcal{T})}({\\bm{x}}_0)$ and a batch of real samples $\\{{{\\bm{y}}_i}\\}_{i=1}^N \\sim q({\\bm{x}}_0)$, we use the unbiased estimator proposed by Gretton et al. \\cite{gretton2012kernel}: \n\\begin{align}\n\\centering\n \\mathcal{L}_{\\mathrm{MMD}^2}^\\text{unbiased} = &\\frac{1}{N(N-1)}\\sum_{i\\neq j}^n k(\\phi({\\bm{x}}_i),\\phi({\\bm{x}}_j)) \\nonumber \\\\ \n & -\\frac{2}{N^2}\\sum_{i=1}^N\\sum_{j=1}^N k(\\phi({\\bm{x}}_i),\\phi({\\bm{y}}_j)) + c.\n\\end{align}\nwhere $N$ is the batch size, $c$ is a constant, and $k$ is a generic positive definite kernel (in our experiments we consider linear, cubic and Gaussian kernels, see Sec. \\ref{sec:ablations}). The loss function is minimized in order to finetune the values of the parameters $\\theta$ of a pretrained denoising neural network composing the diffusion model. Next, we are going to discuss the choice of the feature extraction function $\\phi$. \n\n\\begin{table*}[t]\n \\caption{Unconditional CIFAR-10 generative performance (Inception FID).}\n \n \\label{cifar-res}\n \\begin{center}\n \\begin{tabular}{l c c c c}\n \\hline\n Method & $\\vert \\mathcal{T} \\vert=5$ & $\\vert \\mathcal{T} \\vert=10$ & $\\vert \\mathcal{T} \\vert=15$ & $\\vert \\mathcal{T} \\vert=20$ \\\\%& NFEs=25 \\\\ %\n \\hline\\hline\n DDPM \\cite{ho2020denoising}& 76.3 & 42.1 & 31.4 & 25.9 \\\\\n DDIM~\\cite{song2020denoising} & 32.7 & 13.6 & 9.31 & 7.50 \\\\\n \\textbf{DDIM + MMD-DDM} (Inception-V3) & \\textbf{5.48} & \\textbf{3.80} & \\textbf{4.11} & \\textbf{3.55} \\\\\n \\textbf{DDIM + MMD-DDM} (CLIP) & 6.79 & 4.87 & 4.79 & 4.52 \\\\\n \\hline\n GENIE \\cite{dockhorngenie} & 13.9 & 5.97 & 4.49 & 3.94\\\\% & 3.67 \\\\ %\n PNDM~\\cite{liu2022pseudo} & 35.9 & 10.3 & 6.61 & 5.20 \\\\%& 4.51 \\\\%& 3.30 \\\\\n FastDPM~\\cite{kong2021fast} & - & 9.90 & - & 5.05\\\\% & - \\\\\n Learned Sampler~\\cite{watson2022learning} & 13.8 & 8.22 & 6.12 & 4.72 \\\\%& 4.25 \n Analytic DDIM~\\cite{bao2022analyticdpm} & - & 14.7 & 9.16 & 7.20\\\\% & 5.71 &\\\\% 4.04 \\\\\n DPM-Solver(Type-1) \\cite{lu2022dpm} & - & 6.37 & 3.78 & 4.28 \\\\\n DPM-Solver(Type-2) \\cite{lu2022dpm} & - & 10.2 & 4.17 & 3.72 \\\\\n \\hline\n \\end{tabular}\n \\end{center}\n\\end{table*}\n\n\\begin{table*}[t]\n \\caption{Unconditional CelebA generative performance (Inception FID).}\n \\label{celeb-res}\n \\begin{center}\n \\begin{tabular}{l c c c c}\n \\hline\n Method & $\\vert \\mathcal{T} \\vert=5$ & $\\vert \\mathcal{T} \\vert=10$ & $\\vert \\mathcal{T} \\vert=15$ & $\\vert \\mathcal{T} \\vert=20$ \\\\%& NFEs=25 \\\\ %\n \\hline\\hline\n DDIM~\\cite{song2020denoising} & 22.4 & 17.3 & 16.0 & 13.7 \\\\\n \\textbf{DDIM + MMD-DDM} (Inception-V3) & \\textbf{3.04} & \\textbf{2.58} & \\textbf{2.13} & \\textbf{2.24} \\\\\n \\textbf{DDIM + MMD-DDM} (CLIP) & 4.65 & 3.90 & 3.17 & 3.27 \\\\\n \\hline\n ES+StyleGAN2+DDIM \\cite{lyu2022accelerating} & 9.15 & 6.44 & - & 4.90 \\\\\n PNDM~\\cite{liu2022pseudo} & 11.3 & 7.71 & - & 5.51 \\\\%& 4.51 \\\\%& 3.30 \\\\\n FastDPM~\\cite{kong2021fast} & - & 15.3 & - & 10.7\\\\% & - \\\\\n Diffusion Autoencoder \\cite{preechakul2022diffusion} & - & 12.9 & - & 10.2 \\\\\n Analytic DDPM \\cite{bao2022analyticdpm} & - & 29.0 & 21.8 & 18.1 \\\\\n Analytic DDIM \\cite{bao2022analyticdpm} & - & 15.6 & 12.3 & 10.45 \\\\\n DPM-Solver(Type-1) \\cite{lu2022dpm} & - & 6.92 & 3.05 & 2.82 \\\\\n DPM-Solver(Type-2) \\cite{lu2022dpm} & - & 5.83 & 3.11 & 3.13\\\\\n \\hline\n \\end{tabular}\n \n \\end{center}\n\\end{table*}\n\n\\begin{table}[t]\n \\caption{Unconditional ImageNet generative performance (Inception FID).}\n \n \\label{imagenet-res}\n \\setlength\\tabcolsep{1pt} \n \\begin{center}\n \\begin{tabular}{l c c c }\n \\hline\n Method & $\\vert \\mathcal{T} \\vert=5$ & $\\vert \\mathcal{T} \\vert=10$ & $\\vert \\mathcal{T} \\vert=20$ \\\\\n \\hline\\hline\n DDIM~\\cite{song2020denoising} & 131.5 & 35.2 & 20.7 \\\\\n \\textbf{DDIM + MMD-DDM} (Inc-V3) & 33.1 & 21.1 & \\textbf{12.4} \\\\\n \\textbf{DDIM + MMD-DDM} (CLIP) & \\textbf{27.5} & \\textbf{16.4} & 14.5 \\\\\n \\hline\n Learned Sampler \\cite{watson2022learning} & 55.1 & 37.2 & 24.6\\\\\n Analytic-DDIM \\cite{bao2022analyticdpm} & - & 70.6 & 30.9 \\\\\n Analytic-DDPM \\cite{bao2022analyticdpm} & - & 60.6 & 37.7 \\\\\n DPM-Solver(T2) \\cite{lu2022dpm} & - & 24.4 & 18.53\\\\\n \\hline\n \\end{tabular}\n \n \\end{center}\n\\end{table}\n\n\\begin{table}[t]\n \\caption{Unconditional LSUN-Church Outdoor generative performance (Inception FID).}\n \n \\label{lsun-res}\n \\setlength\\tabcolsep{1pt} \n \\begin{center}\n \\begin{tabular}{l c c c c}\n \\hline\n Method & $\\vert \\mathcal{T} \\vert=5$ & $\\vert \\mathcal{T} \\vert=10$ & $\\vert \\mathcal{T} \\vert=20$ \\\\\n \\hline\\hline\n DDIM~\\cite{song2020denoising} & 49.6 & 19.4 & 12.5 \\\\\n \\textbf{DDIM + MMD-DDM} (Inc-V3) & \\textbf{4.75} & \\textbf{7.55} & \\textbf{6.21} \\\\ \n \\textbf{DDIM + MMD-DDM} (CLIP) & 14.2 & 10.7 & 8.82 \\\\\n \\hline\n S-PNDM \\cite{liu2022pseudo} & 20.5 & 11.8 & 9.20 \\\\\n F-PNDM \\cite{liu2022pseudo} & 14.8 & 8.69 & 9.13 \\\\\n \n \n \\hline\n \\end{tabular}\n \n \\end{center}\n\\end{table}\n\n\\subsection{Perceptually-Relevant Feature Spaces}\n\\label{sec:feature_spaces}\n\nAs we previously mentioned, it is necessary to embed real and generated images in some perceptually-relevant feature space, so that the MMD objective could be effective. The feature mapping network $\\phi$ plays a crucial role in the performance of the method. However, this is not a trivial choice. The most popular choice could be to use the feature space of the penultimate layer of an ImageNet-pretrained Inception-V3 classifier \\cite{szegedy2016rethinking}. This choice is widely used to evaluate performance of generative models, with Inception Score (IS) \\cite{salimans2016improved}, FID \\cite{heusel2017gans} and KID \\cite{binkowski2018demystifying} all using it. \n\nHowever, a recent study \\cite{kynkaanniemi2022role} has examined the effectiveness of using ImageNet-pretrained representations to evaluate generative models, and found that the presence of ImageNet classes has a significant impact on the evaluation. The study highlights some potential pitfalls in using these metrics, and how they can be manipulated by the use of ImageNet pretraining. This suggests that care should be taken when using ImageNet features to optimize generative models as this can potentially distort the FID quality metric and make it unreliable. Indeed, for any image generation method, part of the improvement might lie in the \\textit{perceptual null space} \\cite{kynkaanniemi2022role} of FID, which encompasses all the operations that change the FID without affecting the generated images in a perceptible way. For our finetuning procedure, we have experimentally observed a better overall visual quality of generated images and a consistent gain in FID, when optimizing MMD with Inception features. However, it is hard to quantitatively assess how much of this improvement is due to actual perceptual improvements versus optimizations in the perceptual null space. These considerations apply also to the work of Watson et al. \\cite{watson2022learning}. \n\nOne solution to this problem is to use a different feature space for the feature mapping network, such as one that has not been pretrained on ImageNet. Thus, we propose to optimize the MMD using the feature space of the CLIP image encoder \\cite{radford2021learning}, which has been trained in a self-supervised way and is supposed to have richer representations without exposure to ImageNet classes. Moreover, we also consider the case in which we optimize MMD with Inception features and measure performance with a variant of FID using CLIP features. \nMore comments, details, and a discussion of the various results can be found in Sec. \\ref{sec:feature}.\n\n\\begin{figure*}[t!]\n\\label{figure:res2}\n\\vspace{-0.4cm}\n\\begin{center}\n\n\\begin{tabular}{@{}c@{\\hspace{.1cm}}c@{\\hspace{.1cm}}c@{\\hspace{.1cm}}c@{\\hspace{.35cm}}c@{\\hspace{.1cm}}c@{}}\n & DDIM & MMD-DDM (Inception-V3) & MMD-DDM (CLIP) & Reference & \\\\\n \n \\raisebox{.2cm}{\\rotatebox{90}{\\textit{LSUN-Church}}} & \n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/church_ddim_5.png}} & \n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/church_inception_5.png}} & \n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/church_clip_5.png}} &\n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/church_reference.png}} \\\\ \n \\raisebox{.6cm}{\\rotatebox{90}{\\textit{ImageNet}}} & \n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/imagenet_ddim_10.png}} & \n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/imagenet_inception_10.png}} & \n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/imagenet_clip_10.png}} &\n \\makebox{\\includegraphics[width=.225\\linewidth]{figures\/imagenet_reference.png}} \\\\\n\\end{tabular}\n}\n\\end{center}\n\n\\caption{Generated samples for LSUN-Church (top) and ImageNet (bottom). The samples are obtained using 5 timesteps for LSUN-Church and 10 timesteps for ImageNet, with the DDIM sampling procedure. Results from Standard DDIM (left), the same model finetuned using Inception-V3 features (center-left) and CLIP features (center-right), reference images from the dataset (right). Samples are not cherry-picked.\n\\label{fig:samples2}}\n\\vspace{-0.1cm}\n\\end{figure*}\n\n\\section{Experiments}\n\n\n\\subsection{Setting}\n\\paragraph{Datasets} In order to demonstrate the effectiveness of the proposed solution, we validate it on several datasets with different resolutions. We use CIFAR-10 \\cite{krizhevsky2009learning} at resolution $32\\times32$, CelebA \\cite{liu2015faceattributes} at resolution $64\\times64$, Image-Net \\cite{IMAGENET} at resolution $64\\times64$, and LSUN-Church \\cite{yu2015lsun} at resolution $256\\times256$.\n\n\\paragraph{Models and Sampling} We use the models pretrained by Ho et al. \\cite{ho2020denoising} for the CIFAR-10 and LSUN experiments, the model pretrained by Song et al. \\cite{song2020denoising} for CelebA, and the model pretrained by Nichol and Dhariwal \\cite{nichol2021improved} with the $L_\\text{hybrid}$ objective for ImageNet. All the architectures are based on the modified UNet \\cite{ronneberger2015u} that incorporates self-attention layers \\cite{vaswani2017attention}. We perform our experiments using the efficient sampling strategy of DDIM \\cite{song2020denoising}, as it already has good performance in few-timesteps regime. We fix the timestep schedule in the main experiments to be linear. The MMD kernel is polynomial cubic in all experiments, except the kernel ablation one.\nWe also test the proposed solution with the DDPM \\cite{ho2020denoising} sampling strategy in Sec. \\ref{sec:ablations}.\n\n\\paragraph{Evaluation} We use the FID \\cite{heusel2017gans} to evaluate sample quality. All the values are evaluated by comparing 50k real and generated samples as this is the literature's standard. We also use FID$_\\text{CLIP}$ \\cite{kynkaanniemi2022role} in some experiments to remove the effect of Image-Net classes in the evaluation. Additional evaluation metrics such as Inception Score \\cite{salimans2016improved}, Spatial FID \\cite{nash2021generating}, and Precision and Recall \\cite{sajjadi2018assessing} can be found in the Supplementary Material.\n\n\\paragraph{Implementation Details} For all the experiments we set the batch size equal to 128. We use Adam as optimizer \\cite{kingma2015adam} with $\\beta_1 = 0.9$, $\\beta_2 = 0.999$, $\\epsilon = 1 \\times 10^{-8}$ and learning rate equal to $5\\times 10^{-6}$. When DDIM is used, we set $\\sigma_t=0$. As feature extractors, we use the standard Inception-V3\\footnote{http:\/\/download.tensorflow.org\/models\/image\/imagenet\/inception-2015-12-05.tgz} pretrained on Image-Net, and the ViT-B\/32\\footnote{https:\/\/github.com\/openai\/CLIP} model from CLIP \\cite{radford2021learning}. We use \\textit{torch-fidelity} \\cite{obukhov2020torchfidelity} for the FID evaluation. We train all the models for about 500 iterations. Finetuning with a budget of 5 timesteps required about 10 minutes for CIFAR-10, about 45 minutes for CelebA, and about one hour for ImageNet on a single Nvidia RTX A6000. For LSUN-Church and for the other timesteps budgets, finetuning has been performed on four Nvidia RTX A6000. Finetuning for 5 timesteps of LSUN-Church required about two hours on the mentioned hardware.\n\n\n\\begin{table*}[t!]\n \\caption{Comparison of relative improvements evaluating FID in Inception-V3 feature space versus CLIP feature space.}\n \\vspace{10pt}\n \\label{fid_clip}\n \\begin{center}\n \\begin{tabular}{l c c c c}\n \\hline\n & \\multicolumn{2}{c}{$\\vert \\mathcal{T} \\vert=5$} & \\multicolumn{2}{c}{$\\vert \\mathcal{T} \\vert=10$} \\\\\\hline \\hline\n \\multicolumn{5}{c}{\\textit{CIFAR-10}} \\\\ \\hline\n & FID & FID$_\\text{CLIP}$ & FID & FID$_\\text{CLIP}$ \\\\\n \\hline \n DDIM & 32.7 & 13.7 & 13.6 & 6.87 \\\\\n \\textbf{DDIM + MMD-DDM} (Inception-V3) & 5.48 & 2.11 & 3.80 & 2.01 \\\\\n Improvement & -83.2\\% & -84.4\\% & -72.0\\% & -70.7\\% \\\\ \n \\hline \\hline\n \\multicolumn{5}{c}{\\textit{CelebA}} \\\\\\hline\n & FID & FID$_\\text{CLIP}$ & FID & FID$_\\text{CLIP}$ \\\\\\hline\n DDIM & 22.4 & 12.2 & 17.3 & 9.48 \\\\\n \\textbf{DDIM + MMD-DDM} (Inception-V3) & 3.04 & 4.94 & 2.58 & 4.26 \\\\\n Improvement & -86.4\\% & -59.3\\% & -85.0\\% & -55.0\\% \\\\ \n \\hline \\hline\n \\multicolumn{5}{c}{\\textit{ImageNet}} \\\\\\hline\n & FID & FID$_\\text{CLIP}$ & FID & FID$_\\text{CLIP}$ \\\\\\hline\n DDIM & 131.5 & 29.5 & 35.2 & 11.9 \\\\\n \\textbf{DDIM + MMD-DDM} (Inception-V3) & 33.1 & 15.2 & 21.1 & 9.21 \\\\\n Improvement & -74.8\\% & -56.8\\% & -40.0\\% & -22.6\\% \\\\ \n \\hline\n \\end{tabular}\n \\end{center}\n\\end{table*}\n\\subsection{Image Generation Results}\n\nWe evaluate MMD-DDM using the following timesteps budgets: $\\vert \\mathcal{T} \\vert \\in \\{5, 10, 15, 20\\}$. We report the values of FID on unconditional generation experiments for CIFAR-10 in Table \\ref{cifar-res}, for CelebA in Table \\ref{celeb-res}, for ImageNet in Table \\ref{imagenet-res} and for LSUN-Church in Table \\ref{lsun-res}.\n\nWe compare against several state-of-the-art methods for accelerating DDMs. The tables report the results for MMD-DDM trained with Inception features and we also report results taken from literature for other methods. \nFor all datasets and timesteps budgets, MMD-DDM provides superior or, occasionally, comparable quality to state-of-the art approaches.\nFor the ImageNet experiment, we remark that we report the result of the Learned Sampler approach \\cite{watson2022learning}, which uses an improved version of the model from \\cite{nichol2021improved} trained for 3M iterations, instead of the 1.5M iterations used by our checkpoint, thus making the comparison slightly unfavourable for our method.\nWe do not compare with the progressive distillation method \\cite{salimans2021progressive}, as it cannot be considered a post-training acceleration technique but rather a very computationally-demanding modification of the DDM training procedure.\n\nQualitative comparisons for CIFAR-10 and CelebA are shown in Fig. \\ref{fig:samples} and for LSUN-Church and ImageNet in Fig. \\ref{fig:samples2}. More generated samples, for different numbers of timesteps, can be found in the Supplementary Material. It can be noticed that MMD-DDM provides substantial improvements in visual quality when the number of timesteps is highly constrained. As the available timesteps budget is relaxed to 20 or more, the improvement provided MMD-DDM diminishes, although all approaches start providing high quality samples.\n\n\n\n\\subsection{Feature Space Discussion}\\label{sec:feature}\n\nResults in the previous section were presented with the commonly-used FID metric exploiting Inception features. However, as detailed in Sec. \\ref{sec:feature}, our optimization of Inception features via the MMD loss could raise concerns about the reliability of the FID metric. In this section, we present results using the CLIP feature space in either the MMD loss or the FID metric.\n\nFigs. \\ref{fig:samples} and \\ref{fig:samples2} already show a visual comparison between using the MMD with Inception features and CLIP features and more results are present in the Supplementary Material. It can be noticed that optimizing over CLIP features leads to higher visual quality, including sharper details and clarity, confirming that the CLIP space is a superior embedding of perceptually-relevant features. As a reference, we also report the FID scores obtained by MMD-DDM with CLIP features in Tables \\ref{cifar-res},\\ref{celeb-res},\\ref{imagenet-res},\\ref{lsun-res}. Notice that lower values are observed, possibly due to the reliance of the FID on flawed Inception features and the metric not accurately tracking a genuine improvement in visual quality. \n\nWe further expand the set of results in Table \\ref{fid_clip}, in which we optimize Inception features with the MMD but then measure quality using the FID computed on CLIP features (FID$_\\text{CLIP}$), as proposed in \\cite{kynkaanniemi2022role}. Since the feature space used for evaluation is different from the one used in optimization, the observed gains in FID$_\\text{CLIP}$ suggest us that the quality improvement in quantitatively meaningful and not just an artifact of the metric. Percentage improvements in FID$_\\text{CLIP}$ mostly track those of regular FID, albeit being lower in some cases, suggesting that some overfitting of the perceptual null space does indeed happen when Inception features are used for both MMD and FID.\n\n\n\\subsection{Analysis of Overfitting}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{figures\/NN_celeb_3.png}\\\\\n\\caption{Generated samples by the DDIM model (top) and the finetuned model (bottom) for CelebA. For each generated samples we visualize the top-4 nearest neighbours.}\n\\label{figure:NN}\n\\end{figure}\n\nOne might wonder whether finetuning via the MMD loss leads to images overfitting the features of the training set. This section presents a experiment to dispel this concern.\nTo do so, we looked at the top-$K$ nearest neighbors of generated samples when the CLIP feature space is used for both the optimization with the MMD and the space for nearest neighbor search (Euclidean distance betweeen CLIP features). \nFig. \\ref{figure:NN} provides the results of this experiment for samples generated with both the pretrained model and the finetuned model. \nWe can see that the nearest neighbors of the samples generated after finetuning are not more significantly similar to the generated image than those for the pretrained model. More samples can be found in the Supplementary material.\n\n\n\\subsection{Ablation Studies}\\label{sec:ablations}\nIn this section we consider how the choice of MMD kernel, timestep scheduling and sampling process affect the performance of the proposed method. In all the experiments, unless otherwise specified, we use the DDIM sampling procedure and the Inception-V3 feature space.\nFirst, we ablate the choice of the kernel for the MMD loss by comparing three different kernels: the linear kernel $k^\\text{lin}({\\bm{x}}, {\\bm{y}}) = {\\bm{x}}^\\top{\\bm{y}}$, the polynomial cubic kernel $k^\\text{cub}({\\bm{x}}, {\\bm{y}}) = \\left(\\frac{1}{d}{\\bm{x}}^\\top {\\bm{y}}+1\\right)^3$ \\cite{binkowski2018demystifying} and the Gaussian RBF kernel $k^\\text{rbf}({\\bm{x}},{\\bm{y}}) = \\exp\\left( - \\frac{1}{2 \\sigma^2} \\lVert x - y \\rVert^2 \\right)$. Table \\ref{kernel-abl} reports the results for different kernels in terms of FID, showing a marginal preference for the cubic kernel and overall robustness of MMD-DDM to kernel choice.\n\n\\begin{table\n \\caption{Ablation study for the kernel choice - CIFAR-10.}\n \\vspace{10pt}\n \\label{kernel-abl}\n \\begin{center}\n \\begin{tabular}{l c c c}\n \\hline\n Kernel & $\\vert \\mathcal{T} \\vert=5$ & $\\vert \\mathcal{T} \\vert=10$ & $\\vert \\mathcal{T} \\vert=20$ \\\\\n \\hline\\hline\n Linear & 5.61 & 4.69 & 4.06 \\\\\n Gaussian RBF & 5.89 & 3.88 & 3.61 \\\\\n Cubic & \\textbf{5.48} & \\textbf{3.80} & \\textbf{3.55} \\\\ \n \\hline\n \\end{tabular}\n \\end{center}\n\\end{table}\n\nNext, we ablate the influence of timesteps selection in Table \\ref{timesteps-abl}. We consider the two commonly-used alternatives to select $\\mathcal{T}$: \\textit{linear} $\\tau_i=\\floor{ci}$, and \\textit{quadratic} $\\tau_i = \\floor{ci^2}$, where $c$ is selected to make $\\tau_1 \\approx T$. This experiment does not show a preference for either selection method. However, it is possible that other subset selection strategies such as grid search \\cite{dockhorngenie} or learning the optimal timesteps \\cite{watson2022learning} could further improve results. We remark that MMD-DDM is decoupled from the specific timesteps selection technique.\n\n\\begin{table\n \\caption{Ablation study for the timestep schedule - CIFAR10.}\n \\vspace{10pt}\n \\label{timesteps-abl}\n \\begin{center}\n \\begin{tabular}{l c c c}\n \\hline\n Selection Method & $\\vert \\mathcal{T} \\vert=5$ & $\\vert \\mathcal{T} \\vert=10$ & $\\vert \\mathcal{T} \\vert=20$\\\\\n \\hline\\hline\n Linear & 5.48 & 3.80 & \\textbf{3.55} \\\\\n Quadratic & \\textbf{5.19} & 3.80 & 3.67 \\\\ \n \\hline\n \\end{tabular}\n \\end{center}\n\\end{table}\n\nFinally, we also test MMD-DDM with the DDPM sampling procedure, instead of DDIM. Results are reported in Table \\ref{sampling-abl}. As expected, the DDIM sampling procedure is more powerful and produces better results with a low number of timesteps. However, we notice that MMD-DDM produces significant improvements even when applied to DDPM.\n\n\\begin{table\n \\caption{Ablation study for the sampling procedure - CIFAR10.}\n \\vspace{10pt}\n \\label{sampling-abl}\n \\begin{center}\n \\begin{tabular}{l c c c}\n \\hline\n Sampling & $\\vert \\mathcal{T} \\vert=5$ & $\\vert \\mathcal{T} \\vert=10$ & $\\vert \\mathcal{T} \\vert=20$\\\\\n \\hline\\hline\n DDPM \\cite{ho2020denoising}& 76.3 & 42.1 & 25.9 \\\\\n DDIM~\\cite{song2020denoising} & 32.7 & 13.6 & 7.50 \\\\\n \\hline\n DDPM+MMD-DDM & 6.65 & 5.19 & 4.48 \\\\\n DDIM+MMD-DDM& \\textbf{5.48} & \\textbf{3.80} & \\textbf{3.55} \\\\ \n \\hline\n \n \\end{tabular}\n \\end{center}\n\\end{table}\n\n\n\n\n\\section{Conclusions and Discussion}\nThis paper addressed the problem of inference speed of DDMs. We showed that finetuning a DDM with a constraint on the number of timesteps using the MMD loss provides substantial improvements in visual quality. The limited computational complexity of the finetuning procedure offers a way to quickly obtain an improved tradeoff between inference speed and visual quality for a wide range of DDM designs. A limitation of the current technique lies in the memory requirements when the finetuning needs to be performed over a larger number of timesteps, although gradient checkpoint partially addresses this issue in most practical settings. Furthermore, coupling MMD-DDM with more advanced timestep selection and optimization techniques, possibly via joint optimization, could represent an interesting avenue to further improve speed-quality tradeoffs. Integration with conditional DDMs could also represent a direction for future work. \n\n\n\\paragraph{Broader Impact}\n\nThe goal of our method is accelerate synthesis in DDMs, which can make them more attractive methods for time-critical applications, and also reduce DDMs' environmental footprint by decreasing the computational load during inference. However, it is well known (e.g \\cite{vaccari}) that generative models can have unethical uses and potential bias depending on the context and datasets of the specific use cases. Therefore, practitioners should apply caution and mitigate impacts when using generative modeling for various applications. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{THE MODEL}\nLet us consider the vortex lattice, in which the mean distance between\nvortices is small in comparison with the penetration depth. In this case\nthe magnetic field in the superconductor is almost homogeneous. This model\nmay be appropriate for HTSC. In the field 5 T the mean distance between\nvortices is about 20 nm, while in YBCO the London penetration depth\n$\\lambda_L$ is of the order 150 nm. The diameter of the vortex core is\napproximately equal to the coherence length $\\xi$, which in YBCO is\nabout 2 nm. Therefore, it is possible to neglect possible redistribution\nof the charge and current density in the vortex core and to take it into\naccount only as the source of vortex damping.\nThe damping force ${\\bf F}_D$\nis supposed to be proportional to the vortex velocity ${\\bf v}_L$\nwith frequency independent viscosity coefficient $\\eta$, so that\n\\begin{eqnarray} \\label{eqdamping}\n{\\bf F}_{D} = -{\\eta} {\\bf v}_{L}\n = - {{m _{v}} \\over {\\tau _v}} {\\bf v}_L .\n\\end{eqnarray}\nIntroducing $m_v$ as the vortex mass per unit length, we will use the\nvortex relaxation rate $1\/\\tau_v$ which is given in practical frequency\nunits, rather than the viscosity.\n\\par\nFor pinning we use the simplest model of a parabolic well, so that\nthe pinning force ${\\bf F}_P$ is proportional to vortex displacement\n${\\bf r}_L$:\n\\begin{eqnarray} \\label{eqpinning}\n{\\bf F}_{P} = -{\\kappa} {\\bf r}_{L}\n = - m _{v} {\\alpha} ^{2} {\\bf r}_L\n\\end{eqnarray}\nwith $\\kappa$ and $\\alpha$ being the pinning constant and pinning frequency,\nrespectively.\n\\par\nThe interaction between superconducting fluid moving with velocity\n${\\bf v}_s$ and the vortex system is mediated by the Magnus force\n\\cite{93Ao} given by\n\\begin{eqnarray} \\label{eqmagnusv}\n{\\bf F}_{M} (v) = {{n_s h} \\over 2} ({\\bf v}_s - {\\bf v}_L ) \\times {\\bf z}\n = m _{v} f_s \\Omega ({\\bf v}_s - {\\bf v}_L ) \\times {\\bf z},\n\\end{eqnarray}\nwhere $n_s = f_s n$ is the density of superconducting fluid and ${\\bf\nF}_{M}(v)$ means, that this is the force felt by the vortex. The reaction\nforce ${\\bf F}_{M} (s)$ acting on a superconducting particle is\n\\begin{eqnarray} \\label{eqmagnuss}\n{\\bf F}_{M} (s) = - {{n_v} \\over {n_s}} {\\bf F}_{M} (v)\n = - m \\omega _c ({\\bf v}_s - {\\bf v}_L ) \\times {\\bf z},\n\\end{eqnarray}\nwhere $n_v$ is the vortex density (number of vortices per unit area),\n$ \\omega_c=eB\/m = n_v h \/ 2m$ is the cyclotron frequency in the field\n${\\bf B} = n_v \\Phi _0 {\\bf z}$ caused by the vortex system and\n$ \\Phi_0$ is the magnetic flux quantum.\nIn (\\ref {eqmagnusv}) we introduced the frequency of the cyclotron vortex\nmotion $ \\Omega = {{n h} \/ {2 m_v}} $. It is interesting to note that,\nusing the Hsu's expression for the vortex mass\n$ m _{v} = {{(\\pi}^2} \/4) m k_{F}^{2} {\\xi}^{2} $\n\\cite {93Hsu}, the 2D expression for the Fermi wave vector\n$ k_{F}^{2} = 2 \\pi n $ and\n$ \\xi = {{\\hbar v_F} \/ {\\pi \\Delta}}$ for the coherence length,\nit is possible to show, that\n$\\Omega = \\Delta^2\/E_F$ ($\\Delta$ is the gap and $E_F$ is the Fermi energy),\nwhich is the level separation in the vortex core \\cite {96Golosovsky}.\n\\par\nThe interaction between the vortex system and the normal state fluid may\nbe obtained in the following way. From the Aharonov-Casher Lagrangian \n\\cite {84Aharonov} it can be shown that if the vortex lattice moves with\nvelocity ${\\bf v}_L$, the force imposed by the vortex system on one \nnormal state particle moving with velocity ${\\bf v}_n$ is \n\\begin{eqnarray} \\label{eqLorentzn}\n{\\bf F}_{L} (n) = {{n_v h} \\over {2}} ({\\bf v}_n - {\\bf v}_L ) \\times {\\bf z}\n = m \\omega _c ({\\bf v}_n - {\\bf v}_L ) \\times {\\bf z}.\n\\end{eqnarray}\nAccording to the action-reaction law the vortex lattice\nmust feel the same force with opposite direction.\nIf there are $n_n=f_n n$ normal state particles,\nthe total force per unit length of one vortex is\n\\begin{eqnarray} \\label{eqLorentzv}\n{\\bf F}_{L} (v) = - {{n_n} \\over {n_v}} {\\bf F}_{L} (n)\n = - f_n { {nh} \\over 2} ({\\bf v}_n - {\\bf v}_L ) \\times {\\bf z}\n = - m_v f_n \\Omega ({\\bf v}_n - {\\bf v}_L ) \\times {\\bf z}.\n\\end{eqnarray}\nThis expression is analogous to the Magnus force formula, but has\nopposite sign. It satisfies the invariance requirements, \naccording to which only the relative velocity of the particle \nwith respect to the vortex system\nis decisive. Factor $f_n$ is justified by the fact\nthat the total force is proportional to the number of particles\ninvolved in the interaction\n\\par\nThe questions concerning forces acting on the vortex lattice are still \nseriously controversial. The Lorentz force (\\ref{eqLorentzv}) following from\nthe Aharonov-Casher\nLagrangian is of electrodynamic origin, but similar formula is also used\nto describe interaction of normal state fluid with vortices in neutral systems\n(see e.g. \\cite{97Sonin,96Stone,95Kopnin}).\nUseful comments and replies regarding the spectral flow force and the Iordanskii \nforce can be also found in \n\\cite {98Ao_Kopnin,98Hall_Wexler,98Sonin_Wexler}.\nInteraction of electric charge with moving vortex and the Aharonov-Casher\neffect in two-dimensional superconductors was discussed e.g. by \n\\v{S}im\\'anek \\cite{97Simanek}. Let us note that it would\nnot be correct to consider the Lorentz force \nalso for the superconducting fluid, which would\nexactly cancel the Magnus force. The vortex lattice and the accompanying\nmagnetic field are created by superconducting current, so in this\ncase Lorentz force would mean \"action on itself\".\n\\par\nHaving draw up the interaction forces we will now write the equations\nof motion for the three subsystems. As in the London model,\nthe superconducting fluid is supposed to move without damping,\n\\begin{eqnarray} \\label{eqmotions}\nm {\\dot {\\bf v}}_{s} = e{\\bf E} + {\\bf F}_M (s) ,\n\\end{eqnarray}\nwhile the normal state fluid motion is damped as in the conventional\nDrude model\n\\begin{eqnarray} \\label{eqmotionn}\nm {\\dot {\\bf v}}_{n} = e{\\bf E} + {\\bf F}_L (n) - {m \\over \\tau _n} {\\bf v}_n\n. \\end{eqnarray}\nFinally, for the vortex system we shall use the Newton type equation of\nmotion (of course the vortex mass and also all the forces are considered\nper unit length)\n\\begin{eqnarray} \\label{eqmotionv}\nm_v {\\dot {\\bf v}}_{L} = {\\bf F}_P+{\\bf F}_D + {\\bf F}_M (v)+ {\\bf F}_L (v)\n. \\end{eqnarray}\n\nThe system of three equations of motion (\\ref{eqmotions}-\\ref{eqmotionv})\ntogether with expressions for the interaction forces\n(\\ref{eqmagnusv}-\\ref{eqLorentzv}), damping and pinning force\n(\\ref {eqdamping},\\ref{eqpinning}) form a closed set of equations for\nthe unknown ${\\bf v}_L$, ${\\bf v}_s$ and ${\\bf v}_n$. Assuming a periodic\ntime dependence $e^{i \\omega t}$, the three differential equations reduce\nto the set of three linear equations :\n\\begin{eqnarray} \\label{eqlinearsyst}\nA_{ss} v_s + A_{sv} v_L &=& {e \\over m} E \\nonumber \\\\\n A_{nn} v_n + A_{nv} v_L &=& {e \\over m} E \\\\\nA_{vs} v_s + A_{vn} v_n + A_{vv} v_L &=& 0 \\nonumber\n\\end{eqnarray}\nwith the coefficients\n\\begin{eqnarray} \\label{eqcoeff}\n A_{ss} &=& i(\\omega - \\omega _c ) \\hspace{15mm};\\hspace{5mm}\n A_{sv} = i \\omega _c \\nonumber \\\\\n A_{nn} &=& i(\\omega + \\omega _c - i\/ \\tau _n) \\hspace{4mm};\\hspace{5mm}\n A_{nv} = -i \\omega _c \\nonumber \\\\\n A_{vs} &=& i f_s \\Omega \\hspace{22mm};\\hspace{5mm}\n A_{vn} = - i f_n \\Omega \\\\\n A_{vv} &=& i(\\omega + (f_n - f_s) \\Omega - \\alpha ^2\/ \\omega -\ni\/ \\tau _v ) \\nonumber\n. \\end{eqnarray}\nIf the determinant\n$ D = A_{ss}A_{nn}A_{vv} - A_{nn}A_{sv}A_{vs} - A_{ss}A_{nv}A_{vn} $\nis nonzero, the set can be readily solved to get\n\\begin{eqnarray} \\label{eqveloc}\nv_s \\equiv g_s {eE \\over m} &=& {{ A_{nn}A_{vv}\n + A_{sv}A_{vn} - A_{nv}A_{vn}} \\over D} {eE \\over m} \\nonumber \\\\\nv_n \\equiv g_n {eE \\over m} &=& {{ A_{ss}A_{vv}\n + A_{nv}A_{vs} - A_{sv}A_{vs}} \\over D} {eE \\over m} \\\\\nv_L \\equiv g_L {eE \\over m} &=& {{-A_{ss}A_{vn}\n - A_{nn}A_{vs} } \\over D} {eE \\over m} \\nonumber\n. \\end{eqnarray}\nNow it is straightforward to express the conductivity as\n\\begin{eqnarray} \\label{eqcond}\n\\sigma = {j \\over E} = {e \\over E} (n_s v_s + n_n v_n)\n = \\epsilon _0 \\omega _p ^2 (f_s g_s + f_n g_n)\n, \\end{eqnarray}\nwhere $ \\omega _p = \\sqrt{n e^2 \/ \\epsilon _0 m} $\nis the plasma frequency and the factors $g_s , g_n$ are defined\nby eq. (\\ref{eqveloc}).\n\\par\nAs expected, for physically meaningful parameters\n$( \\tau_v > 0, \\tau_n > 0, \\omega_c \\Omega > 0 )$\nthe real part of conductivity is positive and the Kramers-Kronig\nrelation\n$\\sigma(\\omega)=\\sigma_0 + (\\omega \/ i \\pi)\n \\int _{-\\infty} ^{\\infty} \\!{\\sigma (x) \/ (x^2 - x\\omega )} {dx}$\nas well as the f-sum rule\n$ (1\/ \\pi) \\int _{-\\infty} ^{\\infty}\n \\!{{\\it Re}(\\sigma (\\omega))}{d\\omega} =\n\\epsilon_0 \\omega_p^2 $ are satisfied.\nThe zero frequency limit of the conductivity\n$\\sigma_0 = \\epsilon_0 \\omega_p^2 \\tau_n\n(f_n + i[ \\omega_c \\tau_n (f_s - f_n) +\nf_s\/\\omega_c \\tau_n] )\/(1+\\tau_n^2 \\omega_c^2)$\ndoes not have the delta function component, as the pinning\nconstant is supposed to be finite, while pinning range\nis infinite.\nExpressing the conductivity tensor components as\n$\\sigma_{xx}(\\omega) = (\\sigma(\\omega)+\\sigma(-\\omega))\/2 $ ,\n$\\sigma_{xy}(\\omega) = (\\sigma(\\omega)-\\sigma(-\\omega))\/2 $,\nit is possible to show, that also the Hall sum rule \\cite{97Drew}\n$ (1\/ \\pi) \\int _{-\\infty} ^{\\infty}\n\\!{{\\it Re}(t_H) }{d\\omega} = \\omega_H $, where\n$t_H = \\sigma_{xy}\/\\sigma_{xx} $ and\n$ \\omega_H = \\lim_{\\omega \\rightarrow \\infty} [-i \\omega t_H (\\omega)]\n = \\omega_c(f_s - f_n) $ is satisfied.\nIt is necessary to note, that without normal state fraction\n( $f_n = 0$) the $t_H$ function has a pole at zero frequency, so\nthat in this case the Hall sum rule must be modified to\n$ \\omega_H = (\\alpha^2 \\omega_c \/( \\Omega \\omega_c + \\alpha^2))\n+ (1 \/ \\pi) \\int _{-\\infty} ^{\\infty}\n\\!{{\\it Re}(t_H) }{d\\omega} $.\n\n\\section {Absence of normal state fluid}\nUsually it is considered that at zero temperature all the charge carriers\ncondense, so that normal state fluid is absent. It is not necessary true\nfor all materials, but it is useful to discuss this limit first.\n\\par\nFor free vortices (vortices without pinning and damping) the two equations\nof motion\n$m {\\dot {\\bf v}}_{s} = - m \\omega _c ({\\bf v}_s - {\\bf v}_L ) \\times {\\bf z} $\nfor the superconducting fluid and\n$m {\\dot {\\bf v}}_{L} = m _{v} f_s \\Omega ({\\bf v}_s - {\\bf v}_L ) \\times {\\bf z} $\nfor the vortex system are readily simplified to\n$v_L \/ v_s = \\Omega\/(\\Omega-\\omega)$ and\n$v_L \/ v_s = (\\omega_c - \\omega)\/\\omega_c$, respectively.\nConsequently, two nontrivial solutions exist: for zero frequency\n$v_L=v_s$, while for $\\omega=\\Omega+\\omega_c$ the velocity ratio is\n$v_L \/ v_s =-\\Omega\/\\omega_c$. This means that the superconducting liquid\nand vortices may move either in parallel with constant velocity (this\nsolution is required by Galilean invariance), or may oscillate with\nopposite phase, with the inertial center remaining at rest.\n\\par\nIn general, with $f_n =0$ the coefficient $A_{vn}$ equals zero and the\nconductivity formula (\\ref {eqcond}) reduces to\n\\begin{eqnarray} \\label{eqcondzfn}\n\\sigma(f_n=0) = \\epsilon _0 \\omega _p ^2 {A_{vv} \\over\n {A_{ss} A_{vv} - A_{sv} A_{vs}} }\n. \\end{eqnarray}\nLet us note that in the limit of zero vortex density\n($\\omega_c \\rightarrow 0 $), this formula reduces to the London expression\nfor conductivity $\\sigma = \\epsilon _0 \\omega _p ^2 \/ i \\omega $,\nas expected. For zero pinning (but nonzero damping) the explicit expression\nfor conductivity may be written as:\n\\begin{eqnarray} \\label{eqcondzp}\n\\sigma(f_n=0,\\alpha=0)\n = { {1+i\\tau_v(\\omega - \\Omega)} \\over\n { \\tau_v \\omega(\\omega_c + \\Omega - \\omega) + i(\\omega - \\omega_c)} }\n. \\end{eqnarray}\nIt is clear that in this case the real part of conductivity is nonzero\neven at zero frequency\n$\\sigma_1(f_n=0,\\alpha=0,\\omega=0)=\\epsilon_0 \\omega_p^2 \\tau_v\n\\Omega \/ \\omega_c $.\nContrary to it, for nonzero pinning we get\n$\\sigma(f_n=0,\\alpha\\not=0,\\omega=0)=\\epsilon_0 \\omega_p^2 i\/\\omega_c $\nwith zero real part of conductivity. This result is understandable,\nif we recall that in our simple model the pinning barrier\nis infinite, so that the d.c. transport must be nondissipative.\n\\par\nIn reality the pinning barrier is not infinite. Depending on frequency,\ntemperature, magnetic field, as well as density and strength of pinning\nsites, various regimes as flux creep, flux flow, temperature assisted\nflux flow etc. \\cite {94Blatter} can be recognized. To keep the discussion\nsimple, we will analyze just two simple limits. In the \"full pinning\" (FP)\nlimit the driving field is low, so that each vortex is bound to the individual\npinning valley, making only small oscillations. In this case the pinning\nforce plays an important role. Contrary to it in the limit of high driving\nfield the amplitude of the vortex oscillation is larger than the distance\nbetween the pinning centers, and the averaged pinning force is effectively\nzero (ZP). In the intermediate state the pining force is nonzero, but not\nproportional to the distance from the pinning center which leads to\nnonlinear effects. We will show that, in some frequency range, nonlinear\neffects can be expected even at relatively low fields which are commonly\nused in laboratory experiments.\n\\par\nLet us estimate the realistic values for the parameters of the theory.\nFor coherence length $\\xi=2 \\text{ nm}$, and effective mass $m=4m_e$ using\nthe Hsu's expression for the vortex mass \\cite{93Hsu}, we can\nestimate $\\Omega=2\\hbar\/(\\pi^2 m\\xi^2)=49 \\text{ cm}^{-1}$. The cyclotron\nfrequency in the field 4T is $5.9 \\text{ cm}^{-1}$.\nUsing the expression \\cite{96Golosovsky}\n$\\kappa = (0.01 \\div 0.05) \\mu_0 H^2_c$\nfor the pinning coefficient, and the vortex mass estimation \\cite{91Yeh}\n$m_v=1.6*10^{10} m_e\/m$, the range for pinning frequency\n$\\alpha=19 \\div 95 \\text{ cm}^{-1}$ may be obtained. As\nwe did not select any model for the vortex damping, we leave this parameter\nas free. To make a model calculation we used the following set of parameters:\n$\\omega_c=5$, $\\Omega=50$, $ \\alpha=30$, $\\omega_p=6000$, $1\/\\tau_v=10$\n(all values are in $\\text{cm}^{-1}$). The conductivity calculated\nin FP and ZP limits are displayed in fig.1. The conductivity peaks are\nexpected near the frequencies, where the real or imaginary part of\nthe determinant $D=A_{ss}A_{vv}-A_{sv}A_{vs}$ which appears in the\ndenominator of (\\ref {eqcondzfn}) is zero.\nIn ZP limit the expected peak values of conductivity are\n\\begin{eqnarray} \\label{eqpeaks}\n\\sigma_1(\\omega=0) &=&\n \\epsilon_0 \\omega_p^2 \\tau_v \\Omega \/ \\omega_c \\nonumber \\\\\n\\sigma_1(\\omega=\\Omega+\\omega_c) &=&\n \\epsilon_0 \\omega_p^2 \\tau_v \\omega_c \/ \\Omega \\\\\n\\sigma_1(\\omega=\\omega_c) &=&\n \\epsilon_0 \\omega_p^2\/\\tau_v \\Omega \\omega_c \\nonumber\n. \\end{eqnarray}\n\nIn fig.1. only two sharp peaks are present for the ZP limit (dashed line).\nIt is obvious from (\\ref{eqpeaks}) that, while for low vortex damping\n(large $\\tau_v $) the peaks at eigenfrequencies of the system\n(0 and $\\Omega+\\omega_c$ ) are important, for high vortex damping\nthe peak at $\\omega_c$ will dominate. This is illustrated in fig.2,\nwhere the conductivity for vortex damping $1\/\\tau_v$ from 10 to 200 are\ndisplayed. It is possible to see, how with increasing vortex damping\nthe peak shifts from zero frequency to the cyclotron frequency\n$\\omega_c$. For FP limit, due to the pinning term $\\alpha^2\/\\omega$ the order\nof the determinant D is higher in $\\omega$, so one more peak is expected\nin accord with the model calculation results displayed in fig.1 (solid line).\n\\par\nIn fig.3 the relative value of the vortex oscillation amplitude\n$a_v = |{\\bf r}_L| m \/ eE $ is shown as a function of $\\omega$. It is\nclear that, while at high frequency the oscillation amplitude is low\nso that FP limit is appropriate, at lower frequencies the amplitude is high,\nso that ZP limit must be adopted. In principle, beside the pinning\nfrequency $\\alpha$ determining the pinning force at low\noscillation amplitude, two characteristic lengths $r_1$ and $r_2$,\nthe amplitudes of vortex oscillation, at which the pinning force declines\nfrom the linear law and at which the pinning force is effectively zero,\nshould be introduced. In this way, for a given driving field E,\ntwo crossover frequencies $\\omega_{d_1,d_2} = \\sqrt {eE\/r_{1,2} m} $\nare defined. For $ a_v<1\/\\omega_{d_1}^2$ the FP limit is valid,\nwhile for $a_v>1\/\\omega_{d_2}^2$ the ZP must be used. If neither\ncondition is fulfilled, the system is in a nonlinear region, where\nthe conductivity depends on the driving field strength. It is\ninteresting to note, that for some frequencies both conditions\n$ a_v(FP)<1\/\\omega_{d1}^2$ and $a_v(ZP)>1\/\\omega_{d2}^2$ may be fulfilled\nat one time. This means, that in this frequency region bistability\nmay occur. Depending on the history, at the same experimental conditions\ntwo regimes - the low and high vortex oscillation amplitude, corresponding\nto the low and high resistivity state - may be achieved! All these\npossibilities are illustrated in fig.3. If we estimate the range of\npinning force ($r_d$) to be about $10 \\text{ nm}$ and if the intensity of\nradiation used for measurement is $1 \\text{ mW\/mm}^2$ so that the driving\nfield E is of order $1.7 * 10^4 \\text{ V\/m}$,\nwe get $\\omega_d= 9 \\text{ cm}^{-1}$. For illustration purposes we have\nchosen $\\omega_{d_1}= \\omega_{d_2}=10 \\text{ cm}^{-1}$.\nIt is obvious that, depending on the frequency and intensity of the radiation\nused for the measurements, many interesting nonlinear effects may be expected.\n\n\\section{Influence of normal state fluid}\nFor nonzero temperatures there are two contributions to the real conductivity.\nOne is connected with the normal state charge carriers, the other with\nvortices. As expected, without vortices we get\n$\\sigma(\\omega_c = 0) = \\epsilon_0 \\omega_p^2 [f_s\/i\\omega + f_n\\tau_n\/(1+i\\omega\\tau_n)]$ ,\nwhich is the sum of the London and Drude model contributions.\nThe normal state limit $(f_s \\rightarrow 0 )$ does not have much sense,\nas without superconducting fraction we can not have any vortices.\nHowever, if we simulate external magnetic field by making vortices\nunable to move, we should get the formula for a normal conductor in\nmagnetic field. Indeed, in the limit of vortices fixed to the lattice\n($\\alpha\\rightarrow\\infty$) or of infinite vortex mass ($\\Omega=0$),\nwe get the expected result\n$\\sigma(f_s=0,\\Omega=0)=\\sigma(f_s=0,\\alpha\\rightarrow\\infty) =\n \\epsilon_0 \\omega_p^2\\tau_n \/(1+i(\\omega+\\omega_c)\\tau_n) $.\n\\par\nThe results of model calculations for $f_n=0.5$ in FP and ZP limits\nare displayed in fig.4. To visualize the contribution of vortices,\nthe zero magnetic field conductivity ($\\omega_c= 0$) is also\ndisplayed in these graphs. In FP limit, when the amplitude of vortex\nmotion is small, almost all real part of conductivity originates from\nthe normal state charge carriers - except of the very\nsharp feature near the zero frequency, which is caused by the vortex\nresonance. On the other hand, in the ZP limit the conductivity is much\nlarger and it is almost completely due to the vortex motion,\nwith the normal state fluid playing only a minor role. However, the\nsharp vortex resonance peak is absent. It might be somewhat surprising,\nthat the presence of vortices may slightly decrease the real part of\nconductivity for some frequencies.\n\\par\nIt is instructive to see, how increasing the normal state fraction\ninfluences the conductivity. For the FP limit it is shown in fig.5.\nWe can see that with increasing $f_n$, the central peak\n(connected with the normal state carriers conductivity) gradually\ndevelops, while the side peaks diminish. Recently, Lihn at al.\\cite{96Lihn}\nmeasured far infrared magnetoconductivity tensor in YBaCuO\nthin film. Their data are also displayed in fig.5 (dashed line).\nThe intensity of FIR radiation is usually rather small so the FP limit\ncould be appropriate. It is remarkable, that all the experimentally observed\nfeatures are quite well simulated by the curve with $f_n=0.3$.\nThis seems to indicate the presence of some normal state fraction\n(probably located on CuO chains ) even at the lowest temperature.\nAlternatively it may be due to the enhanced density of quasiparticles\nin an applied magnetic field, as predicted for the d-wave superconductor\n\\cite {{93Volovik},{95Wang}}. It should be noted, that the sharp vortex\nresonance peak on the theoretical curve is at lower frequency than\nthe range accessible by FIR spectroscopy, so it could not be observed\nin the experiment.\n\n\\section {Conclusions}\nVortex lattice together with the superconducting and normal state fluid form three\nsubsystems mutually connected by interaction. Taking into account \nreaction forces by which vortices influence \nsuperconducting and normal state fluid and solving simultaneously\nthe three equations of motion a new, internally consistent\ntheory of vortex dynamics was developed. It was shown that due to the\nfinite range of the pinning force, at some frequencies nonlinear phenomena \nmay be expected even for relatively low driving fields which are\ncommonly used in laboratory experiments. For comparison with experiment,\nthe knowledge of the power of radiation used for the measurements might\nbe crucial. The presented theory can qualitatively explain recent\nmeasurements of far infrared magnetoconductivity tensor made by\nLihn at.al \\cite{96Lihn}. The d.c. conductivity calculated in the framework \nof this model enables to explain theoretically controversial,\nbut experimentally firmly established Hall voltage sign reversal\n\\cite{Kolacek,98Kolacek}.\n\n\\acknowledgements\nThe authors are grateful to H.D.Drew and H.Lihn for providing\nthe original data from their magnetooptical measurement, as well as to\nE.H.Brandt, E.\\v{S}im\\'anek and E.Sonin for helpful discussions.\nThis work was supported by grants\nGA\\v{C}R $\\sharp202\/96\/0864$ and M\\v{S}MT KONTAKT ME 160.\nOne of us (J.K.) thanks the Japanese International Superconductivity Center\n(ISTEC) and New Energy and Industrial Technology Development Organization\n(NEDO) for the fellowship, during which part of this work was done.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIt is generally assumed that type Ia supernovae (SNe Ia) are thermonuclear\nexplosions of degenerate white dwarfs near the Chandrasekhar limit (cf.\nWheeler \\& Harkness 1990), or perhaps mergers of white dwarfs (WDs) in a\nbinary system (Paczynski 1985). Either detonation or deflagration models\n(Arnett 1969; Nomoto {\\it et al.}\\ 1976) then produce the visible energy release\nthat characterize the SN light and velocity curves. Detailed models show\nconsiderable differences in these scenarios (Khokhlov {\\it et al.}\\ 1993), but\ntheir large intrinsic luminosity coupled with the assumed universal\nphysics involved in the Chandrasekhar mass limit have led to strong\nstatements concerning the use of the magnitude at maximum (M$_B$(max)) of\nSNe Ia as distance indicators (Branch \\& Tammann 1992).\n\nThere has been some recent evidence that a moderate to large dispersion\nexists in M$_B$(max), however, and perhaps in the intrinsic color of these\nobjects at maximum. This evidence has been revealed by the extensive efforts of\nseveral groups to obtain high quality and frequently sampled observations\nof a large number of SNe. In particular, Phillips (1993) has\nsummarized high quality SN Ia measurements in nearby galaxies, with the\nresult that the intrinsic dispersion in M$_ B$(max) appears to be\n$\\sim 0.^m8$, and correlated with the decay time of the light curve.\nFurthermore, the underluminous nature of SN1991bg (Filippenko\n{\\it et al.}\\ 1992; Leibundgut {\\it et al.}\\ 1993) is striking. This apparent dispersion\nin M$_B$(max) has led to some discussion of different models of the origin \nof SN Ia explosions. Unfortunately, current measurements are insufficient to\ndiscriminate in detail between various explosion models (detonation versus\ndeflagration, etc.) or origin scenarios (see Wheeler \\& Harkness 1990;\nIben \\& Tutukov 1991; Khokhlov {\\it et al.}\\ 1993). Woosley \\& Weaver (1994) and\nLivne \\& Arnett (1995), among others, have discussed explosions of\nsub-Chandrasekhar limit WDs. Pinto \\& Eastman (1997) examine the physics\nof SN Ia light curves in detail and use an analytic model to study the\nsensitivity of the resultant light curves to various properties of\nsupernova explosions. They find that a variation in total mass can lead to\na sequence of light curves that reproduces the luminosity -- decline rate\nrelation. Other possible parameters (explosion energy, $^{ 56}$Ni mass,\nand opacity) lead to relations between luminosity and light curve shape\nthat are opposite to the observed behavior. They conclude that the total\nmass of the explosion is a natural and simple explanation of the\nobservations.\n\nIf a wider range of masses could be contributing to SN Ia explosions, then\nthe possibility arises that different stellar populations would produce\ndifferent origin functions (see also Kenyon {\\it et al.}\\ 1993). Moreover, since\nstellar populations do evolve, SN Ia luminosities may depend on the\nparticular form of the WDMF which should evolve with redshift. In\nprinciple, these dependencies can produce biases and selection effects in\nsurveys for distant SNe which become important to evaluate and remove.\nThese include the standard Malmquist bias concerns, as well as concerns\nabout galaxy type and position dependencies in the {\\it detected} SN\nsamples.\\footnote{For example, a radial gradient in luminosities of SNe in\ngalaxies could easily exist because of abundance and age gradients, and\ncombined with the reddening distribution, systematic luminosity\ndifferences for SNe in the outer regions of spiral or irregular galaxies\nmight dominate the observed samples.} These concerns are standard ones\nwith any extragalactic sample, although they have not been extensively\ninvestigated in the SN studies to date, partly due to the small sample\nsize (see Ruiz-Lapuente {\\it et al.}\\ 1995).\n\nAt issue here is whether the progenitors of SNe Ia have a significant\nrange in mass, that in turn produces a range in SN Ia luminosities, or if\nSNe Ia principally come fron Chandrasekhar mass white dwarfs. In this\npaper, we focus attention on the first possibility and produce a series of\nmodels which produce different white dwarf mass functions (WDMFs) for\ndiffering star formation histories. These models have predictive power\nfor both the range in SN Ia luminosities as well as the mean SN Ia\nluminosity for a given mean stellar population age.\n\nHowever, regardless of the physics that produces SNe Ia, it is now well-established\nthat empirical corrections to their luminosity based on the form of the\nlight curve (e.g., Riess {\\it et al.}\\ 1995,1996; Hamuy {\\it et al.}\\ 1995, 1996b)\nproduces a Hubble diagram which is linear out to z = 0.1 with a scatter of\n$\\leq 0^{m}.15$. To first order, this argues that the intrinsic range of\nSN Ia luminosities is irrelevant as the multi-color light curve (MLCS)\nand\/or luminosity-decline correlations empirically correct for this range.\nIn fact, these empirical corrections may be so good that systematic\ndifferences between galactic stellar populations may now be revealed.\nThere is already some observational evidence bearing on this (Hamuy {\\it et al.}\\ \n1996) and so the principle task of our modelling procedure is to\ndemonstrate how systematic differences in galactic stellar populations\ndirectly lead to systematic galaxy-galaxy differences in SN Ia\nluminosities; thus the observed galaxy correlations {\\it may} have the\npower to discriminate between different progenitor models.\n\nIn fact, we will demonstrate that our model predicts evolutionary\ncorrections to SN Ia luminosities that, at z = 0.5, are an important\npercentage of the total cosmological signal differential between q$_o$ = 0\nand q$_o$ = 0.5. In light of the concerted efforts being made in the\ndetection of SNe Ia at redshifts $\\geq$ 0.3 (Perlmutter {\\it et al.}\\ 1995) and\nthe expectation that fundamental cosmological parameters can be determined\nit seems especially important to understand, in as much detail as\npossible, the dependence of mean SN Ia luminosity on the underlying\nstellar population.\n\nThe thrust of this paper evaluates the WDMF and its variation\nas a function of stellar population and evolutionary state, under the\nassumption that the dispersion of M$_B$(SN Ia) is correlated with the mass\nof the WD progenitor. In particular we focus on one issue: how does the\nWDMF depend on its parent stellar population (Section 2)? We then apply a\nsimple parameterization of that dependence to two cases of interest: SNe\nIa arising in different galaxy types (populations with different star\nformation histories), and the dependence of progenitor mass on\ncosmological look back time (Section 3). We evaluate these effects on the\ndetermination of $q_o$ from $z \\geq 0.3$ SN Ia detections. Our concern is\nin the {\\it scatter} in the candle, and the zero point of the flux scale\nis irrelevant for this discussion (but relevant for $H_o$).\n\n\\section{A Simple Parameterization of the White Dwarf Mass Function}\n\nWe consider the luminosity distribution of SNe Ia to be a separable\nfunction, $ \\Lambda $, which can be written as\n\n\\begin{equation}\n \\Lambda = G(m) \\ N_{wd} \\ L(m_{wd}),\n\\end{equation}\n\n\\noindent\nwhere $G(m)$ is a source function, i.e. a restriction beyond the stellar\npopulation inputs on the mass range of WDs which can become SNe Ia and\nwhich would include various pathways (binary formation, etc.), $N_{wd}$ is\nthe number distribution given by the WDMF, and $L(m_{wd})$ is the\nconversion from WD mass to luminosity (essentially the Ni mass core of the\nexploding WD). For this paper we will assume that $G(m) = 1$ (i.e., no\nadditional restrictions beyond those which we model); for the\nChandrasekhar mass ignition model, this function would be a delta function\nat $1.4 M_{\\sun}$.\n\nWe build a simple model of the WD mass distribution as a function of\npopulation age based on prescriptions for the WD initial mass -- final\nmass relation, theoretical stellar lifetimes, and a star formation rate\n(SFR) parameterization. We assume that the rate of formation of stars of\na given mass at a given time can be characterized by a separable initial\nmass function (IMF) and SFR, as\n\n\\begin{equation}\n R(m,t) = \\Phi(m) \\ (A\/t_s) \\ e^{-t\/t_d},\n\\end{equation}\n\n\\noindent\nwhere {\\em m} is mass in solar units, {\\em A} is a dimensionless\nnormalization, $t_s$ is the dimensional time unit, $t_d$ is the decay time\nof the SFR, and $\\Phi(m)$ is the IMF of the form\n\n\\begin{equation}\n \\Phi(m) = N_o \\ m^{\\alpha}.\n\\end{equation}\n\nThe number of stars which will leave the main sequence to become WDs in a\ngiven mass interval, $dm$, and in a given time interval, $dt$, is\n\n\\begin{equation}\n dN_{evol}(m,t) = \\Theta(t - \\tau(m)) \\ R(t - \\tau(m))\\, dm\\, dt,\n\\end{equation}\n\n\\noindent\nwhere $\\tau(m)$ is the timescale of evolution for a star of mass $m$, and\n$\\Theta$ is a step function equal to $1$ for $t \\geq \\tau(m)$ and $0$\notherwise, and which allows the two cases of $t < \\tau(m)$ and $t \\geq\n\\tau(m)$ to be compactly written. To determine the number of stars which\nhave evolved into WDs by a given time, $t$, the above equation is\nintegrated over $t$ to yield\n\n\\begin{equation}\t\t \n N_{evol}(m,t) = \\Theta(t-\\tau(m)) \\int_{\\tau(m)}^t dt \\, \\Phi(m) \\ (A\/t_s) \\ e^{-(t-\\tau(m))\/t_d} \\ dm.\n\\end{equation}\n\n\\noindent\nLetting $ t' = t - \\tau(m)$, then\n\n\\begin{eqnarray}\n N_{evol}(m,t)& = &\\Theta(t-\\tau(m)) \\ \\Phi(m) dm \\int_0^{t-\\tau(m)} dt' (A\/t_s) \\ e^{-t'\/t_d} \\\\\n & = &\\Theta(t-\\tau(m)) \\ \\Phi(m) dm \\ A \\ (t_d\/t_s) \\ [1 - e^{-(t-\\tau(m))\/t_d}].\n\\end{eqnarray}\n\nTo convert $N_{evol}(m,t)$ to $N_{wd}(m,t)$ requires an initial mass --\nfinal mass relation, which we achieve from a quadratic parameterization of\nempirical relation ``A'' of Weidemann \\& Koester (1983):\n\n\\begin{equation}\t\t \n M_{wd} = 0.48 - 0.016 \\ m + 0.016 \\ m^2,\n\\end{equation}\n\n\\noindent\nwhere $m$ is the initial main sequence mass of a star as used above, and\n$M_{wd}$ is the mass of the resulting WD. This agrees with observations,\nwhich are very limited, and gives a $1.376 M_{\\sun}$ WD for $m = 8\nM_{\\sun}$, which we assume to be the highest mass star that produces a WD\nremnant. Clearly, all results we obtain subsequently stem from this\nparameterization, and so, to the extent that it can be justified by the\nobservations, we have a reasonably firm foundation.\n\nWe also require $\\tau(m)$, the pre-WD evolutionary timescales as a\nfunction of mass, which we achieve from a re-parameterization of the\nequations given in Eggleton {\\it et al.}\\ (1989). We simplify their\nparameterization as we require only lifetimes for stars with masses from\n$1$ to $8 M_{\\sun}$, whereas their equations are valid for $1$ to $80\nM_{\\sun}$. Additionally, we renormalize their stellar lifetimes so that a\n$1 M_{\\sun}$ star has a main sequence lifetime of $10^{10}$ years.\nFollowing Eggleton {\\it et al.}\\ (1989) we also take the post main sequence\nlifetime to be 15\\% of the main sequence lifetime. The resulting\nparameterization is then\n\n\\begin{equation}\t\t \n \\tau(m) = t_o \\ m^{-2.8},\n\\end{equation}\n \n\\noindent\nwith $t_o = 1.15 \\times 10^{10}$ years, for $m = 1$ -- $8 M_{\\sun}$. For\nan $8 M_{\\sun}$ star, the timescale of evolution is $34$ Myrs.\n\nTo use the above equations we set A = $1$ (arbitrary normalization) and\n$t_s = 1$ Gyr (i.e., all time units in Gyrs). We then choose various SFR\nmodels with $t_d = 1, 3, 5, 10,$ and $100$ Gyrs to simulate the range from\nsingle age ellipticals to constant star formation spirals. We explore the\nrange $\\alpha = 0$ to $-3$ ($\\alpha = -2.35$ is the Salpeter value) for\nthe slope of the IMF and then calculate $N_{evol}$ over the range of time\nuntil t = $12$ Gyrs. Finally, we transform $N_{evol}$ to $N_{wd}$ via the\ninitial mass -- final mass relation.\n\nFigure 1 shows the resultant WDMF for different values of the mass\nfunction slope ($\\alpha = -3, -2, -1, 0$) in each panel, for two different\nSFR decay times ($1$ Gyr, essentially a burst; and $100$ Gyrs, almost\nconstant SF), and for $6$ different ages ($0.5, 1, 2, 4, 8,$ and $12$\nGyrs) since the onset of SF.\n\nOne immediate test of these models is a comparison with the solar\nneighborhood WD mass function. Figure 2 shows the observed mass function\nof Bergeron {\\it et al.}\\ (1992). As they note, this magnitude-limited survey\nselects against the fainter, low radius (high mass) WDs. Additional\nselection may also be caused by the quicker cooling of higher mass WDs,\nand possible scale height inflation that would preferentially select\nagainst all stars of higher mass than the current turn-off mass of the\ndisk population. Plotted on this distribution is our $10$ Gyr, steady\nSFR, $\\alpha = -2.35$, model with an arbitrary normalization. The\nagreement is satisfactory after noting that Bergeron {\\it et al.}\\ (1992)\ninterpret the lowest mass WDs (first several bins) as likely results of\nbinary evolution. On this basis, we believe that our models produce WDMFs\nthat are astrophysically reasonable.\n\n\\section{Luminosity Functions: Predictions, Samples, and Biases}\n\nWoosley \\& Weaver (1994) explored the details of $0.6$ -- $0.9 M_{\\sun}$\nWDs accreting from a companion post main sequence star. They found a\nnumber of scenarios where $0.1$ to $0.2 M_{\\sun}$ of material (He) could\nbe accreted before a thermal runaway in the surface layers occurred.\nSince these thermal runaways propagate more rapidly around the surface of\nthe WD than the resulting shock wave propagates into the interior of the\nWD, the shock wave is focused in the deep interior, often resulting in a\ndetonation. We use their models as the basis of our parameterization of\nthe amount of light given off by the supernova (based on $^{56}$Ni\nproduction) as a function of mass of the accreting white dwarf. The\nWoosley \\& Weaver models are meant to explore a range of accretion rates\nand metallicities, and we parameterize their results as model A, which has\na mass accretion rate of $2.5 \\times 10^{-8} M_{\\sun}$ yr$^{-1}$, and\nmodel B, which has a mass accretion rate of $3.5 \\times 10^{-8} M_{\\sun}$\nyr$^{-1}$. Model A creates SN Ia type explosions for a pre-accretion mass\nas low as $0.6 M_{\\sun}$, whereas Model B creates SNe Ia for masses as low\nas $0.7 M_{\\sun}$. The upper mass limit of their pre-accretion WDs is\n$0.9 M_{\\sun}$, but we will assume that this relation can be extrapolated\nup to $1.1 M_{\\sun}$, which is a likely upper limit to C-O WDs (Iben \\&\nWebbink 1989; but see Kippenhahn \\& Weigert 1990). Our extrapolation and\nthe unknown upper limit is overly simplistic, but is sufficient for our\npurposes. Our parameterizations of these two models are then\n\n\\begin{equation}\t\t \n L_a(m_{wd}) \\sim m^{ni}_a(m_{wd}) = -1.2 + 2.4 \\ m_{wd} \\ for \\ m_{wd} \\geq 0.6 \\ and\n\\end{equation}\n\n\\begin{equation}\t\t \n L_b(m_{wd}) \\sim m^{ni}_b(m_{wd}) = -1.3 + 2.3 \\ m_{wd} \\ for \\ m_{wd} \\geq 0.7,\n\\end{equation}\n\n\\noindent\nwhere $^{56}$Ni masses in excess of $1.376 M_{\\sun}$ are set to $1.376\nM_{\\sun}$. This adjustment only affects WDs in the incremental mass range\n$1.07$ -- $1.10 M_{\\sun}$, and only for model A.\n\nThe resultant luminosity functions, $\\Lambda$, from the product of $N(m)$\nand $L(m_{wd})$, are shown in Figure 3 for several combinations of age,\n$\\alpha$, and SFR parameterizations. The $\\Lambda$ functions are very\nflat, as expected from the nature of the almost power law mass functions\nand the simple linear relation between the Ni mass and the WD progenitor\nmass. These luminosity functions are strongly non-gaussian, which is\nlikely to be the result in general if the wide mass range assumption we\nhave made (essentially the Woosley \\& Weaver models) are not given any\nfeatures by the source function, $G(m)$.\n\nIdeally we would now like to rigorously compare Figure 3 with the observed\nSN Ia luminosity function. However, it is our contention that the\nobserved LF is poorly known. Essentially all extant surveys have to be\ncorrected for completeness. These incompleteness corrections depend on\nthe assumed intrinsic form for $\\Lambda$\\footnote{So $\\Lambda_{obs} =\nS(\\Lambda)$, where $S$ is a selection function which depends on the\nmagnitude limit and other properties of the survey.}. As such these\ncorrections usually have the flavor of self-fulfilling prophecies: the\nderived ``$\\sigma$'' will depend on the assumed dispersion. Since the\nnumber of well-measured SNe Ia occurring in host galaxies with well\nmeasured distances is quite low (e.g., the 9 objects in Phillips 1993),\nneither the intrinsic LF nor a reliable estimate of the mean M$_B$(SN Ia)\ncan be made from extant data.\n\nAs a result of data paucity, the construction of the proper SN Ia LF is\ncurrently an ambiguous and contentious issue which remains unresolved.\nFigure 4a shows the B luminosity function for 29 SNe Ia from the\nCal\\'an\/Tololo survey (Hamuy {\\it et al.}\\ 1995, 1996a), plus the 9 objects from\nPhillips (1993). One of our referees argued that the 9 SNe from Phillips\n(1993) ``over represents'' peculiar SNe Ia. We feel this reasoning was\ncircular since the criteria for inclusion in the Phillips sample is only\nthat a good distance to the galaxy has been derived (from surface\nbrightness fluctuations or Tully-Fisher measurements). How could \nselection based on the existence of an\nindependent distance estimate cause an over representation of anomalous\nSNe Ia? In fact, the Phillips criteria is exactly what should be used in\nthe construction of a representative LF as long as no identifiable bias\nexists in the distance determinations to these 9 galaxies. We also\nchoose the Cal\\'an\/Tololo sample because it is the largest collection of\nSNe Ia with homogeneous (although still not quantified) selection\ncriteria. Figure 4a presents the observed LF, uncorrected for any\nprobable selection effects. The Phillips (1993) sample of nearby SNe Ia\nin galaxies has unknown selection effects, while the Cal\\'an\/Tololo sample\nof southern SNe has some galaxy type dependencies with distance that are\nstill being explored. For this sample, absolute magnitudes of SNe Ia are\nassigned using redshift as the distance indicator. The most significant\naspect of Figure 4a is not its shape or mean value but rather the total\nluminosity range that is exhibited. The very faint object evident in this\nfigure is SN1991bg, a very red SN that has been universally tagged as\nbeing an anomalous SN.\n\nFigure 4b shows the SN Ia LF for all the SNe from Vaughan {\\it et al.}\\ (1995)\n(hereinafter VBMP) with data obtained after 1970 ($30$ SNe). If we\nexclude from the first sample SN1991bg, the two distributions are\nessentially identical. For the 30 objects in Figure 4b, the mean\nB-magnitude is $-18.50 \\pm 0.49$ while the mean B-magnitude for the $37$\nobjects in Figure 4a is $-18.50 \\pm 0.4$. These dispersions are\nrelatively large, and obviously uncorrected SNe Ia would not appear to be\na premier distance indicator. VBMP claim to be able to lower this\ndispersion by identifying and removing SNe with deviant red or blue\ncolor. Since the intrinsic spectral energy distribution of SNe Ia is not\nyet well known from theory, a color-based rejection criteria is at best\nrisky. If we examine the 20 objects in the VBMP sample that were\ndiscovered after 1980 and reject SN1991bg and SN1986G (as obvious\ndeviants), the mean magnitude is $-18.40 \\pm 0.46$ (a change of $-0.^m1$\nin the mean is significant in the cosmological context). VBMP reject $3$\nmore objects from this sample, including the very well studied object\nSN1989B. VBMP specify the observed color (B$-$V = $0.30$) of SN1989B as\nbeing anomalous but Wells {\\it et al.}\\ (1994) attribute its color to a large\nreddening, specifically E(B$-$V) = $0.37$. Removing this single object\nfrom the 18 most recent SNe in the VBMP sample lowers the dispersion from\n$0.46$ to $0.34$ mag! Yet if reddening is the reason for the anomalous\ncolor, then obviously the absolute magnitude of SN1989B is substantially\nbrighter than the value listed in VBMP. After trimming of the anomalous\nobjects in Figure 4b, VBMP find a distribution with a mean magnitude of\n$-18.54 \\pm 0.35$. This mean is very similar to the values we derive for\nFigure 4a.\n\nWe contend that the SN Ia LF is simply not yet well-determined due to\nlimited sample sizes and survey volumes, and the difficulty of determining\ndirect and independent distances to many of the host galaxies. While we\nmay have a reasonable estimate for the maximum brightness of SNe Ia, we do\nnot know the entire LF. Furthermore, the SN Ia LF of VBMP is not\nrepresentative of the {\\it whole distribution} of SN Ia luminosities but\nrather of VBMP's selected sub-sample in which they have chosen to ignore\nor reject a fair percentage of the faintest SNe. Are these rejected\nobjects not, therefore, SNe Ia? Without an adequate explanation of the\nmechanism that causes the rejected objects to be anomalously\nunderluminous, it seems premature to a priori exclude them when specifying\nthe intrinsic range of SN Ia luminosities and then claim that the sample\nof distant SNe Ia is {\\it identical} to the selected sub-sample.\n\nFor example, we know that the Cal\\'an\/Tololo sample has obvious selection\neffects; SNe Ia fainter than $-18.0$ will not be found in at least half\nthe surveyed volume, since they fall below the apparent magnitude cutoff\nof the survey. Some of these selection effects are discussed in Hamuy\n{\\it et al.}\\ (1994). While we are not prepared here to analyze completeness of\nextant SNe samples, we schematically illustrate\nour concerns in Figure 5, which shows the redshift distribution\nof the Cal\\'an\/Tololo SN Ia sample. This distribution is extremely flat,\nwith a median recessional velocity of $\\sim$ 14,000 km s$^{-1}$, but\nextending out past 30,000 km s$^{-1}$. We have included lines in this\nfigure to demonstrate the expected increase in the sample due to volume\neffects. The dashed line is a normalization assuming the survey is\ncomplete out to 4,000 km s$^{-1}$ (which {\\it may} be representative of\nthe lower luminosity SNe Ia), while the dotted line assumes it is complete\nout to 14,000 km s$^{-1}$ (representative of the brighter SNe Ia). In\neither case, we conclude that the sample is severely incomplete through\nmuch of its volume; in the first case it is 98\\% {\\it incomplete} at the\nmedian redshift of 14,000 km s$^{-1}$. If the deeper completeness\nnormalization is assumed, the excess of low redshift SNe are those of\nfainter absolute magnitude. These comments about incompleteness in the\nsamples are a reflection of our concerns about the completeness and\naccuracy of the extant SN samples which have been used to construct the SN\nIa LF.\n\nOur simple model of the {\\it range} of SN Ia luminosities attempts to\nexplore the systematic connection between this range and galaxy type.\nIndeed, it may be very difficult to judge if the very distant SNe Ia have\na broad or narrow distribution of absolute magnitudes, due to selection\neffects and cosmological corrections. We are exploring a scenario that\nmakes certain predictions that can be tested on both local and distant\nsamples.\n\n\\subsection{Galaxy Population Dependencies}\n\nFor the purposes of this discussion, we consider that our simple models\nmay be assigned to galaxy types based on stellar population type. Thus we\nwill assume that we can characterize elliptical galaxies as single burst\nmodels, with the majority of stars formed $12$ Gyrs in the past, and with\na $1$ Gyr exponential decay time. We will also assume a IMF slope of\n$\\alpha = -2$ for this single burst model. We will assume an actively\nstar forming galaxy (SFG) can be characterized by star formation starting\napproximately $8$ Gyrs ago (the approximate age of the Galactic disk) with\neffectively continuous star formation ($t_d = 100$ Gyrs) and with several\nIMF slopes. Clearly, these assumptions can be challenged, but they\ncorrectly predict the average UBV color differences between spiral and\nelliptical galaxies (see Larson \\& Tinsley 1978; Bothun 1982). For the\ncosmological parameters we will assume $H_o = 50$ km s$^{-1}$ Mpc$^{-1}$\n(after all this is a paper on SNe) and $q_o$ = 0.5. This universe is less\nthan $14$ Gyrs old.\n\nAlthough the power law nature of the LFs shown in Figure 3 precludes the\ncalculation of a reliable mean SN Ia luminosity as a function of galaxy\ntype, we can use these means to make rough estimates. Figure 6 shows the\nmean SN Ia luminosities that are obtained by integrating over the Ni mass\ndistribution as normalized by total number of SN Ia events. This figure\ndemonstrates that our model LFs do not change shape after $\\sim 0.5$ Gyrs,\nwhich is the evolutionary timescale of the lowest mass progenitors ($3\nM_{\\sun}$) which explode as SNe Ia. We caution, however, that our model\ndoes not include the unknown, but possibly large, time delays inherent in\nthe binary mechanism before mass transfer begins. Thus the large change in\nthe SN Ia LF shape evident at early times should take place over a greater\ntime period, making the SN Ia LF more sensitive to stellar population age\nthan this figure implies. Figure 6 demonstrates that, in the case in\nwhich ellipticals and spirals have the same IMF slope $\\alpha$, the\ndifferences in mean SN Ia luminosity are small ($\\leq 0.04$ mag). This\nsmall difference is not surprising as star formation histories with\nsimilar $\\alpha$ will produce very similar WDMFs once the mean age is\ngreater than the evolutionary timescale of the WD progenitors. The\ndifference in mean SN Ia magnitude between an $\\alpha = -2$ elliptical and\nan $\\alpha = 0$ spiral is $0^m.22$. (An even more extreme difference\noccurs with large lookback times.) The data from the SN samples support\nthese dependencies on the underlying stellar population. Hamuy {\\it et al.}\\ \n(1996a) find that the brightest SNe occur in late type galaxies (see their\nFigure 3) and even more strikingly, that a strong correlation exists\nbetween the decline rate of the SN Ia light curves and the host galaxy\nmorphology (see their Figure 4). Branch {\\it et al.}\\ (1996) find that SNe Ia\nthat occur in ``red'' galaxies are 0.3 magnitudes less luminous than those\nthat occur in ``blue'' galaxies. While our model calculations are meant\nto be illustrative only, they do show that differences in mean stellar\npopulation age and\/or slope of the IMF can produce significant differences\nin mean M$_B$(SN Ia) that are approximately the same size as the effects\nseen in existing data samples.\n\nHowever, variations in mean M$_B$(SN Ia) between galaxy types are not the\nrelevant quantity with respect to distance measurements. Rather, the\ntotal range of M$_B$(SN Ia) is important, especially when considering the\neffects of Malmquist bias. In Figure 7 we plot the initial mass -- final\nmass relation (equation 8) together with nickel mass production. The\nmodels indicate a range of $4$ -- $5.5$ in Ni mass, which indicates a\nrange in SN Ia luminosities of up to $1^m.8$. We terminate our masses at\na $1.4 M_{\\sun}$ WD, but if WD mergers at all masses are a possible\nchannel, then the functions should be continued up to the possible sum of\n$2.8 M_{\\sun}$, giving a total possible range of $2.6$ mags for SN Ia\nluminosities.\n\nOur predicted range is similar to the observed range shown by Phillips\n(1993). However, this comparison is only indirect. Our predicted results\nare for the SN Ia luminosity range in a single galaxy for a specific WDMF,\nwhereas in comparing to observations, we are sampling over a range of\ngalaxy types. Still, the rough agreement between the results based on our\nmodel parameterization and the available observations has an alarming\nimplication: an order of magnitude range in M$_B$(SN Ia) immediately\nsuggests that Malmquist corrections are large for any distant\nextragalactic sample of SNe. We believe the current observations are\neffectively sampling this range. That the method of Riess {\\it et al.}\\ (1996)\ncan lead to Hubble diagrams with such low dispersions indicates that the\nlight-curve corrections to SN Ia luminosities are very effective at\ncompressing this intrinsic luminosity range. If these corrections\ncontinue to work well in larger samples, then it becomes clear that the\nintrinsic luminosity range of SNe Ia is essentially irrelevant with\nrespect to determining distances. All that is required is a secure\ncalibration of these light-curve corrected luminosities.\n\nFor the simple case in which the Ni mass is proportional to the WD mass\n(e.g., equations 10 and 11), our models predict a spread in SN Ia\nluminosities of at least $1^m.5$. However, invoking variations in WD\nprogenitor mass as the sole cause of the SN Ia luminosity spread may not\nbe necessary. Various explosion scenarios can easily give 50\\% variation\nin the energy release (Khokhlov {\\it et al.}\\ 1993; H\\\"oflich {\\it et al.}\\ 1996).\nWhether these models would have a systematic dependency on the WDMF or on\nthe evolutionary state or metal abundance undoubtedly depends in detail on\nthe nature of the explosion. We have also ignored any effects of changes\nin the WDMF on the binary frequency or whatever progenitors are the SN Ia\nsource $G(m)$ (cf.\\ Kenyon {\\it et al.}\\ 1993). While the wide binary source\nfunction may be independent to zeroth order of the details of the\nindividual stellar mass function, the SN Ia source function probably\nevolved in a more complicated manner than a simple dependence on mean WD\nmass. Hence, several physical effects can cause the SN Ia LF to depart\nsignificantly from a delta function.\n\n\\subsection{Cosmology and Evolutionary Corrections}\n\nThe redshift -- magnitude relation in standard form yields the equation\n\n\\begin{equation}\n m = M + 25 - 5 \\ log \\ H_o + 5 \\ log \\ cz + 1.086 \\ (1-q_o) \\ z + ....\n\\end{equation}\n\n\\noindent\nAt z$ = 0.5$, for $H_o = 50$ km s$^{-1}$ Mpc$^{-1}$, the difference\nbetween $q_o$ of $0.0$ and $0.5$ (empty versus critical mass models) is\n$0.^m27$ (assuming a zero cosmological constant). With photometric\naccuracies of $\\sim 0.^{m}1$ per SN event, statistics of a sample of $10$\nwell-measured objects would permit an $\\sim 8 \\sigma$ discrimination\nbetween empty and critical models. Clearly, however, systematic errors\nor biases at the $10$\\% level become very significant and lead to an\neffective $q_o$ measurement.\\footnote{We would measure $q_o^{eff} = q_o -\n(dL\/dt)\/ L \/ H_o$, in the case of luminosity evolution, for example.}\n\nFor these cosmological parameters the look-back time at $z = 0.5$ is\n$3.75$ Gyrs. Inspecting our models we see little change in our E galaxy\nsources. However, the actively SFG shows significant evolutionary effects\nin the sense that the WDMF is populated toward the more massive objects in\nthe past, and thus the $\\Lambda$ function produces brighter SNe. The WDMF\nalso depends sensitively on the assumed age parameter for the SFG; if the\ngalaxies are assumed to initiate star formation $8$ Gyrs ago, at $z = 0.5$\nthe mean luminosity can be as large as $0^{m}.31$ brighter for the $\\alpha\n= 0$ case.\\footnote{We ignore here the K-correction issue, which can be\ncomplicated at the few percent level for complex spectral types such as\nSNe. For example, different K-corrections are probably necessary at\ndifferent epochs for a given supernova event, because of the significant\nevolution of the spectral energy distribution. Clearly, shifting the\nobserved bandpass with redshift is an important consideration (cf.\\ Hamuy\n{\\it et al.}\\ 1993; Kim {\\it et al.}\\ 1996).}\n\nThe nature of the general galaxy population at $z = 0.5$ is still poorly\ndetermined. At a minimum, however, we expect the fraction of young or\nstarbursting galaxies in some random field to be significantly higher than\nis observed at the present epoch. The SN Ia rate at some epoch, z, is a\nfunction of the total number of young stars that exist at that epoch since\nmassive white dwarfs are produced by short-lived stars. Hence, a single,\nmassive starbursting galaxy could completely dominate the rate.\nAdditionally, early star formation may be characterized by a far different\nIMF slope than we observe today. This would be particularly troublesome\nas the SN Ia LF changes dramatically with IMF slope (see Figure 6). For\nour purposes, we pose one specific question: how much of a star formation\nburst is required to create a significant number of SNe Ia from the burst\npopulation relative to a single-age $5$ -- $8$ Gyr (elliptical) galaxy?\nSNe that occur in ellipticals (or the old population in a spiral which we\nassume is negligible at these redshifts) can be thought of as being the\nbackground SN population against which SNe occurring in star bursting\ngalaxies are detected. The relative contribution of the SF and background\npopulations can be parameterized as\n\n\\begin{equation}\n {No. \\ burst \\ SNe \\over No. \\ background \\ SNe} = {n_{burst} \\ exp (-t\/t_o)\n \\over n_{back} \\ exp (-t\/t_o)} = {n_{burst } \\ exp (-1 \\ or \\ -0.5)\n \\over n_{back} \\ exp (-8 \\ or \\ -5)}\n\\end{equation}\n\n\\noindent\nFor population age, t = $5$ Gyrs,\n\n\\begin{equation}\n (SN_{burst}\/SN_{back}) = (n_{burst}\/n_{back}) \\ (55 - 90)\n\\end{equation}\n\n\\noindent\nand for population age, t = $8$ Gyrs,\n\n\\begin{equation}\n (SN_{burst}\/SN_{back}) = (n_{burst}\/n_{back}) \\ (1100 - 1800).\n\\end{equation}\n\nTable 1 provides the percentage of SNe resulting from a burst as a\nfunction of the percentage of mass in the burst population relative to\nunderlying stellar population. Again, the values in this table come from\nconsidering one single age elliptical with one starbursting spiral.\nColumn 1 lists the percentage of total SNe that come from the starbursting\nspiral, while columns 2 and 3 list the starbursting mass fraction for the\nt = $5$ and $8$ Gyrs cases.\n\nFor the case of low burst strength (e.g., $\\leq 2$\\%), we find the\nexpected result that since the field contains two galaxies, $50$\\% of the\nSNe come from one of the two galaxies. However, for a star formation\nburst of $10$ -- $20$\\%, $90$\\% of the SNe will come from that one\nstarburst galaxy. The situation is even more extreme if we consider a\ntrue starburst galaxy (burst strength $\\geq 100$\\%) in which case $99$\\%\nof the SNe come from that one galaxy. These results indicate that if a\nfield at $z = 0.5$ contained $90$\\% ellipticals and $10$\\% starburst\nspirals with burst amplitudes of $10$ -- $20$\\%, then $50$\\% of the total\nSNe generated by these galaxies would come from the minority population.\nIf, however, these galaxies are preferentially dusty, then extinction\neffects may reduce the detection of SNe Ia from these hosts. Thus,\nsamples of distant SNe might be dominated by host galaxies which have\nyoung mean ages.\n\n\\section{Conclusions and Caveats}\n\nOverall, our simple model of the dependence of the SN Ia luminosity on the\nunderlying WDMF allows us to make the following predictions:\n\n1. In the mean, the SNe Ia occurring in spiral galaxies should be more\nluminous than those occurring in elliptical galaxies; bright SNe Ia in E\ngalaxies should be very rare. This effect can be seen in the data\ncompilations of Phillips (1993) and in the Cal\\'an\/Tololo survey (Hamuy\n{\\it et al.}\\ 1995, 1996a).\n\n2. A correlation should exist between SN Ia luminosity and the color of\nthe host galaxy population, with brighter SNe present in bluer galaxies.\n\n3. The SNe in the disks of spiral galaxies should be more luminous in the\nmean than those in the bulges. This effect may be hard to observe because\nof reddening effects. A reddening independent light curve parameter (such\nas $\\Delta$m$_{15}$) should correlate with position in a spiral galaxy,\nwith the broader light curves (smaller $\\Delta$m$_{15}$ values)\npreferentially in the disks or outer regions of the spirals.\n\n4. More distant SNe should show slower light curve decay (smaller\n$\\Delta$m$_{15}$ values) than the nearby sample because these SNe are\npreferentially more luminous. This prediction is a consequence of both\nstarbursting galaxies dominating distant samples and the Malmquist bias\nthat directly results from the large range in intrinsic SN Ia\nluminosities.\n\n5. Distant SNe are expected to come predominantly from bright, blue,\nspiral or irregular galaxy hosts, most of which are in an elevated state\nof star formation. The mean age of these hosts will be younger than\nthe mean age of most z = 0 calibrating galaxies, making it important that\nstarburst galaxies like NGC 5253 are included in the local calibrating \nsample. We have already shown that $M_B$(max) is sensitive to the mean age of\nthe stellar population. Thus correcting for this mean age effect requires \ndetailed knowledge of the nature of the stellar populations in distant \ngalaxies. The predicted difference in $M_B$(max) obtained under modest \nassumptions about the star formation history of galaxies is an appreciable \nfraction of the cosmological signal that\ndistinguishes $q_o = 0$ from $q_o = 0.5$.\n\n6. The form of the LF for SNe Ia should be approximately a power law (see\nFigure 3), if we assume the binary formation function introduces no strong\nfeatures. Larger samples of low redshift SNe will be needed to determine\nthis function.\n\n7. In general, the form of the WDMF predicts a range of 1.5--2.5$^m$ in SN\nIa luminosities. We have argued that current SN Ia samples have\neffectively sampled this range in luminosity and that their usefulness as\na distance indicator depends critically on the universality of the light\ncurve correction algorithms (e.g., MLCS) in compressing this luminosity\nrange.\n\nIf many of these predictions are borne out, we would contend that such\nobservational evidence favors the sub-Chandrasekhar mass hypothesis as the\nmain SN Ia progenitor. In fairness, our results and modelling procedure\nand its application to the SN Ia distance scale are subject to a number of\ncaveats and we close this paper by discussing them.\n\nIn converting WD masses into SN Ia luminosities we use the recent\ncalculations for sub-Chandrasekhar explosions by Woosley \\& Weaver\n(1994). These models accurately reproduce the observed correlation\nbetween decline rates of the light curves and luminosity, and are able to\nproduce more $^{44}$Ti and $^{48}$Cr than other types of models (see\ndiscussion in Livne \\& Arnett 1995), which is important in matching solar\nabundances. The 1D treatment of Woosley \\& Weaver (1994) yields similar\nresults to the 2D treatment of Livne \\& Arnett (1995). We choose the\nWoosley \\& Weaver models because they have clear predictive power, not\nbecause we consider these models to be definitive. While models of\nChandrasekhar mass explosions can also yield a range of luminosity (e.g.,\nH\\\"oflich {\\it et al.}\\ 1995; 1996), based on the nature and degree of turbulence\nin the explosion, we contend that, because the dispersion in SN Ia\nluminosities is not small and seems to be correlated with galaxy\nmorphology, effects in addition to explosion physics most likely produce\nthe observed LF. It is not our intention to delve into SN explosion\nphysics or discuss which SN models in the literature are more nearly\ncorrect. Instead we have argued that a major part of the observed\nluminosity range for SNe Ia {\\it can} result from a dependence of mean SN\nIa luminosity upon the mean stellar population of the host galaxy.\n\nIn this case, the mixture of host galaxy types in any SN Ia sample\ndetermines the LF for that sample. Thus, it is not surprising that there\nis disagreement over the form of a typical host galaxy in\nthe SN Ia sample at $z = 0$. Moreover,\nour models clearly show the importance of starbursting galaxies in distant\nsamples. The higher SN Ia rate in these galaxies allows the minority\npopulation to dominate the observed frequency. Since these galaxies have\nyounger mean ages and hence more extended WDMFs, the range of SN Ia\nluminosities is larger than that in a $z = 0$ spiral or elliptical.\nIndeed, the distribution of SN Ia host galaxies in the nearby Universe\nshows some curious properties which makes it hard to determine if the\ntypical host is a spiral or an elliptical. For instance, the modestly\nstar forming galaxy M 100 has had four detected SNe since 1901 (one of\nwhich is a type II), whereas the megastar elliptical M 87 has had zero.\nNGC 5253, a low mass but actively star forming galaxy, has had two\ndetected SNe Ia in the last 100 years. By comparison, the Coma cluster,\nhome to $\\sim$ $10,00$ gas poor L* galaxies (e.g., $10^4$ NGC 5253\nmasses), has not had a single SN Ia event detected for the last 22 years.\n\nIn contrast to this anecdotal evidence, which suggests that spiral hosts\ndominate over elliptical hosts, the 30 or so SNe Ia that have been\ndetected in the Cal\\'an\/Tololo survey show nearly equal numbers of\nelliptical and spiral hosts beyond $z = 0.033$, demonstrating an {\\it\nanti-}Malmquist bias. The dominance of nearby spiral hosts [at redshifts $\nz \\leq 0.033$ (Hamuy {\\it et al.}\\ 1996a)], may be a result of the avoidance of\nnearby clusters in the search fields. However, in the distant half of the\nsample no {\\it a priori} selection against clusters existed, and hence,\nproportionately more ellipticals should be in that sample, causing some of\nthe variation. Thus variation in the S\/E host ratio could reflect these\nselection criteria, as well as the low space density of relatively\nunreddened starburst spirals\nin the local universe ($z \\leq 0.1$). It is unlikely, however, that\nsimilar circumstances would continue to hold at larger redshifts.\n\nFinally, we comment on the use of light curve corrections to SN Ia\nluminosities in the context of our model. Astrophysical measurements,\nbased on either light curve parameters (Phillips 1993; Riess {\\it et al.}\\ \n1995,1996; Hamuy {\\it et al.}\\ 1996a) or spectroscopic analysis (Nugent {\\it et al.}\\ \n1995), appear to correlate well with peak SN luminosity. Hamuy {\\it et al.}\\ \n(1996a) show that high quality light curves exhibit a characteristic shape\nand luminosity -- decay time relation (Phillips 1993; hereafter the\nPhillips relation) that produce significantly improved peak magnitudes and\nmuch more accurate relative distances than the use of a single absolute\nmagnitude calibration (see also Riess {\\it et al.}\\ 1995,1996). In the model\nexplored here, the Phillips relation represents a stellar mass sequence.\nAlthough we do not attempt to derive the relation between light curve\nparameters and mass explicitly, the luminosity -- mass relation is itself\nlinear.\n\nIntrinsic dispersion around the Phillips relation or the MLCS relation of\nRiess {\\it et al.}\\ (1996) would be caused by additional parameters (e.g.,\nmetallicity), which may or may not correlate with the host galaxy stellar\npopulation and the WDMF. It is the dispersion around these relations\nthat become directly relevant to correcting distant samples for Malmquist\nbias. Moreover, corrections to SN Ia peak luminosities using $z = 0$\nlight curves may not be strictly applicable to distant SNe in\nstar-bursting galaxies as such galaxies will be rare (or perhaps\nnon-existent) in the nearby calibrating sample. At the very least, our\nmodels show that knowledge of the WDMF as inferred from the nature of the\nstellar population of the host galaxy is critical in order to determine\npotential systematic differences in light-curve shape between the distant\nhost galaxy and the calibrating sample. Minimizing these differences may\nwell validate the approaches of Hamuy {\\it et al.}\\ (1995, 1996b) and Riess {\\it et al.}\\ \n(1995,1996) that have produced linear Hubble relations out to $z = 0.1$\nwith a scatter of approximately 0$^{m}.13$ -- $0^{m}.17$ (Hamuy {\\it et al.}\\ \n1996b, equations $7$ -- $9$) which raise the expectation that $q_o$ can be\ndetermined from such data.\n\n\\acknowledgments\n\nWe acknowledge helpful discussions with Karl Fisher, Philip Pinto, and\nMichael Richmond. We also acknowledge Mark Phillips, Mario Hamuy, and\nNick Suntzeff for inspiring us to investigate the possible connection\nbetween SN Ia luminosities and the underlying stellar population.\nFinally, we wish to dedicate this paper to Marc Aaronson, who would have\nwanted a thorough investigation of the reliability of SNe Ia as standard\ncandles.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}