diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzphrx" "b/data_all_eng_slimpj/shuffled/split2/finalzzphrx" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzphrx" @@ -0,0 +1,5 @@ +{"text":"\\section{On Diagram $\\text{\\bfseries\\sffamily{C}}$}\n\nSimilar as in the proof of {\\sc Lemma}\\;\\ref{lem:N}, in every model for\nDiagram\\;$\\text{\\bfseries\\sffamily{C}}$, where $\\mathfrak{m}=\\card A$, \nwe have that the cardinality $\\mathfrak{m}$ is transfinite.\n\n\\begin{lem}\\label{lem:C}\nIf\\\/ $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and\\\/ $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ for some\\\/ $\\mathfrak{m}\\ge\\mathfrak{1}$, \nthen\\\/ $\\aleph_0\\le\\mathfrak{m}$.\n\\end{lem}\n\n\\begin{proof} Assume that $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ \nfor some cardinal $\\mathfrak{m}\\ge\\mathfrak{1}$\nand let $A$ be a necessarily infinite set with $\\card{A}=\\mathfrak{m}$. \nLet $f:A^2\\to [A]^2$ and $g:\\fin(A)\\to\\iseq(A)$ be injections.\nThe goal is to construct with the functions $f$ and $g$ an injection $h:\\omega\\to A$.\nWe first construct a countably infinite set of pairwise disjoint \nnon-empty finite subsets of $A$. For this, we first\nchoose an element $a_0\\in A$, let $E_0:=\\{a_0\\}$, and let $\\nE_0:=\\{E_0\\}$.\n\nAssume that for some $n\\in\\omega$ \nwe have already constructed an $(n+1)$-element set $\\nE_n:=\\blcb E_i:i\\le n\\brcb$\nof pairwise disjoint non-empty finite subsets of $A$. \nLet $$E_{n+1}:=\\bigcup_{i,j\\le n}\\!\\Big{\\{}\\{x,y\\}:\\exists a\\in E_i\\,\\exists b\\in E_j\n\\bigl(f(\\langle a,b\\rangle)=\\{x,y\\}\\bigr)\\Big{\\}}\\setminus\\bigcup_{i\\le n}E_i\\,,$$\nand let $\\nE_{n+1}:=\\nE_n\\cup \\{E_{n+1}\\}$. Notice that for $k:=\\bigcup_{i\\le n}E_i$,\nwe have $$\\bigg{|}\\bigcup_{i\\le n}E_i{\\,}^2\\bigg{|}\\;=\\;k^2\\,>\\,\\binom k2\n\\;=\\;\\bigg{|}\\biggl[\\,\\bigcup_{i\\le n}E_i\\biggr]^2\\bigg{|}\\,,$$ \nwhich implies that $E_{n+1}\\neq\\emptyset$. \nProceeding this way, $\\blcb E_n:n\\in\\omega\\brcb$ is a countably infinite \nset of pairwise disjoint non-empty finite subsets of $A$.\n\nNow, we apply the function $g$. For every $n\\in\\omega$, let $S_n:=g(E_n)$. \nFurthermore, let $\\nS_0:=S_0$, and in general, for $n\\in\\omega$ let $\\nS_{n+1}:=\n\\nS_n{}^\\frown S_{n+1}$. This way, we obtain an infinite sequence\n$\\nS_\\infty$ of elements of~$A$. Since $g$ is injective and the sets finite sets\n$E_n$ are pairwise disjoint, the sequence $\\nS_\\infty$ must contain infinitely\nmany pairwise distinct elements of~$A$. Now, let $h$ be the enumeration of these\npairwise distinct elements in the order they appear in $\\nS_\\infty$. Then\n$h:\\omega\\to A$ is an injection.\n\\end{proof}\n\nAs a consequence of {\\sc Proposition}\\;\\ref{prp:fin-to-one} and {\\sc Lemma}\\;\\ref{lem:C} we get\n\n\\begin{cor}\nIf\\\/ $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and\\\/ $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ \nfor some\\\/ $\\mathfrak{m}=\\card A\\ge\\mathfrak{1}$, then there \nexists a finite-to-one function\\\/ $g:\\seq(A)\\to\\fin(A)$.\n\\end{cor}\n\n\n\n\\begin{comment}\n\nAs a last result we show that in every model for Diagram\\;{\\text{\\bfseries\\sffamily{Z}}}, where $\\mathfrak{m}=\\card A$, \nthere is a finite-to-one function from $\\seq(A)$ to~$\\fin(A)$.\n\n\\begin{prp}\\label{prp:finite-to-one}\nIf\\\/ $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$ and\\\/ $\\fin(\\mathfrak{m})\\le\\iseq(\\mathfrak{m})$ for some infinite \ncardinal\\\/ $\\mathfrak{m}=\\card A$, then there exists a finite-to-one function\\\/ $g:\\seq(A)\n\\to\\fin(A)$.\n\\end{prp}\n\n\\begin{proof}\nBy {\\sc Lemma}\\;\\ref{lem:C}, there exists an injection $h:\\omega\\to A$\nand for each $i\\in\\omega$, let $x_i:=h(i)$, let $B=\\{x_{i}:i\\in\\omega\\}$, and let\n$C:=\\{x_{2i}:i\\in\\omega\\}$. Notice that \n$$\\iota(a)=\n\\begin{cases}\na &\\text{if $a\\in A\\setminus B$,}\\\\\nx_{2i+1} &\\text{if $a=x_i$,}\n\\end{cases}$$\nis a bijection between $A$ and $A\\setminus C$. \nSince $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$, there exists \nan injection $f:A^2\\to [A]^2$, and without loss of generality we may assume that\n$f:A^2\\to [A\\setminus C]^2$. Now, we define the function $g:\\seq(A)\n\\to\\fin(A)$ as follows: $g(\\langle\\;\\rangle):=\\emptyset$, \n$g(\\langle a\\rangle):=\\{a\\}$, and for every\nsequence $s=\\langle a_1,\\ldots,a_l\\rangle\\in\\seq(A)$ of length at least~$2$, let\n$$g(s):=\\{x_{4l}\\}\\cup\\bigcup \\blcb f(\\langle a_k,x_{2k}\\rangle:1\\le k\\le l\\brcb.$$\nLet now $E=g(s)$ for some sequence $s\\in\\iseq(A)$ of length at least~$2$.\n\nIt is not hard to see that for any finite set $E\\subs A$, there are just finitely \nmany sequences $s\\in\\seq(A)$ such that $g(s)=E$, which shows that \nthe function $g$ is finite-to-one.\n\\end{proof}\n\n\n\n\n\\begin{proof} \nSince $\\mathfrak{m}^2\\le[\\mathfrak{m}]^2$, there exists \nan injection $f:A^2\\to [A]^2$. Now, we define the function $g:\\iseq(A)\n\\to\\fin(A)$ as follows: $g(\\langle\\;\\rangle):=\\emptyset$, \n$g(\\langle a\\rangle):=\\{a\\}$, and for every\nsequence $s=\\langle a_1,\\ldots,a_l\\rangle\\in\\iseq(A)$ of length at least~$2$, let\n$$g(s):=\\bigcup \\blcb f(\\langle a_k,a_{k+1}\\rangle):1\\le k}[rr]& &\\seq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\\]\nis consistent with\\\/ $ZF$.\n\\end{conc}\n\\end{comment}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{\\reflectbox{C}}}} \n\nWe show that Diagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}} holds in a permutation model \n$\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ which is similar to the {\\it Second Fraenkel Model}, \nwhere $\\mathfrak{m}$ is the cardinality of the set of atoms of $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$. \n\nThe permutation model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ is constructed as follows\n(see also \\cite[p.\\,197]{cst}):\nThe set of atoms of the model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ consists of\ncountably many mutually disjoint, cyclically ordered $3$-element sets. \nMore formally,\n\\[\nA=\\bigcup_{n\\in\\omega} P_n,\n\\quad\n\\mbox{where }P_n=\\{a_n,b_n,c_n\\}\\ (\\mbox{for }n\\in\\omega),\n\\]\nand the cyclic ordering on $P_n$ is illustrated by the following figure:\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=.9]{circle.eps}\n\\end{figure}\n\nOn each triple $P_n$, we define the cyclic distance between two elements\nby stipulating $$\\cy{a_n}{b_n}=\\cy{b_n}{c_n}=\\cy{c_n}{a_n}=1$$ and \n$$\\cy{a_n}{c_n}=\\cy{b_n}{a_n}=\\cy{c_n}{b_n}=2\\,.$$\n\nLet $G$ be the group of those permutations of $A$ which preserve the\ntriples $P_n$ (\\ie, $\\pi P_n=P_n$ for $\\pi\\inG$ and $n\\in\\omega$)\nand their cyclic ordering.\nThe sets in $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}$ are those with finite support.\n\n\\begin{prp} Let\\\/ $A$ be the set of\natoms of\\\/ $\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}}$ and let\\\/ $\\mathfrak{m}:=|A|$. Then\n$$\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{C}}}}}\\models [\\mathfrak{m}]^2<\\mathfrak{m}^2<\\iseq(\\mathfrak{m})<\\fin(\\mathfrak{m})\\,.$$ \n\\end{prp}\n\n\\begin{proof} We first show that $[\\mathfrak{m}]^2\\le\\mathfrak{m}^2$, $\\mathfrak{m}^2\\le\\iseq(\\mathfrak{m})$,\nand $\\iseq(\\mathfrak{m})\\le\\fin(\\mathfrak{m})$, and then we show that\n$[\\mathfrak{m}]^2\\neq\\mathfrak{m}^2$, $\\mathfrak{m}^2\\neq\\iseq(\\mathfrak{m})$, and $\\iseq(\\mathfrak{m})\\neq\\fin(\\mathfrak{m})$.\n\\medskip\n\n\\noindent $[\\mathfrak{m}]^2\\le\\mathfrak{m}^2$: We define an injective function $f_1:[A]^2\\to A^2$.\nLet $\\{x,y\\}\\in [A]^2$ and $m,n\\in\\omega$ be \nsuch that $x\\in P_m$ and $y\\in P_n$. Without loss of generality we may assume\nthat $m\\le n$. If $m}[u]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{N}}\n\nWe first show that in every model for Diagram\\;$\\text{\\bfseries\\sffamily{N}}$, we have that \nthe cardinality $\\mathfrak{m}$ is transfinite.\n\n\\begin{lem}\\label{lem:N}\nIf\\\/ $\\fin(\\mathfrak{m})\\le\\mathfrak{m}^2$ for some\\\/ $\\mathfrak{m}\\ge\\mathfrak{5}$, then\\\/ $\\aleph_0\\le\\mathfrak{m}$.\n\\end{lem}\n\n\\begin{proof} Let $A$ be a set of cardinality $\\mathfrak{m}\\ge\\mathfrak{5}$ and assume that\n$h:\\fin(A)\\to A^2$ is an injection. First we choose a\n$5$-sequence $S_5:=\\langle a_1,\\ldots,a_5\\rangle$ of pairwise distinct elements of~$A$. \nThe ordering of $S_5$ induces an ordering on $P_5:=\\fin(\\{a_1,\\ldots,a_5\\})$, and since\n$|h[P_5]|=2^5$ and $2^5>5^2$, there exists a first set $u\\in P_5$ such that for \n$\\pair xy=h(u)$, the set $D_6:=\\{x,y\\}\\setminus \\{a_1,\\ldots,a_5\\}$ is non-empty. \nIf $x\\in D_6$, let $a_6:=x$, otherwise, let $a_6:=y$. Now, let $S_6:=\\langle a_1,\\ldots,a_6\\rangle$\nand $P_6:=\\fin(\\{a_1,\\ldots,a_6\\})$. As above, we find a $u\\in P_6$ such that for \n$\\pair xy=h(u)$, the set $D_7:=\\{x,y\\}\\setminus \\{a_1,\\ldots,a_7\\}$ is non-empty.\nIf $x\\in D_7$, let $a_7:=x$, otherwise, let $a_7:=y$. Proceeding this way, we \nfinally have an injection from $\\omega$ into $A$, which shows that $\\aleph_0\\le\\mathfrak{m}$.\n\\end{proof}\n\n\\begin{prp}\\label{prp:fin-to-one}\nIf\\\/ $\\aleph_0\\le\\mathfrak{m}$ for some infinite \ncardinal\\\/ $\\mathfrak{m}=\\card A$, then there exists a finite-to-one function\\\/ $g:\\seq(A)\n\\to\\fin(A)$.\n\\end{prp}\n\n\\begin{proof}\nBy {\\sc Lemma}\\;\\ref{lem:N}, there exists an injection $h:\\omega\\to A$,\nand for each $i\\in\\omega$, let $x_i:=h(i)$, let $B=\\{x_{i}:i\\in\\omega\\}$, and let\n$C:=\\{x_{2i}:i\\in\\omega\\}$. Notice that \n$$\\iota(a)=\n\\begin{cases}\na &\\text{if $a\\in A\\setminus B$,}\\\\\nx_{2i+1} &\\text{if $a=x_i$,}\n\\end{cases}$$\nis a bijection between $A$ and $A\\setminus C$. Thus,\nit is enough to construct a finite-to-one function $g:\\seq(A\\setminus C)\n\\to\\fin(A)$. Let $s=\\langle a_0,\\ldots,a_{n-1}\\rangle\\in \\seq(A\\setminus C)$ and let\n$\\ran(s):=\\{a_0,\\ldots,a_{n-1}\\}$. The sequence $s$ gives us in a natural way\nan enumeration of $\\ran(s)$, and with respect to this enumeration we can encode the \nsequence $s$ by a natural number $i_s\\in\\omega$. Now, let\n$g(s):=\\ran(s)\\cup\\{x_{2i_s}\\}$. Then, since there are just finitely many \nenumerations of $\\ran(s)$, $g$ is a finite-to-one function.\n\\end{proof}\n\nThe following result is just a consequence of {\\sc Proposition}\\;\\ref{prp:fin-to-one} \nand {\\sc Lemma}\\;\\ref{lem:N}.\n\n\\begin{cor}\nIf\\\/ $\\fin(\\mathfrak{m})\\le\\mathfrak{m}^2$ for some\\\/ $\\mathfrak{m}=\\card A\\ge\\mathfrak{5}$, then there \nexists a finite-to-one function\\\/ $g:\\seq(A)\\to\\fin(A)$.\n\\end{cor}\n\nWe now introduce the technique we intend to use in order to build a permutation \nmodel from which it will follow that for some infinite cardinal $\\mathfrak{m}$,\nthe relation $\\fin(\\mathfrak{m})<\\mathfrak{m}^2$ is consistent with $\\ZF$. Notice that this \nrelation is the main feature of Diagram~$\\text{\\bfseries\\sffamily{N}}$ and that this relation implies\nthat $\\aleph_0\\le\\mathfrak{m}$.\nIn the next section, we shall use a similar permutation model in order to show \nthe consistency of Diagram~$\\text{\\bfseries\\sffamily{Z}}$ with $\\ZF$. \n\nLet $K$ be the class of all the pairs $(A,h)$ such that $A$ is a (possibly empty) set\nand $h$ is an injection $h\\colon\\fin(A)\\to A^2$. We will also refer to the elements \nof $K$ as models. We define a partial ordering $\\leq$ on $K$ by stipulating\n$$(A,h)\\leq (B,f)\\iff\nA\\subs B\\;\\wedge\\;h\\subs f\\;\\wedge\\;\n\\ran\\left(f|_{\\fin(B)\\setminus\\fin(A)}\\right)\\subs(B\\setminus A)^2\\,.$$ \nWhen the functions involved are clear from the context, with a slight abuse \nof notation we will just write $A\\leq B$ instead of $(A,h)\\leq(B,f)$ \nand $A\\in K$ instead of $(A,h)\\in K$.\nNotice that the last condition in the definition of $(A,h)\\leq (B,f)$\nimplies that $(B\\setminus A)\\in K$.\n\n\\begin{prp}[CH]\\label{prp:limit}\nThere is a model\\\/ $M_*$ of cardinality\\\/ $\\mathfrak{c}$ in $K$ such that:\n\\begin{itemize}\n \\item $M_*$ is\\\/ $\\aleph_1$-universal, i.e., if\\\/ $N\\in K$ is countable then\\\/ $N$ \n is isomorphic to some\\\/ $N_*\\leq M_*$.\n \\item $M_*$ is\\\/ $\\aleph_1$-homogeneous, i.e., if\\\/ $N_1,N_2\\leq M_*$ are \n countable and\\\/ $\\pi\\colon N_1\\to N_2$ is an isomorphism then there exists \n an automorphism\\\/ $\\pi_*$ of $M_*$ such that\\\/ $\\pi\\subs\\pi_*$.\n \\item If\\\/ $N\\leq M_*$ and\\\/ $A\\subs M_*$ are countable, then there \n is an automorphism\\\/ $\\pi$ of\\\/ $M_*$ that fixes\\\/ $N$ pointwise, \n such that\\\/ $\\pi(A)\\setminus N$ is disjoint from $A$. \n\\end{itemize}\n\\end{prp}\n\n\\begin{proof}\nWe construct the model $M_*$ by induction on~$\\omega_1$, where we assume that $\\omega_1=\\mathfrak{c}$.\nLet $M_0=\\emptyset$. When $M_\\alpha$ is already defined for some $\\alpha\\in\\omega_1$, we can define \n$$C_\\alpha:=\\blcb N\\leq M_\\alpha:N\\in K\\;\\textrm{and $N$ is countable}\\brcb.$$\n\nThe construction of $M_{\\alpha+1}$, starting from $M_\\alpha$, consists of a disjoint \nunion of two differently built sets of models. First, for each element $N\\in C_\\alpha$, \nlet $S_N$ be a system of representatives for the $\\textit{strong}$ isomorphism classes \nof all the models $M\\in K$ such that $N\\leq M$, $M$ is countable,\nand for all $M_1,M_2\\in S_N$ we have $M_1\\cap M_2=N$. \nHere, by $\\textit{strong}$ we mean that, \nfor two models $M_1$ and $M_2$ with $N\\le M_1,M_2$,\nit is not enough to be isomorphic in order to belong to \nthe same class, but we require that there exists an isomorphism between $M_1$ and $M_2$ \nthat fixes $N$ pointwise, which we can express by saying that \\textit{$M_1$ is isomorphic to $M_2$ \nover $N$}. We first extend $M_\\alpha$ by the set\n$$M_\\alpha'=\\bigsqcup_{N\\in C_\\alpha}\\bigsqcup_{M\\in S_N}M\\setminus N,$$\nwhere {``}\\,$\\bigsqcup$\\,{''} indicates that we have a \\textit{disjoint union}.\nNow, we extend $M_\\alpha$ by a second set $M_\\alpha''$, where $M_\\alpha''$ is defined as follows: \nfor each pair $\\{M_1,M_2\\}\\in[C_\\alpha]^2$, we fix a way of constructing a common extension $M_1*M_2$ \nwith $M_1,M_2\\leq M_1*M_2$. Let $f^{M_1}$ and $f^{M_2}$ be the functions coming with \nthe models $M_1,M_2$, let $N_0=M_1\\cup M_2$, let $f_0=f^{M_1}\\cup f^{M_2}$, and, \nfor each finite set $E$ in \\hbox{$\\nE_0:=\\textrm{fin}(N_0)\\setminus\\textrm{dom}(f_0)$,} choose\na pair $p_E=\\langle a_1,a_2\\rangle$ such that $\\{a_1,a_2\\}\\cap N_0=\\emptyset$ and for \nall $E,E'\\in\\nE_0$, either $E=E'$ or $\\ran(p_E)\\cap\\ran(p_{E'})=\\emptyset$,\nwhere for $p_E=\\langle a_1,a_2\\rangle$, \\hbox{$\\ran(p_E):=\\{a_1,a_2\\}$.}\nNow, by induction on~$\\omega$, define \n$$N_{i+1}=N_i\\sqcup\\bigsqcup_{E\\in\\nE_i}\\ran(p_E),$$\ntogether with \n$$f_{i+1}=f_i\\cup\\bigcup_{E\\in\\nE_i}\\langle E,p_E\\rangle\\qquad\\text{and}\\qquad\n\\nE_{i+1}=\\textrm{fin}(N'_{i+1})\\setminus\\textrm{dom}(f_{i+1}),$$ and set \n$M_1*M_2:=\\bigcup_{i\\in\\omega}N_i$ with $f^{M_1*M_2}:=\\bigcup_{i\\in\\omega}f_i$. Now, let\n$$M_\\alpha''\\;:=\\bigsqcup_{\\{M_1,M_2\\}\\in[C_\\alpha]^2}M_1*M_2\\,\\setminus(M_1\\cup M_2).$$ \n\nFinally, let $M_{\\alpha+1}=M_\\alpha\\sqcup M_\\alpha'\\sqcup M_\\alpha''$,\nfor non-empty limit ordinals $\\delta$ define \n$M_\\delta=\\cup_{\\alpha\\in\\delta}M_\\alpha$, and let\n$$M_*=\\bigcup_{\\alpha\\in\\omega_1}M_\\alpha.$$ \n\nIt remains to show that the model $M_*$ has the required properties:\nFirst we notice that $M_*$ has cardinality $|M_*|=\\mathfrak{c}$, as required,\nand since, by construction, $M_1$ is $\\aleph_1$-universal,\n$M_*$ is also $\\aleph_1$-universal. In order to show that $M_*$ \nis $\\aleph_1$-homogeneous, let $N_1,N_2\\leq M_*$ be countable models and \n$\\pi\\colon N_1\\to N_2$ an isomorphism. Let $\\{x_\\alpha:\\alpha\\in\\omega_1\\}$ be \nan enumeration of the elements of $M_*$ and let $I_0:=N_1$. If $x_{\\delta_1}$ \nis the first element (w.r.t. this enumeration) in $M_*\\setminus I_0$, \nthen, by construction, there exists a model \n$I_1\\leq M_*$ such that $I_0\\leq I_1$ and $x_{\\delta_1}\\in I_1$. Again by \nconstruction, there is a model $J_1$ with $N_2\\leq J_1\\leq M_*$ such that \nthere exists an isomorphism $\\pi_1\\colon I_1\\to J_1$ with $\\pi\\subseteq\\pi_1$. \nIn fact, we have that $J_1$ and $I_1$ are isomorphic over $N_2$.\nProceed inductively with $x_{\\delta_{\\alpha+1}}$ being the first element in \n$M_*\\setminus I_\\alpha$, we find models $I_\\alpha\\leq I_{\\alpha+1}\\leq M_*$, \n$J_\\alpha\\leq J_{\\alpha+1}\\leq M_*$ and isomorphisms \n$\\pi_{\\alpha+1}\\colon I_{\\alpha+1}\\to J_{\\alpha+1}$, and finally we obtain\n$\\pi_*=\\cup_{\\alpha\\in\\omega_1}\\pi_\\alpha$, which is the required automorphism.\n\nTo show the last property of the theorem, let $N\\leq M_*$ and $A\\subs M_*$ \nbe both countable. Since the cofinality of $\\omega_1$ is greater than $\\omega$, \nwe can find by construction both a countable model $M$ satisfying the properties \n$A\\subseteq M$, $N\\leq M\\leq M_*$ and a further model $M'$ with $N\\leq M'\\leq M_*$ \nsuch that $M'\\cap(A\\setminus N)=\\emptyset$, and such that there exists an \nisomorphism $i\\colon M\\to M'$ with $i$ fixing $N$ pointwise. Now, by $M_*$ \nbeing $\\aleph_1$-homogeneous we obtain an automorphism $i_*$ extending $i$, \nas required.\n\\end{proof}\n\nAs anticipated, the construction of the previous theorem does not exploit any \nparticular property of the functions $h\\colon\\fin(A)\\to A^2$. In fact, \nthe construction is an analogue of a Fra\\\"{\\i}ss\\'e limit as it relies \non similar properties, like, for example, a modified version of \nthe {\\sl Disjoint Amalgamation Property\\\/} (DAP) of~$K$, where we require that embeddings \nbetween structures $f\\colon(A,h)\\to(B,g)$ are allowed only when, according to our \nprevious definition, $A\\leq B$. Indeed, exactly the same construction can be carried \nout in the alternative framework of models $(A,f,g,h)$, where $A$ is a set and we have \nthree injections $f\\colon A^2\\to[A]^2$, $g\\colon[A]^2\\to\\iseq(A)$ and \n$h\\colon\\iseq(A)\\to\\fin(A)$, which will be used below \nto show the consistency of Diagram~$\\text{\\bfseries\\sffamily{Z}}$ with~$\\ZF$.\n\nGiven {\\sc Proposition}\\;\\ref{prp:limit}, we consider the permutation \nmodel $\\modcV_\\text{\\bfseries\\sffamily{N}}$ that arises \nnaturally by considering the elements of the $\\aleph_1$-universal \nand $\\aleph_1$-homogeneous model $M_*$ \nas the set of atoms and its automorphisms $\\operatorname{Aut}(M_*)$ as the \ngroup $G$ of permutations. In particular,\neach permutation of $M_*$ preserves the injection \n$h\\colon\\fin(M_*)\\to M_*^2$ \nthat the model $(M_*,h)$ comes with.\n\nWe are now ready to prove the following result.\n\n\\begin{thm}\nLet\\\/ $M_*$ be the set of\natoms of\\\/ $\\modcV_\\text{\\bfseries\\sffamily{N}}$ and let\\\/ $\\mathfrak{m}=|M_*|$. Then\n$$\\modcV_\\text{\\bfseries\\sffamily{N}}\\models [\\mathfrak{m}]^2<\\fin(\\mathfrak{m})<\\mathfrak{m}^2<\\iseq(\\mathfrak{m})\\,.$$ \n\\end{thm}\n\n\\begin{proof}\nThe existence of an injection $h\\colon\\fin(M_*)\\to M_*^2$ in $\\modcV_\\text{\\bfseries\\sffamily{N}}$ follows \ndirectly from the definition of the specific permutation model. So, we only need to \nprove that in $\\modcV_\\text{\\bfseries\\sffamily{N}}$, there is no reverse injection from $M_*^2$ into $\\fin (M_*)$, \nand that there are no injections from $\\fin(M_*)$ into $[M_*]^2$ \nor from $\\iseq(M_*)$ into $M_*^2$.\n\nIn order to show that there is neither an injection from $M_*^2$ into \n$\\fin (M_*)$, nor an injection from $\\iseq(M_*)$ into $M_*^2$,\nassume towards a contradiction that $\\modcV_\\text{\\bfseries\\sffamily{N}}$ contains an injection\n$f_1\\colon M_*^2\\to\\fin (M_*)$ or an injection $f_2\\colon\\iseq(M_*)\\to M_*^2$. Let $S$ \nbe a finite support of both functions~$f_1$ and $f_2$ (if they exists). In other words, \n$S\\in\\fin (M_*)$ and for each automorphism $\\pi\\in\\operatorname{Fix}_{G}(S)$ \nwe have $\\pi(f_1)=f_1$ and $\\pi(f_2)=f_2$, respectively. \nLet $N_1$ be a countable model in $K$ with $S\\subs N_1\\leq M_*$. \nLet $(N_2,g)$ be a countable model in $K$ \nsuch that $(N_1,h|_{N_1})\\le (N_2,g)$,\nconstructed as follows: The domain of $N_2$ is \nthe disjoint union $$N_2=N_1\\sqcup\\{x,y,z\\}\\sqcup\\{a_i:i\\in\\omega\\}\\,.$$\nFurthermore, we define the injection $g\\colon\\fin(N_2)\\to N_2^2$ such that $g\\supseteq h|_{N_1}$\nand for $E\\in\\fin (N_2)\\setminus\\fin (N_1)$ we define $g(E)=\\langle e_1,e_2\\rangle$ such\nthat $g$ is injective and satisfies the following conditions (recall the since $N_2$ is \ncountable, also $\\fin(N_2)$ is countable):\n\\begin{itemize}\n \\item If $E\\cap\\{x,y,z\\}=\\emptyset$ then $\\langle e_1,e_2\\rangle=\\langle a_n,a_m\\rangle$ for some \n $n,m\\in\\omega$.\n \n \\item If $|E\\cap\\{x,y,z\\}|=1$, then $\\langle e_1,e_2\\rangle =\\langle u,a_k\\rangle$ \n for some $k\\in\\omega$, where $u$ is the unique element in $E\\cap\\{x,y,z\\}$. \n \n \\item If $|E\\cap\\{x,y,z\\}|=2$, then $\\langle e_1,e_2\\rangle=\\langle v,a_k\\rangle$ for some $k\\in\\omega$, \n where $v$ is the unique element in $\\{x,y,z\\}\\setminus(E\\cap\\{x,y,z\\})$.\n \\item If $|E\\cap\\{x,y,z\\}|=3$ then $\\langle e_1,e_2\\rangle=\\langle a_n,a_m\\rangle$ for some $n,m\\in\\omega$. \n\\end{itemize}\nNotice that there are automorphisms of $(N_2,g)$ that just permute ${x,y,z}$ and\nfix all other elements of $N_2$ pointwise.\nBy construction of $M_*$, we find a model $N_2'\\in K$ such that \n$N_1\\leq N_2'\\leq M_*$ and $N_2'$ is isomorphic to $N_2$ over~$N_1$.\nFor this reason we can refer to $N_2$ as a legit submodel of $M_*$ that extends $N_1$ \nin the way we described. Let us now consider $f_1(\\langle x,y\\rangle)$, where we assumed in $\\modcV_\\text{\\bfseries\\sffamily{N}}$\nthe existence of an injection $f_1\\colon M_*^2\\to\\fin (M_*)$ with finite support~$S$. \nIf $f_1(\\langle x,y\\rangle)\\nsubseteq N_2\\setminus N_1$, then we can\napply the third property of {\\sc Proposition}\\;\\ref{prp:limit} with respect to $f_1(\\langle x,y\\rangle)$ \nand $N_1$ and~$N_2$, which gives us a contradiction.\nIf $\\{x,y\\}\\subs f_1(\\langle x,y\\rangle)$ or $\\{x,y\\}\\cap f_1(\\langle x,y\\rangle)=\\emptyset$,\nwe could swap $x$ and $y$ while fixing every other element of $N_2$ \npointwise and get $f(\\langle x,y\\rangle)=f_1(\\langle y,x\\rangle)$, which would imply that\n$f_1$ is not injective. So, assume that\n$|\\{x,y\\}\\cap f_1(\\langle x,y\\rangle)|=1$ and without loss\nof generality assume that $\\{x,y\\}\\cap f_1(\\langle x,y\\rangle)=\\{x\\}$. \nNow, if $z\\in f_1(\\langle x,y\\rangle)$, {\\ie}, $\\{x,z\\}\\subs f_1(\\langle x,y\\rangle)$, \nwe similarly obtain a contradiction by swapping $z$ and $x$, \nwhile if $z\\notin f_1(\\langle x,y\\rangle)$ we get a contradiction by swapping $z$ and $y$. \nThis shows that $f_1$ cannot belong to $\\modcV_\\text{\\bfseries\\sffamily{N}}$.\n\nFor what concerns $f_2$, let us consider the set $\\nS$ consisting of sequences\nwithout repetition of $\\{x,y,z\\}$ of length~$2$ or~$3$.\nNotice that $|\\nS|=12$. Now, for each element $s\\in\\nS$, if $f_2(s)=\\langle a,b\\rangle$, then $a$ and\n$b$ are such that $a\\neq b$ and $\\langle a,b\\rangle\\in\\{x,y,z\\}^2$\\,---\\,notice that otherwise, for example, \nif $\\{a,b\\}\\cap\\{x,y\\}=\\emptyset$, then we can swap $x$ and $y$ and hence move $s$ \nwithout moving $\\langle a,b\\rangle$, which is not consistent with $S$ being a support of $f_2$. \nWe get the conclusion by noticing that, because of this restriction, \nthere are only six possible images of elements of $E$, which implies \nthat $f_2$ cannot be an injection.\n\nIt remains to show that in $\\modcV_\\text{\\bfseries\\sffamily{N}}$ there are no injections \nfrom $\\fin(M_*)$ into $[M_*]^2$. For this, assume towards a contradiction \nthat there exists such a function $f_3$\nin $\\modcV_\\text{\\bfseries\\sffamily{N}}$ and assume that $S$ is a finite support of~$f_3$. Then, \nlet $N_1$ be a countable model in $K$ with \n$S\\subs N_1\\leq M_*$. We will construct a countable model $(N_2,g)\\in K$ \nsatisfying $(N_1,h|_{N_1})\\leq (N_2,g)\\leq M_*$ with a finite subset \n$u\\in\\fin(N_2\\setminus N_1)$ such that, for all $\\langle x,y\\rangle\\in(N_2\\setminus N_1)^2$, \none of the following holds:\n\\begin{itemize}\n\\item there is no finite set $E\\in\\fin(N_2)$ with $h(E)=\\langle x,y\\rangle$;\n\\item there exists an automorphism $\\pi$ of $N_2$ over $N_1$ with $\\pi(u)\\neq u$ \nand $\\pi\\{x,y\\}=\\{x,y\\}$.\n\\end{itemize}\nLet $u=\\{a_0,b_0,c_0\\}$ be disjoint from $N_1$ and define $G^1_0=N_1\\sqcup\\{a_0,b_0,c_0\\}$. \nNow, for each finite set $E\\in\\fin(G^1_0)$ which is not in the domain of \n$h_0=h|_{N_1}$, that is, for each finite set $E\\in\\fin(G^1_0)$ with \n$E\\cap\\{a_0,b_0,c_0\\}\\neq\\emptyset$, let $\\{x_E,y_E\\}$ be a pair of new elements \nand define $$G^*_0\\;:=\\;G^1_0\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_0)\\setminus\\dom(h_0)}\n\\hspace{-2ex}\\{x_E,y_E\\}\\qquad\\text{and}\\qquad\nh^1_0:=h_0\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_0)\\setminus\n\\dom(h_0)}\\hspace{-2ex}\\blcb\\big{\\langle} E,\\langle x_E,y_E\\rangle\\big{\\rangle}\\brcb.$$ \nLet now $G^2_0$ be an extension of $G^*_0$ \nby adding a copy of $G^1_0\\setminus N_1$, where the ``copy function'' \nis denoted $\\tau_0$. Notice that at this stage, $G^1_0\\setminus N_1=\\{a_0,b_0,c_0\\}$.\nMore formally, $G^2_0=G^*_0\\sqcup\\{\\tau_0(a):a\\in \nG^1_0\\setminus N_1\\}$, together with an extension of $h^1_0$ defined as \n$$h^2_0:=h^1_0\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_0)\\setminus\\dom(h_0)}\n\\hspace{-2ex}\\blcb\\big{\\langle} \\tau_0(E),\\langle y_E,x_E\\rangle\\big{\\rangle}\\brcb,$$ \nwhere, given $E\\in\\fin(G^1_0)\\setminus\\dom(h_0)$, $\\tau_0(E)$ is defined \nas $$\\tau_0(E):=(E\\cap N_1)\\sqcup\\{\\tau_0(a):a\\in E\\setminus N_1\\}.$$ \nNotice that if $a\\in G^1_0\\setminus N_1$, then $\\tau_0(a)\\in G^2_0\\setminus G^*_0$. \nThe construction carried out so far is actually the first of countably many \nanalogous extension steps we will consequently apply in order to consider \nthe union of all the progressive extensions. \nThat is, assume that for some \n$i\\in\\omega$ we have already defined $G^2_i$ and $h^2_i$. Define $G^1_{i+1}=G^2_i$ \nand, for each finite set $E\\in\\fin(G^1_{i+1})$ which is not in the domain of $h^2_i$, \nconsider a pair of new elements $\\{x_E,y_E\\}$ and define\n$$G^*_{i+1}:=G^1_{i+1}\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_{i+1})\n\\setminus\\dom(h^2_i)}\\hspace{-2ex}\\{x_E,y_E\\}\\qquad\\text{and}\\qquad\nh^1_{i+1}:=h^2_i\\;\\sqcup\\hspace{-2ex}\n\\bigsqcup_{E\\in\\fin(G^1_{i+1})\\setminus\\dom(h^2_i)}\\hspace{-2ex}\n\\blcb\\big{\\langle}E,\\langle x_E,y_E\\rangle\\big{\\rangle}\\brcb.$$ \nLet now $G^2_{i+1}$ be an extension of $G^*_{i+1}$ by adding a copy of \n$G^1_{i+1}\\setminus N_1$, where the ``copy function'' is now $\\tau_{i+1}$. \nMore formally, $G^2_{i+1}:=G^*_{i+1}\\;\\sqcup\\{a_{i+1}:a\\in G^1_{i+1}\\setminus N_1\\}$, \ntogether with an extension of $h^1_{i+1}$ defined as \n$$h^2_{i+1}:=h^1_{i+1}\\;\\sqcup\\hspace{-2ex}\\bigsqcup_{E\\in\\fin(G^1_{i+1})\n\\setminus\\dom(h^2_i)}\\hspace{-2ex}\\blcb\\big{\\langle}\\tau_{i+1}(E),\\langle y_E,x_E\\rangle\\big{\\rangle}\\brcb,$$ \nwhere, again, given $E\\in\\fin(G^1_{i+1})\\setminus\\dom(h^2_i)$, $\\tau_{i+1}(E)$ \nis defined as $\\tau_{i+1}(E):=(E\\cap N_1)\\sqcup\\{\\tau_{i+1}(a):\na\\in E\\setminus N_1\\}$, for which we newly remark that if \n$a\\in G^1_{i+1}\\setminus N_1$, then $\\tau_{i+1}(a)\\in G^2_{i+1}\\setminus \nG^*_{i+1}$. \nNotice that every automorphism of $G_{i+1}^1$ which moves a finite set $E$ to\n$E'$, moves the pair $\\{x_E,y_E\\}$ to $\\{x_{E'},y_{E'}\\}$, and consequently, it\nmoves $\\big{\\langle} E,\\langle x_E,y_E\\rangle\\big{\\rangle}$ to $\\big{\\langle} E',\\langle x_{E'},y_{E'}\\rangle\\big{\\rangle}$.\nIn particular, every automorphism of $(G_{i+1}^1,h_i^2)$ can be extended to an automorphism\nof $(G_{i+1}^*,h_{i+1}^1)$. Moreover, every automorphism of $(G_{i+1}^*,h_{i+1}^1)$ can be extended to an automorphism of $(G_{i+1}^2,h_{i+1}^2)$.\n\nNow, let $$N_2:=\\bigcup_{i\\in\\omega}\nG^1_i\\qquad\\text{and}\\qquad g:=\\bigcup_{i\\in\\omega}h^1_i\\,.$$\nWe claim that $(N_2,g)$ satisfies the required properties. \nIndeed, if $\\langle x,y\\rangle\\in(N_2\\setminus N_1)^2$ and there is some \nfinite set $E\\in\\fin(N_2)$ with $g(E)=h(E)=\\langle x,y\\rangle$, then by \nconstruction there exists some index $n\\in\\omega$ such that \n$\\langle x,y\\rangle\\in (G^*_n\\setminus G^1_n)^2$, which implies that there \nexists an automorphism $\\pi$ of $N_2$ over $N_1$ acting as follows:\n$\\pi\\langle x,y\\rangle=\\langle y,x\\rangle$ and \n$\\pi u=\\pi\\{a_0,b_0,c_0\\}=\\blcb\\tau_n(a_0),\\tau_n(b_0),\\tau_n(c_0)\\brcb$, \nwhich in particular means $\\pi\\{x,y\\}=\\{x,y\\}$ and $\\pi u\\neq u$, \nas desired. We can finally consider the image $f_3(u)=\\{x,y\\}$: \nIf $\\{x,y\\}\\nsubseteq N_2\\setminus N_1$, then we can\napply the third property of {\\sc Proposition}\\;\\ref{prp:limit} with respect to $\\{x,y\\}$ \nand $N_1$ and~$N_2$, which gives us a contradiction.\nThus $\\{x,y\\}\\subseteq N_2\\setminus N_1$, and if there exists some finite \nset $E\\in\\fin(N_2)$ with $g(E)=h(E)=\\langle x,y\\rangle$, then by the reasoning \nabove we find that some automorphism of $N_2$ over $N_1$ does not \npreserve $f_3$, a contradiction. In every other case, we consider \n$\\textrm{cl}(N_1\\cup\\{x,y\\},M_*)$ and notice that, since for no \n$E\\in\\fin(N_2)$ we have $h(E)=\\langle x,y\\rangle$ or $h(E)=\\langle y,x\\rangle$, then $u$ \ncannot be a subset of $\\textrm{cl}(N_1\\cup\\{x,y\\},M_*)$, which allows \nus to fix $\\textrm{cl}(N_1\\cup\\{x,y\\},M_*)$ pointwise, while not \npreserving $u$, a contradiction as well.\n\\end{proof}\n\nSo, the model $\\modcV_{\\text{\\bfseries\\sffamily{N}}}$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{->}[drr]& &\\seq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{->}[u]& &\\mathfrak{m}^2\\ar@{->}[u]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{\\reflectbox{N}}}} \n\nWe show that Diagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}} holds in the model constructed\nin~\\cite{weird} (see also~\\cite[p.\\,209\\,ff]{cst}), \nwhere $\\mathfrak{m}$ is the cardinality of the set of atoms of that model.\n\nThe atoms of the permutation model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ for \nDiagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}} are constructed as follows:\n\\begin{itemize}\n\\setlength{\\itemsep}{.5ex}\n\\item[($\\alpha$)] Let $A_0$ be an arbitrary infinite set.\n\\item[($\\beta$)] $G_0$ is the group of {\\it all\\\/} permutations of $A_0$.\n\\item[($\\gamma$)] $A_{n+1}:=A_n\\cup\\blcb (n+1,p,\\varepsilon):\np\\in\\bigcup_{k=0}^{n+1} A_n^k\\wedge\\varepsilon\\in\\{0,1\\}\\brcb$.\n\\item[($\\delta$)] $G_{n+1}$ is the subgroup of the\npermutation group of $A_{n+1}$ containing all permutations \n$\\sigma$ for which there are\n$\\pi_\\sigma\\inG_n$ and $\\varepsilon_{\\sigma,p}\\in\\{0,1\\}$ such that\n\\begin{eqnarray*}\n\\sigma(x)=\n\\begin{cases}\n\\pi_\\sigma(x) &\\mbox{if $x\\in A_n$,}\\cr\\noalign{\\vspace{3pt}}\n(n+1,\\pi_\\sigma(p),\\varepsilon_{\\sigma,p}+_2\\varepsilon) \n&\\mbox{if $x=(n+1,p,\\varepsilon)$,}\n\\end{cases}\n\\end{eqnarray*}\nwhere for $p=\\langle p_0,\\ldots,p_{l-1}\\rangle\\in\n\\bigcup_{0\\le k\\le n+1} A_n^k$, \n$\\pi_\\sigma(p):=\\langle \\pi_\\sigma(p_0),\\ldots,\n\\pi_\\sigma(p_{l-1})\\rangle$\nand $+_2$ denotes addition modulo $2$.\n\\end{itemize}\n\nLet $A:=\\bigcup\\{A_n:n\\in\\omega\\}$ be the set of atoms and let \n$\\operatorname{Aut}(A)$ be the group of all permutations of $A$. Then\n\\[\nG:=\\blcb H\\in\\operatorname{Aut}(A):\\forall n\\in\\omega\\,(H\\res{A_n}\\inG_n)\\brcb\n\\]\nis a group of permutations of $A$. \nThe sets in $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ are those with finite support.\n\n\\begin{prp} Let\\\/ $A$ be the set of\natoms of\\\/ $\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}}$ and let\\\/ $\\mathfrak{m}:=|A|$. Then\n$$\\modcV_{\\text{\\rm\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}}\\models \\mathfrak{m}^2<\\iseq(\\mathfrak{m})<[\\mathfrak{m}]^2<\\fin(\\mathfrak{m})\\,.$$ \n\\end{prp}\n\n\\begin{proof}\nIn~\\cite{weird} it is shown that $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}\\models\\seq(\\mathfrak{m})<[\\mathfrak{m}]^2$, \nwhich implies that $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}\\models\\iseq(\\mathfrak{m})<[\\mathfrak{m}]^2$. Thus, since\n$\\mathfrak{m}^2\\le \\iseq(\\mathfrak{m})$ and $[\\mathfrak{m}]^2\\le\\fin(\\mathfrak{m})$, it remains to show that\nin $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ we have $\\mathfrak{m}^2\\neq\\iseq(\\mathfrak{m})$ and $[\\mathfrak{m}]^2\\neq\\fin(\\mathfrak{m})$.\n\\medskip\n\n\\noindent $\\mathfrak{m}^2\\neq\\iseq(\\mathfrak{m})$: We show that there is no\ninjection $g_1:\\iseq(A)\\to A^2$. Assume towards a contradiction\nthat there is such an injection with finite support $E_1$.\n\nSince $E_1$ is finite, there is an integer $n_1\\in\\omega$ \nsuch that $E_1\\subs A_{n_1}$. \nBy extending $E_1$ if necessary, we may assume that if \n\\hbox{$(n+1,\\langle a_0,\\ldots,a_{l-1}\\rangle,\\varepsilon)\\in E_1$}, \nthen also $a_0,\\ldots,a_{l-1}$ belong to $E_1$ as well as the\natom $(n+1,\\langle a_0,\\ldots,a_{l-1}\\rangle,1-\\varepsilon)$.\n\nFor a large enough number $k\\in\\omega$ choose a $k$-element set\n$X\\subs A_0\\setminus E_1$ such that\n$|\\iseq\\left(X\\right)|>\\big{|}\\left(E_1\\cup X\\right)^2\\big{|}$.\nNotice that $|\\iseq\\left(X\\right)|\\ge k!$ and \nthat $\\big{|}\\left(E_1\\cup X\\right)^2\\big{|}=\\left(|E_1|+k\\right)^2$.\nThus, we find a sequence $s\\in\\iseq(X)$ such that $g_1(s)\\notin \n\\left(E_1\\cup X\\right)^2$. So, there exists a $\\pi\\in\\operatorname{Fix}_{G}(E_1\\cup X)$ \nsuch that $\\pi g_1(s)\\neq g_1(s)$ but $\\pi s=s$, which contradicts \nthe fact that $E_1$ is a support of~$g_1$.\\smallskip\n\n\\noindent $[\\mathfrak{m}]^2\\neq\\fin(\\mathfrak{m})$: We show that there is no\ninjection $g_2:\\fin(A)\\to [A]^2$. Assume towards a contradiction\nthat there is such an injection with finite support~$E_2$.\nAfter extending $E_2$ in the same way as above, if necessary, \nfor a large enough number $k\\in\\omega$ we choose again a $k$-element set\n$X\\subs A_0\\setminus E_2$ such that\n$|\\fin\\left(X\\right)|>\\big{|}\\left[E_2\\cup X\\right]^2\\big{|}$.\nThen, by similar arguments as above, we can show that $E_2$ is not\na support of~$g_2$, which gives us the desired contradiction.\n\\end{proof}\n\nSo, the permutation model $\\modcV_{\\text{\\bfseries\\sffamily{\\reflectbox{N}}}}$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{<-}[urr]& &\\mathfrak{m}^2\\ar@{->}[u]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\section{Permutation Models}\n\nIn order to show, for example, that for some infinite cardinals $\\mathfrak{m}$ and $\\mathfrak{n}$, \n\\hbox{$\\mathfrak{m}<\\mathfrak{n}$} is consistent with $\\ZF$, \nby the {\\sc Jech-Sochor Embedding Theorem} \n(see, for example, \\cite[Thm.\\,6.1]{Jechchoice} or~\\cite[Thm.\\,17.2]{cst}), it is\nenough to construct a permutation model in which this statement holds.\nThe underlying idea of permutation models, which will be models of \nset theory with atoms ($\\ZFA$),\nis the fact that a model $\\modcV\\models\\ZFA$ does not distinguish between the\natoms, where atoms are objects which do not have\nany elements but which are distinct from the empty set. \nThe theory $\\ZFA$ is essentially the same as that of $\\ZF$\n(except for the definition of ordinals, where we have to require that\nan ordinal does not have atoms among its elements). Let $A$ be a set.\nThen by transfinite recursion on the ordinals $\\alpha\\in\\Omega$ we can define\nthe $\\alpha$-power ${\\pwr}^\\alpha(A)$ of~$A$ and\n${\\pwr}^\\infty(A):=\\bigcup_{\\alpha\\in\\Omega}{\\pwr}^\\alpha(A)$. Like\nfor the cumulative hierarchy of sets in $\\ZF$, one can show that if\n$\\modcM$ is a model of $\\ZFA$ and $A$ is the set of atoms of\n$\\modcM$, then $\\modcM:={\\pwr}^\\infty(A)$. The\nclass $M_0:={\\pwr}^\\infty(\\emptyset)$ is a model of $\\ZF$\nand is called the {\\bf kernel}. Notice that all ordinals belong to the kernel.\nBy construction we obtain that every permutation of the set of atoms induces an\nautomorphism of $\\modcV$, where the sets in the kernel are fixed.\n\nPermutation models were first\nintroduced by Adolf~Fraenkel and, in a precise version (with\nsupports), by Andrzej~Mostowski. The version with filters, which we will follow\nbelow, is due to Ernst~Specker (a detailed introduction to permutation models \ncan be found, for example, in~\\cite[Ch.\\,8]{cst} or~\\cite{Jechchoice}).\n\nIn order to construct a permutation model, we usually start with \na set of atoms $A$ and then define a group $G$ of permutations or \nautomorphisms of $A$. \n\nThe permutation models we construct below are of the following simple\ntype: For each finite set $E\\in\\fin(A)$, let $$\\operatorname{Fix}_{G}(E):=\\blcb\\pi\\inG:\n\\forall a\\in E\\,(\\pi a=a)\\brcb\\,,$$ and let $\\nF$ be the \nfilter of subgroups on $G$ generated by the subgroups \n$\\{\\operatorname{Fix}_{G}(E): E\\in\\fin(A)\\}$. In other words, $\\nF$ is the set\nof all subgroups $H\\leG$, such that the there exists a finite set\n$E\\in\\fin(A)$, such that $\\operatorname{Fix}_G(E)\\le H$.\n\nFor a set $x$, let \n$$\\Sym_{G}(x):=\\{\\pi \\inG :\\pi x\\,=\\,x\\}$$ \nwhere $$\\pi x=\\begin{cases}\n\\ \\emptyset &\\text{if $x=\\emptyset$,}\\\\[1ex]\n\\ \\pi a&\\text{if $x=a$ for some $a\\in A$,}\\\\[1ex]\n\\blcb\\pi y:y\\in x\\brcb&\\text{otherwise.}\n\\end{cases}$$\nThen, a set $x$ is {\\it symmetric\\\/} if and only if\nthere exists a set of atoms $E_{x}\\in\\fin(A)$, such that\n$$\\operatorname{Fix}_{G}(E_x)\\le\\Sym_{G}(x).$$ We say that $E_x$ is a {\\it\nsupport\\\/} of $x$. Finally, let $\\modcV$ be the class \nof all hereditarily symmetric objects; then\n$\\modcV$ is a transitive model of $\\ZFA$. \nWe call $\\modcV$ a {\\it permutation model}.\nSo, a set $x$ belongs to the permutation model\n$\\modcV$ (with respect to $G$ and $\\nF$), \nif and only if $x$ has a finite support $E_x\\in\\fin(A)$.\nBecause every $a\\in A$ is symmetric, we get that each atom $a\\in A$\nbelongs to $\\modcV$.\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{Z}}}\n\nWe are now going to set an analogue framework to the one for Diagram~$\\text{\\bfseries\\sffamily{N}}$, \njust with the definitions adapted, in order to show the consistency of \nDiagram~$\\text{\\bfseries\\sffamily{Z}}$. In fact, as mentioned above, \nwe can state the same proposition, guaranteeing the existence of a suitable \n$\\aleph_1$-universal and $\\aleph_1$-homogeneous model.\n\nLet $K$ be the class of all the pairs $(A,f,g,h)$ such that $A$ is a (possibly empty) \nset and $f,g,h$ are the following three injections:\n$$f\\colon A^2\\to[A]^2\\qquad \ng\\colon[A]^2\\to\\textrm{seq}^{1-1}(A)\\qquad h\\colon\\textrm{seq}^{1-1}(A)\\to\\textrm{fin}(A)$$\nAs before, we define a partial ordering $\\leq$ on $K$ \nby stipulating $(A,f_1,g_1,h_1)\\leq (B,f_2,g_2,h_2)$ if and only if \n\\begin{itemize}\n\\item $A\\subseteq B$, \n\\item $f_1\\subseteq f_2$,\n$\\ran\\left(f_2|_{B^2\\setminus A^2}\\right)\\subs [B\\setminus A]^2$, \n\\item $g_1\\subseteq g_2$,\n$\\ran\\left(g_2|_{[B]^2\\setminus [A]^2}\\right)\\subs\\iseq(B\\setminus A)$,\n\\item $h_1\\subseteq h_2$,\n$\\ran\\left(h_2|_{\\iseq(B)\\setminus \\iseq(A)}\\right)\\subs\\fin(B\\setminus A)$.\n\\end{itemize}\n\n\\begin{prp}[CH]\\label{prp:AtomsZ}\nThere is a model\\\/ $M_*$ of cardinality\\\/ $\\mathfrak{c}$ in $K$ such that:\n\\begin{itemize}\n \\item $M_*$ is\\\/ $\\aleph_1$-universal, i.e., if\\\/ $N\\in K$ is countable \n then\\\/ $N$ is isomorphic to some\\\/ $N_*\\leq M_*$.\n \\item $M_*$ is\\\/ $\\aleph_1$-homogeneous, i.e., if\\\/ $N_1,N_2\\leq M_*$ are \n countable and\\\/ $\\pi\\colon N_1\\to N_2$ is an isomorphism then there exists \n an automorphism\\\/ $\\pi_*$ of\\\/ $M_*$ such that\\\/ $\\pi\\subs\\pi_*$.\n \\item If\\\/ $N\\leq M_*$ and\\\/ $A\\subs M_*$ are countable, then there is an \n automorphism\\\/ $\\pi$ of\\\/ $M_*$ over\\\/ $N$ such that\\\/ \n $\\pi(A)\\setminus N$ is disjoint from\\\/ $A$. \n\\end{itemize}\n\\end{prp}\n\\begin{proof}\nThe proof is essentially the same as the one of {\\sc Proposition}\\;\\ref{prp:limit}.\n\\end{proof}\n\nWe define $\\modcV_\\text{\\bfseries\\sffamily{Z}}$ as the permutation model obtained by setting \nthe elements of the $\\aleph_1$-universal and $\\aleph_1$-homogeneous model $M_*$ \nas the set of atoms and its automorphisms $\\operatorname{Aut}(M_*)$ as the group $G$ of permutations. \nIn particular, each permutation of $M_*$ preserves\nthe injections $f,g,h$ that the model $(M_*,f,g,h)$ comes with.\n\n\\begin{thm}\nLet\\\/ $M_*$ be the set of\natoms of\\\/ $\\modcV_\\text{\\bfseries\\sffamily{Z}}$ and let\\\/ $\\mathfrak{m}=|M_*|$. Then\n$$\\modcV_\\text{\\bfseries\\sffamily{Z}}\\models \\mathfrak{m}^2<[\\mathfrak{m}]^2<\\iseq(\\mathfrak{m})<\\fin(\\mathfrak{m})\\,.$$ \n\\end{thm}\n\\begin{proof}\nThe existence of the required injections is clear by the definition of the \nmodel. Thus, it remains to prove that there are no reverse injections. \nFirst, we give two preliminary definitions. Given a model $(M,f,g,h)$ \nand a countable subset $A\\subs M$, \nwe define the \\textit{closure\\\/} $\\textrm{cl}(A,M)$ as \nthe smallest superset of $A$ that is closed under $f,g,h$ and pre-images \nwith respect to the same functions. Constructively, we can characterize \n$\\textrm{cl}(A,M)$ as a countable union as follows: \nDefine $\\textrm{cl}_0=\\textrm{cl}_0(A,M):=A$ and, for all $i\\in\\omega$,\n\\begin{align*}\n \\textrm{cl}_{i+1}= \\textrm{ cl}_i &\\;\\sqcup \\bigsqcup_{p\\in(\\textrm{cl}_i)^2}f(p)\\; \n \\sqcup \\bigsqcup_{q\\in[\\textrm{cl}_i]^2}\\ran(g(q))\\;\\sqcup\n \\bigsqcup_{s\\in\\iseq(\\textrm{cl}_i)}\\hspace{-2ex}h(s) \\\\[1.0ex]\n & \\sqcup \\bigsqcup_{q\\in[\\textrm{cl}_i]^2} \\ran(f^{-1}(q)) \\;\\sqcup \n \\bigsqcup_{s\\in\\textrm{seq}^{1-1}(\\textrm{cl}_i)}\\hspace{-2ex}g^{-1}(s) \\;\\sqcup \n \\bigsqcup_{r\\in\\textrm{fin}(\\textrm{cl}_i)}\\hspace{-.8ex}\\ran(h^{-1}(r))\\,,\n\\end{align*}\nin order to finally define $\\textrm{cl}(A,M)=\\cup_{i\\in\\omega}\\textrm{ cl}_i$. \nFurthermore, we set a standardized way \nto extend a \\textit{partial} model $(A,f,g,h)$, where $f,g,h$ are only \npartial functions, to an element of $K$: Consider $(A,f',g',h')$, \nwhere $A$ is a countable set and $f,g,h$ are injections with \n\\begin{center}\n$$ \n\\begin{array}{ccc}\n \\dom(f)\\subseteq A^2,&\\dom(g)\\subseteq [A]^2, & \\dom(h)\\subseteq\\textrm{seq}^{1-1}(A)\\\\[1.4ex]\n \\ran(f)\\subseteq [A]^2, &\\ran(g)\\subseteq\\textrm{seq}^{1-1}(A),& \\ran(h)\\subseteq\\textrm{fin}(A).\n\\end{array}\n$$\n\\end{center}\nLet $(M_0,f'_0,g'_0,h'_0)=(A,f',g',h')$ and, for $j\\in\\omega$, define inductively \n$(M_{j+1},f'_{j+1},g'_{j+1},h'_{j+1})$ as follows: $M_{j+1}$ is the fully \ndisjoint union $$M_j\\quad\\sqcup\\bigsqcup_{P\\in M_j^2\\setminus\\dom(f_j)}\n\\hspace{-3ex}\\{a_P,b_P\\}\\quad\\sqcup\\bigsqcup_{Q\\in[M_j]^2\\setminus\\dom(g_j)}\\hspace{-3.4ex}\\{a_Q,b_Q,c_Q\\}\n\\quad\\sqcup\\bigsqcup_{R\\in\\iseq(M_j)\\setminus\\dom(h_j)}\\hspace{-5ex}\\{a_R,b_R,c_R\\}.$$ \nFor what concerns the injections $f'_{j+1},g'_{j+1},h'_{j+1}$, we naturally \nrequire the inclusions $f'_j\\subseteq f'_{j+1}$, $g'_j\\subseteq g'_{j+1}$, and $h'_j\\subseteq h'_{j+1}$, \nas well as the equalities $\\dom(f'_{j+1})=M_j^2$, $\\dom(g'_{j+1})=[M_j]^2$, \nand $\\dom (h'_{j+1})=\\textrm{seq}^{1-1}(M_j)$, respectively, where for\n$P\\in M_j^2\\setminus\\dom (f_j)$, \n$Q\\in[M_j]^2\\setminus\\dom (g_j)$, and $R\\in\\textrm{seq}^{1-1}(M_j)\n\\setminus\\dom (h_j)$, we define $$f'_{j+1}(P):=\\{a_P,b_P\\}\\,,\\qquad g'_{j+1}(Q):=\n\\langle a_Q,b_Q,c_Q\\rangle\\,,\\qquad h'_{j+1}(R):=\\{a_R,b_R,c_R\\}\\,.$$ \nWe are now in the position of defining the plain extension of $(A,f',g',h')$ as \n$$(M,f,g,h)=\\left(\\bigcup_{j\\in\\omega}M_j,\\;\\bigcup_{j\\in\\omega}f'_j,\\;\\bigcup_{j\\in\\omega}g'_j,\\;\n\\bigcup_{j\\in\\omega}h'_j\\right),$$ and we can finally prove, in three analogous steps, \nthat neither of the three injections of the model $(M_*,f,g,h)$ \nadmits a reverse injection.\n\nAssume there is an injection $i\\colon [A]^2\\to A^2$ with finite support \n$S$, where $A$ is the set of atoms. Let $N_1\\in K$ \nbe a countable model such that $N_1\\leq M_*$ and $S\\subs N_1$. \nLet $\\{x,y\\}\\in [A]^2$ with $N_1\\cap\\{x,y\\}=\\emptyset$, \nlet $M_0=N_1\\sqcup\\{x,y\\}$, and let $N_2$ be the plain extension of $M_0$.\nWithout loss of generality we can assume that $N_2\\leq M_*$. \nConsider $\\langle a,b\\rangle=i(\\{x,y\\})$. \nThen $\\{a,b\\}\\nsubseteq N_1$ and $\\{a,b\\}\\subs N_2$, since otherwise, we could \napply the third property of {\\sc Proposition}\\;\\ref{prp:AtomsZ} with respect to $\\{a,b\\}$ \nand $N_1$ and $N_2$, \nrespectively. Moreover, $\\{a,b\\}\\cap\\{x,y\\}=\\emptyset$, since otherwise,\n({\\eg}, $a=x$), we can swap $x$ and $y$ which would imply that $b=y$,\nand since $x\\neq y$, we get $i(\\{x,y\\})=\\langle x,y\\rangle\\neq\\langle y,x\\rangle=i(\\{y,x\\})$\nwhich is a contradiction to the assumption that $i$ is a function.\nFurthermore, we have \n\\begin{equation}\n\\{x,y\\}\\;\\subs\\;\\textrm{cl}(N_1\\cup\\{a,b\\},M_*)\\tag{$*$}\n\\end{equation}\nsince otherwise, we could \napply the third property of {\\sc Proposition}\\;\\ref{prp:AtomsZ} with respect to $\\{x,y\\}$ and\n\\hbox{$\\textrm{cl}\\bigl(N_1\\cup\\{a,b\\},M_*\\bigr)\\leq M_*$.} \nNow, this last inclusion implies that \n$a\\neq b$ and that $\\{a,b\\}=f(\\langle x',y'\\rangle)$ for \nsome $x',y'\\in N_2\\setminus N_1$. \nTo see this, notice first that since $N_1\\cup \\{a,b\\}\\subs N_2$,\nwe build the closure \n\\hbox{$\\textrm{cl}\\bigl(N_1\\cup\\{a,b\\},M_*\\bigr)$} within the\nplain extension~$N_2$, and recall that for $\\{u,v\\}\\in [N_2]^2\\setminus [M_0]^2$\nwe have $g(\\{u,v\\})=\\langle x_1,x_2,x_3\\rangle$ where $\\langle x_1,x_2,x_3\\rangle\\in \n\\iseq(N_2\\setminus M_0)$, and that for $\\langle x_1,\\ldots,x_n\\rangle\\in\\iseq(N_2)\\setminus\n\\iseq(M_0)$ we have $h(\\langle x_1,\\ldots,x_n\\rangle)\\in [N_2\\setminus M_0]^3$.\nIf there are no $x',y'$ such that $f(\\langle x',y'\\rangle)=\\{a,b\\}$, then, since\n$\\{a,b\\}\\subs N_2$, $\\{a,b\\}$ is a proper subset of $\\ran\\bigl(g(\\{u,v\\})\\bigr)$ for some $u,v$, \nor of $h(\\langle x_1,\\ldots,x_n\\rangle)$ for some $x_1,\\ldots,x_n$.\nIn both cases we have that $\\{x,y\\}\\nsubseteq\\textrm{cl}(N_1\\cup\\{a,b\\},M_*)$,\nwhich is a contradiction to~$(*)$.\nNow, since $f(\\langle x',y'\\rangle)=\\{a,b\\}$, by the construction of the plain extension $N_2$\nwe find an automorphism $\\pi$ of $N_2$ \nthat fixes $N_1\\cup\\{x,y\\}$ pointwise and for which we have $\\pi(a)=b$ and $\\pi(b)=a$.\nHence, $i(\\pi\\{x,y\\})=\\langle a,b\\rangle\\neq\\langle b,a\\rangle=\\pi i(\\{x,y\\})$, which is a\ncontradiction.\n\nAssume there is an injection $i\\colon \\mathrm{seq}^{1-1}(A)\\to [A]^2$ with \nfinite support $S$. Let $N_1\\in K$ be a countable model such that\n$N_1\\leq M_*$ and $S\\subs N_1$. Let $\\langle x,y,z\\rangle\\in\\iseq(A)$ with \n$N_1\\cap\\{x,y,z\\}=\\emptyset$, let $M_0=N_1\\sqcup\\{x,y,z\\}$, and \nlet $N_2$ be the plain extension of $M_0$. Finally, let $\\{a,b\\}=i(\\langle x,y,z\\rangle)$. \nThen, a contradiction follows by noticing\\,---\\,with similar arguments as \nabove\\,---\\,that necessarily \n$\\{x,y,z\\}\\not\\subs\\textrm{cl}\\bigl(N_1\\cup\\{a,b\\},M_*\\bigr)$.\nSo, similarly as above, there is an automorphism $\\pi$ of $M_*$ which fixes \n$\\textrm{cl}(N_1\\cup\\{a,b\\},M_*)$ pointwise, but $\\pi(\\{x,y,z\\})\\neq\\{x,y,z\\}$.\n\nFinally, assume there is an injection $i\\colon\\fin(A)\\to\\mathrm{seq}^{1-1}(A)$ \nwith finite support $S$. Let $N_1\\in K$ \nbe a countable model such that $N_1\\leq M_*$ and $S\\subs N_1$. \nLet $\\{x,y,z\\}\\in [A]^3$ be such that \\hbox{$N_1\\cap\\{x,y,z\\}=\\emptyset$,} \nlet $M_0=N_1\\sqcup\\{x,y,z\\}$, and let $N_2$ be the plain extension of $M_0$.\nConsider $\\langle a_j:j\\in n\\rangle=i(\\{x,y,z\\})$ for some $n\\in\\omega$. It is easy \nto see that we must have $\\{a_j:j\\in n\\}\\cap (N_2\\setminus M_0)\\neq\\emptyset$, \nand as before it must also hold $\\{x,y,z\\}\\subs\n\\textrm{cl}\\bigl(N_1\\cup\\{a_j:j\\in n\\},M_*\\bigr)$. \nThe last inclusion implies that a $3$-cycle $\\pi$ applied to $\\{x,y,z\\}$ \ncannot leave $\\{a_j:j\\in n\\}\\cap (N_2\\setminus M_0)$ unchanged, since $\\pi$ \nmoves every unordered pair, ordered pair and injective sequence with values \nin $\\{x,y,z\\}$. We conclude the proof by noticing that we can easily find \nan automorphism of $N_2$ that fixes $N_1$ pointwise and that acts on \n$\\{x,y,z\\}$ as a $3$-cycle.\n\\end{proof}\n\nSo, the model $\\modcV_{\\text{\\bfseries\\sffamily{Z}}}$ witnesses the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{->}[urr]& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\n\\subsection{A Model for Diagram \\protect\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}} \n\nAs mention above, Diagram~\\protect{\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}}\nholds in the {\\it Ordered Mostowski Model\\\/}, \nwhere $\\mathfrak{m}$ is the set of atoms (see, for example, \n\\cite[Related\\;Result\\,48,\\;p.\\,217]{cst}). This\nleads to the following\n\n\\begin{conc} \nThe existence of an infinite cardinal\\\/ $\\mathfrak{m}$ satisfying\n\\[\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{->}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{<-}[ll]\\ar@{->}[ull]\n}\n\\]\nis consistent with\\\/ $\\ZF$.\n\\end{conc}\n\\section{Introduction}\n\nLet $M$ be a set. Then $\\fin(M)$ denotes the set of all finite subsets of $M$,\n$M^2$ denotes the Cartesian product $M\\times M$, \n$[M]^2$ denotes the set of all $2$-element subsets of $M$,\n$\\iseq(M)$ denotes the set of all finite sequences without repetitions\nwhich can be formed with elements of $M$, and $\\seq(M)$\ndenotes the set of all finite sequences which can be formed with elements of $M$\n(where repetitions are allowed).\n\nFurthermore, for a set $A$, let $\\card{A}$ denote the cardinality of $A$. \nWe write $\\card A=\\card B$, if there exists a bijection between $A$ and $B$,\nand we write $\\card A\\le\\card B$, if there exists a bijection between $A$\nand a subset $B'\\subs B$ (\\ie, $\\card A\\le\\card B$ if and only if there \nexists an injection from $A$ into $B$). Finally, we write $\\card A<\\card B$\nif $\\card A\\le\\card B$ and $\\card A\\neq\\card B$. By the \n{\\sc Cantor-Bernstein Theorem},\nwhich is provable in $\\ZF$ only (\\ie, without using the Axiom of Choice),\nwe get that $\\card A\\le\\card B$ and $\\card A\\ge\\card B$ implies \n$\\card A=\\card B$. \n\nLet $\\mathfrak{m}:=|M|$, and let $[\\mathfrak{m}]^2:=\\card{[M]^2}$, $\\mathfrak{m}^2:=\\card{M^2}$, \n$\\fin(\\mathfrak{m}):=\\card{\\fin(M)}$, $\\iseq(\\mathfrak{m}):=\\card{\\iseq(M)}$,\nand $\\seq(\\mathfrak{m}):=\\card{\\seq(M)}$. \nConcerning these cardinalities, in $\\ZF$ we obviously have $\\iseq(\\mathfrak{m})\\le \n\\seq(\\mathfrak{m})$, $[\\mathfrak{m}]^2\\le \\fin(\\mathfrak{m})$ and $\\mathfrak{m}^2\\le\\iseq(\\mathfrak{m})$, \nwhere the latter relations are visualized\nby the following diagram (in the diagram, $\\mathfrak{n}_1$ is below \n$\\mathfrak{n}_2$ if $\\mathfrak{n}_1\\le\\mathfrak{n}_2$):\n\n{\\small\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{-}[d]& &\\iseq(\\mathfrak{m})\\ar@{-}[d]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\n}\n\\]\n\nMoreover, for \\emph{finite\\\/} cardinals $\\mathfrak{m}$ with $\\mathfrak{m}\\ge\\mathfrak{5}$ we have\n$$[\\mathfrak{m}]^2<\\mathfrak{m}^2<\\fin(\\mathfrak{m})<\\iseq(\\mathfrak{m})\\,,$$\nand in the presence of the Axiom of Choice ({\\ie}, in $\\ZFC$),\nfor every infinite cardinal $\\mathfrak{m}$ we have\n$$[\\mathfrak{m}]^2=\\mathfrak{m}^2=\\fin(\\mathfrak{m})=\\iseq(\\mathfrak{m})\\,.$$\nIt is natural to ask whether some of these equalities can be proved also\nin $\\ZF$, {\\ie}, without the aid of $\\AC$. Surprisingly, this is not the\ncase. In~\\cite{carla}, a permutation model was constructed in which for\nan infinite cardinal $\\mathfrak{m}$ we have $\\seq(\\mathfrak{m})<\\fin(\\mathfrak{m})$\n(see~\\cite[Thm.\\,2]{carla} or~\\cite[Prp.\\,7.17]{cst1st}). \nAs a consequence we obtain that the existence\nof an infinite cardinal $\\mathfrak{m}$ such that $\\iseq(\\mathfrak{m})<\\fin(\\mathfrak{m})$ is \nconsistent with $\\ZF$. This consistency result was modified\nto the existence of an infinite cardinal \n$\\mathfrak{m}$ for which $\\mathfrak{m}^2<[\\mathfrak{m}]^2$ (see~\\cite[Prp.\\,7.18]{cst1st}), \nand later, it was strengthened to the existence of an infinite cardinal \n$\\mathfrak{m}$ for which $\\seq(\\mathfrak{m})<[\\mathfrak{m}]^2$ (see~\\cite{weird} or~\\cite[Prp.\\;8.28]{cst}).\nThe consistency of $\\fin(\\mathfrak{m})<\\iseq(\\mathfrak{m})$ for infinite cardinals $\\mathfrak{m}$ can be obtained\nwith the {\\it Ordered Mostowski Model\\\/} (see, for example, \n\\cite[Related\\;Result\\,48,\\;p.\\,217]{cst}), in which there is an \ninfinite cardinal $\\mathfrak{m}$ with $$[\\mathfrak{m}]^2<\\mathfrak{m}^2<\\fin(\\mathfrak{m})<\\iseq(\\mathfrak{m})\\,.$$\nConsistency results as well as $\\ZF$-results concerning the relations between these \ncardinals with other cardinals can be found, for example, \nin~\\cite{Shen1,Shen2} or~\\cite{berlin}.\n\nConcerning the four cardinalities $[\\mathfrak{m}]^2$, $\\mathfrak{m}^2$, $\\fin(\\mathfrak{m})$, and $\\iseq(\\mathfrak{m})$, \na question which arises naturally is whether for some infinite cardinal $\\mathfrak{m}$, \n$\\fin(\\mathfrak{m})<\\mathfrak{m}^2$ is consistent with $\\ZF$ (see \\cite[Related\\;Result\\,20,\\;p.\\,133]{cst}).\nMoreover, assuming that $\\mathfrak{m}$ is infinite and the four cardinalities \n$[\\mathfrak{m}]^2$, $\\mathfrak{m}^2$, $\\fin(\\mathfrak{m})$, $\\iseq(\\mathfrak{m})$ are pairwise distinct and pairwise \ncomparable in~$\\ZF$, one may ask which linear orderings on these four cardinalities\nare consistent with~$\\ZF$.\n\nSince for cardinals $\\mathfrak{m}\\ge\\mathfrak{5}$,\nwe cannot have $[\\mathfrak{m}]^2>\\fin(\\mathfrak{m})$ or $\\mathfrak{m}^2>\\iseq(\\mathfrak{m})$, there are only the following\nsix linear orderings on these four cardinalities which might be consistent with $\\ZF$\n(where for two cardinals $\\mathfrak{n}_1$ and $\\mathfrak{n}_2$, \n$\\mathfrak{n}_1\\longrightarrow\\mathfrak{n}_2$ means $\\mathfrak{n}_1<\\mathfrak{n}_2$). \n\n\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]\\ar@{->}[drr]& &\\iseq(\\mathfrak{m})\\ar@{<-}[d]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{N}}$}\n}\n\\hfill\n\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]\\ar@{->}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{C}}$}\n}\n\\hfill\n\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2\\ar@{->}[urr]& &\\mathfrak{m}^2\\ar@{->}[ll]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{Z}}$}\n}\n\\hfill{\\ }\n\n\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[d]& &\\iseq(\\mathfrak{m})\\ar@{<-}[d]\\ar@{->}[dll]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{\\reflectbox{N}}}$}\n}\n\\hfill\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{<-}[rr]& &\\iseq(\\mathfrak{m})\\ar@{<-}[d]\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{<-}[ll]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{\\reflectbox{C}}}$}\n}\n\\hfill\n{\\ }\\hfill\\parbox{.3\\textwidth}{\n\\[\n\\xymatrix@R=10mm@C=3mm{\n\\fin(\\mathfrak{m})\\ar@{->}[rr]& &\\iseq(\\mathfrak{m})\\\\ \n[\\mathfrak{m}]^2& &\\mathfrak{m}^2\\ar@{<-}[ll]\\ar@{->}[ull]\n}\n\\]\n\\centerline{\\small{\\it Diagram\\;}$\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}$}\n}\n\\hfill{\\ }\n\nBelow we show that each of the five diagrams $\\text{\\bfseries\\sffamily{N}}$, $\\text{\\bfseries\\sffamily{Z}}$, $\\text{\\bfseries\\sffamily{\\reflectbox{N}}}$, $\\text{\\bfseries\\sffamily{\\reflectbox{C}}}$, \n$\\text{\\bfseries\\sffamily{\\reflectbox{Z}}}$ is consistent with $\\ZF$. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nEntanglement is the most important resource in quantum information, with a vast array of practical uses \\cite{HHHH09}. In physics more generally, understanding the entanglement structure in physical systems is becoming increasingly important, e.g., in condensed matter theory where quantum phase transitions are signalled by long-range entanglement \\cite{OAFF01,ON02,VLRK03,AFOV08,LR09}. In high energy physics, quantifying the entanglement of states of a quantum field has applications to a variety of problems \\cite{SW85,Summers08,KST17}, from the AdS\/CFT correspondence \\cite{RT06}, through to detecting spacetime curvature by probing vacuum entanglement \\cite{SM09} or harvesting this entanglement by locally coupling small systems to the field \\cite{Valentini91,Reznik03,RRS05,SR07,PM15,PM16,SM17,SMM17}.\n\nQuantifying entanglement in states of quantum fields is a nontrivial task due to the UV dependence of quantum entanglement near a boundary, leading to naive divergences \\cite{HLW94,CC04,CC06}. Many ad hoc approaches have been developed to deal with these divergences, usually relying on subtracting the UV divergent piece \\cite{HLW94}. Of course, operationally, there are no such divergences in the entanglement we can measure because any apparatus we can build to extract entanglement from the field vacuum would only use a finite amount of energy.\n\nSurprisingly little work has been done in the quantum information literature on the problem of quantifying accessible entanglement subject to an energy constraint. Here we build a theory for this problem and use it to understand the energy cost of extracting entanglement via local operations and classical communication (LOCC). We use the term extraction rather than one-shot distillation \\cite{BD10}, as we want to emphasise that we are not necessarily distilling all the entanglement from a state. In contrast, we wish to quantify the \\textit{optimal} energy cost per EPR pair extracted. While individual protocols for entanglement extraction are interesting, we are primarily concerned with the protocol that minimises the energy cost.\n\nThere are some very interesting related ideas in the literature:\\ in \\cite{CY17}, general quantum operations costing \\textit{zero} energy are studied. Also, the energy cost of \\textit{creating} entanglement in specific many-body systems was calculated in \\cite{GL09}. Similarly, in the setting of quantum thermodynamics, the energy\/work cost of creating correlations in quantum systems was studied in \\cite{HPHSKBA15,BPFHH15,FHP16}. These give useful strategies for creating correlations between finite dimensional systems or a pair of bosons or fermions using energy-conserving (global) unitary operations in the presence of heat baths. In \\cite{DKPSSS17}, entanglement distillation is considered (also in the presence of a heat bath) with an energy constraint:\\ asymptotically many entangled pairs are distilled into EPR pairs, with the constraint that the energy before and after is \\textit{equal}. In \\cite{MBDK15}, using a specific local entanglement harvesting protocol (called entanglement farming), the energy cost in the low energy regime was calculated. In contrast, here we are interested in how the \\textit{optimal} energy cost scales with the number of EPR pairs extracted and in the overall entanglement structure of states, which is a rather different question.\n\nIn this article, we first provide the setting and define the energy cost of a quantum operation. Then we introduce entanglement extraction subject to an energy constraint. Following this, we explore the idea using a toy model, which is chosen to share many of the features of the vacuum state of a quantum field but to also be relatively simple. Next, we introduce the entanglement temperature, which relates the amount of entanglement extracted to the energy cost. In the following section, we look at the energy cost of entanglement extraction in quantum field theories using physical arguments. Then we discuss some general methods for quantifying the energy cost of entanglement extraction. In particular, one of these methods uses matrix product states to numerically bound the energy cost of entanglement extraction for some condensed matter systems and to plot the entanglement temperature. We conclude with an outlook.\n\n\\section{Preliminaries}\nWe focus on a simplified setup exemplifying the core features of our problem. To whit, we focus on the setting where two players, Alice and Bob, have access to a bipartition of a common system with Hilbert space $\\mathcal{H}_{AB}$. This system, which we refer to as the \\textit{physical system}, has a Hamiltonian $H_{AB}$, which neither Alice nor Bob can modify. Alice and Bob also have access to local ancillary degrees of freedom $A'B'$, which they can use to store the entanglement they extract from the physical system. Thus, the total Hilbert space for the system is given by \n\\begin{equation}\n\t\\mathcal{H}_{AA'BB'} \\equiv \\mathcal{H}_{AB}\\otimes \\mathcal{H}_{A'B'}. \n\\end{equation} \nWe assume that Alice and Bob can carry out any local operation they like on the ancillary degrees of freedom with no energy cost. Thus we associate to the total system+ancilla system the Hamiltonian $H=H_{AB}\\otimes\\openone_{A'B'}$.\n\\begin{figure}[ht!]\n \\resizebox{8.5cm}{!}{\\includegraphics{Intro_fig_01.png}}\n \\footnotesize{\\caption[Basic Idea]{Energy cost of entanglement extraction:\\ Alice and Bob have access to two parts of a quantum system in an entangled state $\\ket{\\Omega}_{AB}$ with Hamiltonian $H_{AB}$. Using local operations and classical communication (LOCC), they extract $m$ EPR pairs into their ancillary systems $A^{\\prime}$ and $B^{\\prime}$, leaving the physical system in the final state $\\ket{\\psi}_{AB}$. Dropping subscripts, the energy cost is then $\\Delta E = \\bra{\\psi}H\\ket{\\psi}-\\bra{\\Omega}H\\ket{\\Omega}$. \\label{fig:intro}}}\n\\end{figure}\n\nWe assume that Alice and Bob can only perform local operations and classical communication (LOCC). We also suppose that Alice and Bob are working in the one-shot regime, which is natural if, for example, we are thinking of understanding the entanglement structure of vacuum states in quantum field theory where there is only one copy of the system available. In contrast, in the asymptotic many-copy regime we could use entanglement distillation protocols \\cite{HHHH09}. We will also comment on the energy cost in the asymptotic regime.\n\nWe quantify the energy cost as follows. Suppose we have a completely positive trace-preserving map $\\mathcal{E}:\\mathcal{S}(\\mathcal{H})\\rightarrow \\mathcal{S}(\\mathcal{H})$ acting on the space $\\mathcal{S}(\\mathcal{H})$ of density operators on a Hilbert space $\\mathcal{H}$. Suppose we also have a Hamiltonian $H \\in \\mathcal{B}(\\mathcal{H})$. Given a state $\\rho \\in \\mathcal{S}(\\mathcal{H})$, the operation $\\mathcal{E}$ induces an energy change \n\\begin{equation}\n\t\\Delta(\\mathcal{E}, \\rho) = \\operatorname{tr}(\\mathcal{E}(\\rho)H) - \\operatorname{tr}(\\rho H)\n\\end{equation}\nwhen acting on $\\rho$ (this can be negative). This is the energy cost when we apply the channel $\\mathcal{E}$ to the state $\\rho$. We may also define the \\textit{energy cost for the channel} $\\mathcal{E}$, which involves an optimisation:\\ we imagine that an adversary prepares the system in the state $\\rho$ which leads to the \\textit{largest} possible change in energy after application of $\\mathcal{E}$:\n\\begin{equation}\n\t\\Delta(\\mathcal{E}) = \\sup_{\\rho \\in \\mathcal{S}(\\mathcal{H})} \\operatorname{tr}(\\mathcal{E}(\\rho)H) - \\operatorname{tr}(\\rho H).\n\\end{equation}\nExploiting the variational definition of the operator norm $\\|\\cdot\\|_\\infty$, we notice that\n\\begin{equation}\n\t\\Delta(\\mathcal{E}) = \\|H-\\mathcal{E}^*(H)\\|_\\infty,\n\\end{equation}\nwhere $\\mathcal{E}^*$ is the dual of $\\mathcal{E}$ acting in the Heisenberg picture. In the following, we will typically be interested in $\\Delta(\\mathcal{E},\\rho)$ for a specific state and channel, or the set of LOCC channels with $\\Delta(\\mathcal{E},\\rho)\\leq \\Delta E$. We denote this set by $\\mathcal{C}_{\\text{LOCC}}(\\Delta E)$.\n\n\\section{Extracting entanglement subject to an energy constraint}\nHere we propose a definition for the entanglement accessible to Alice and Bob when they have access only to operations costing less than $\\Delta E$. \n\nWe imagine that the system $AB$ starts in a state $\\sigma_{ABA^{\\prime}B^{\\prime}}=|\\Omega\\rangle_{AB}\\bra{\\Omega}\\otimes|00\\rangle_{A'B'}\\!\\bra{00}$ where $|0\\rangle$ is a convenient fiducial state of the ancilla and $|\\Omega\\rangle_{AB}$ is the initial state of the physical system. Alice and Bob are now allowed to carry out LOCC operations costing less than $\\Delta E$ in total to maximise the quantum entanglement between $A'$ and $B'$. Suppose that $\\mathcal{E}\\in \\mathcal{C}_{\\text{LOCC}}(\\Delta E)$, then we write $\\rho_{AA'BB'} = \\mathcal{E}(\\sigma_{ABA^{\\prime}B^{\\prime}})$. Thus we define the \\emph{entanglement accessible with energy $\\Delta E$} to be\n\\begin{equation}\n\t\\text{Ent}_{\\Delta E}(|\\Omega_{AB}\\rangle) \\equiv \\sup_{\\mathcal{E}\\in \\mathcal{C}_{\\text{LOCC}}(\\Delta E)} \\text{Ent} (\\rho_{A'B'}),\n\\end{equation}\nwhere $\\text{Ent}$ is some convenient entanglement measure.\n\nWe also define the \\emph{energy cost of extracting $m$ EPR pairs} to be $\\Delta E=\\min\\Delta\\left(\\mathcal{E},\\sigma_{ABA^{\\prime}B^{\\prime}}\\right)$, where the minimum is over all LOCC channels satisfying $\\rho_{A'B'}=\\ket{\\phi^{+\\otimes m}}_{A^{\\prime}B^{\\prime}}\\bra{\\phi^{+\\otimes m}}$, and $\\ket{\\phi^{+}}=\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle)$ is a maximally entangled state of two qubits.\n\nIt is well possible that after extracting entanglement the energy of the system can \\textit{go down}, i.e., extracting entanglement can cool the system. This all depends on the state $|\\Omega_{AB}\\rangle$, i.e., whether it is an excited state or ground state. Since the emphasis in this paper is on ground states, we assume henceforth that $|\\Omega_{AB}\\rangle$ is the ground state of $H_{AB}$.\n\nA key ingredient in any entanglement extraction protocol is the strength of the interaction between Alice's and Bob's systems. If we write $H_{AB}=H_A\\otimes \\openone_B + \\openone_A \\otimes H_B + V_{AB}$, then the limitations on how much entanglement Alice and Bob can extract using LOCC are determined by $V_{AB}$. Indeed, if $V_{AB}=0$, the ground state $\\ket{\\Omega}$ will have no entanglement between Alice's and Bob's systems.\n\nThere is a useful naive protocol for entanglement extraction:\\ Alice and Bob first swap the states of their primed and non-primed systems. Then they can prepare a state of the physical $AB$ system (using LOCC) with minimal \\textit{local} energy, meaning Alice\/Bob prepares $\\ket{\\psi_{A\/B}}$, such that $\\bra{\\psi_{A\/B}} H_{A\/B} \\ket{\\psi_{A\/B}}$ is minimised. The total energy change is, with $\\ket{\\psi}=\\ket{\\psi_A}\\ket{\\psi_B}$,\n\\begin{equation}\n\\begin{split}\n & \\bra{\\psi} H \\ket{\\psi}- \\bra{\\Omega} H\\ket{\\Omega}\\\\\n \\leq & \\bra{\\psi} V_{AB} \\ket{\\psi}- \\bra{\\Omega} V_{AB}\\ket{\\Omega} \\leq 2\\|V_{AB}\\|_{\\infty}.\n \\end{split}\n\\end{equation}\nTherefore, when the coupling is sufficiently weak, Alice and Bob can safely extract all the entanglement whilst only incurring a small energy cost.\n\nIn contrast, for strong couplings the situation is entirely different, which is exactly the case for quantum field theories, where extracting all the entanglement costs a divergent amount of energy. For the example of a free fermion field, we see in the appendix that all product states $\\ket{\\psi}$ satisfy $\\bra{\\psi}H\\ket{\\psi}\\geq 1\/a$, where $a$ is the regulator (the lattice spacing in this case). Thus, the energy diverges as $a\\rightarrow 0$ for any product state, meaning that extracting all the entanglement costs a diverging amount of energy. In general, the energy cost for extracting all the entanglement will diverge in quantum field theory. Again using a lattice regulator, the energy contained in the interaction terms between a region $A$ and the rest scales like $(\\partial A\/a^{d})$, where $\\partial A$ is the boundary of $A$, which is also shown in the appendix.\n\n\\section{A toy model}\nIn this section we discuss an idealised model, which exemplifies many of the features of the quantum field vacuum. It has high entanglement and a high energy cost for extracting all this entanglement, as we will see. \n\nSuppose that the system $AB$ is actually composed of $2n$ qubits, with $n$ qubits in $A$ and $n$ qubits in $B$. We call the qubits $A_j$ (respectively, $B_j$), for $j=1,2, \\ldots, n$. We suppose that $H_{AB}$ is given by \n\\begin{equation}\n\tH_{AB} = \\sum_{j=1}^n (\\mathbb{I}-P_{A_jB_j}),\n\\end{equation}\nwhere $P_{A_jB_j}$ is the projector onto the maximally entangled state $\\ket{\\Phi^{+}}=\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle)$ of qubits $A_j$ and $B_j$. The ground state $|\\Omega_{AB}\\rangle$ of $H_{AB}$ is thus a product of maximally entangled pairs, i.e., it is a maximally entangled state between $A$ and $B$.\n\nIf Alice and Bob could do arbitrary LOCC, then they could easily extract $n$ EPR pairs. However, if they are only allowed an energy cost of $\\Delta E$, then naively they should only be able to extract $O(\\Delta E)$ EPR pairs. \n\nIn the most extreme case, Alice and Bob fully extract all the EPR pairs. Then, in order that this entanglement is between ancilla degrees of freedom in $A'$ and $B'$, it must be that $A$ and $B$ are in a separable state $\\sigma_{AB}$. Since the energy depends linearly on $\\sigma_{AB}$, we may as well suppose that $\\sigma_{AB}$ is an extreme point of the convex set of separable states, namely, a product state $|\\phi\\rangle_A|\\psi\\rangle_B$. The energy of our initial state $|\\Omega\\rangle_{AB}$ was zero, so the energy cost of any entanglement extraction procedure must be greater than \n\\begin{equation}\n\t\\inf_{|\\phi\\rangle_A|\\psi\\rangle_B} \\sum_{j=1}^n (1-\\langle \\phi_A|\\langle \\psi_B|P_{A_jB_j}|\\phi_A\\rangle|\\psi_B\\rangle).\n\\end{equation}\nThis infimum is achieved by finding the supremum:\n\\begin{equation}\n \t\\sup_{|\\phi\\rangle_A|\\psi\\rangle_B} \\langle \\phi_A|\\langle \\psi_B|P_{A_jB_j}|\\phi_A\\rangle|\\psi_B\\rangle,\n\\end{equation} \nwhich is equal to $1\/2$. (E.g., setting each pair to $\\ket{00}_{A_iB_i}$ will do the job.) Thus, the energy cost is given by\n\\begin{equation}\n\t\\Delta E \\ge \\frac{1}{2}\\sum_{j=1}^n 1 = n\/2.\n\\end{equation}\n\nMore generally, suppose Alice and Bob extract fewer EPR pairs (say $m$). One option is to use the following simple protocol. They swap the states of the first $m$ EPR pairs of the physical system into their ancilla systems, which they can do using local operations. The first $m$ pairs of qubits of the physical system are now in a product pure state. Then they can apply local unitaries mapping each of these qubit pairs to the state $\\ket{00}_{A_iB_i}$, getting the final energy cost\n\\begin{equation}\n\t\\Delta E = \\frac{1}{2}\\sum_{j=1}^m 1 = m\/2.\n\\end{equation}\nThus, the total energy cost is $1\/2$ per EPR pair extracted.\n\nOf course, there may be a protocol extracting the same amount of entanglement but costing less energy. Here we will argue that the simple protocol given above is, in fact, optimal.\n\nWe assume that after applying their operations, Alice and Bob get $m$ Bell states in the ancilla $\\ket{\\Phi^{+\\otimes m}}_{A'B'}$ and some pure state in the physical system $\\ket{\\psi}_{AB}$. Denote the Schmidt values (decreasingly ordered) of the initial state $\\ket{\\Omega}_{AB}\\ket{00}_{A'B'}$ by $\\alpha_i$, and note that the Schmidt rank is $2^n$. For it to be possible to transform this state into the new state $\\ket{\\psi}_{AB}\\ket{\\Phi^{+\\otimes m}}_{A'B'}$, with Schmidt values $\\beta_i$, the majorization condition \\cite{NC00} must be satisfied:\n\\begin{equation}\n \\forall K\\geq 1\\ \\ \\ \\ \\sum_{i=1}^{K}\\alpha_i\\leq \\sum_{i=1}^{K}\\beta_i.\n\\end{equation}\nFor this to be possible, the Schmidt rank of the resulting state must be smaller. This implies that the Schmidt rank of the new state of the physical system $\\ket{\\psi}_{AB}$ can be at most $2^{n-m}$.\n\n\\begin{figure}[ht!]\n\n \\setlength{\\figheight}{5cm}\n \\setlength{\\figwidth}{8.5cm}\n \\def.875{.875}\n\n\n \\input{.\/Toy.tikz} \n \\footnotesize{\\caption[Toy Model]{This figure shows the minimum energy cost of extracting entanglement for the toy model. More precisely, the figure shows the minimum change in energy $\\Delta E$ when there is a decrease in zero entropy $\\Delta S_0$ (with $S_0$ equal to $\\log_2$ of the Schmidt rank) of the state of the physical system.\\label{fig:toymodel} When $\\Delta S_0$ is an integer $m$ (which corresponds to extracting $m$ EPR pairs from the system), then the plot shows $\\Delta E =0.5m$. The calculation was performed using DMRG by restricting the bond dimension between Alice and Bob's systems.}}\n\\end{figure}\n\nFigure \\ref{fig:toymodel} shows numerics from a DMRG calculation of the minimum increase in energy as the Schmidt rank of the state of the physical system decreases. Based on these numerical results, we see that the minimum increase in energy when the Schmidt rank decreases by a factor of $2^m$ is $0.5 m$. This can be achieved by the simple protocol of the previous section, indicating that this protocol is optimal. Actually, this whole argument also goes through even if Alice and Bob have some additional shared entanglement that can be used as a catalyst, as in \\cite{JP99}.\n\nIn terms of entanglement distillation in the asymptotic setting, this is not optimal. In that case, one can distil entanglement at a lower energy cost, which we show later. In practice, however, we only have access to one copy of a quantum field or condensed matter system, so it is crucial to consider the one-shot setting. Furthermore, entanglement distillation protocols rely on projecting onto typical subspaces defined by the singular vectors of the initial state \\cite{NC00}, which for extremely complex systems would be practically impossible.\n\n\\section{The entanglement temperature}\nIn the previous section, the total energy cost was $1\/2$ per EPR pair extracted. To relate the change in entanglement entropy $\\Delta S$ to the energy cost $\\Delta E$, we define the \\textit{entanglement temperature} $T_{\\mathrm{ent}}$ by\n\\begin{equation}\n \\Delta E = T_{\\mathrm{ent}} \\Delta S.\n\\end{equation}\n(The name entanglement temperature is chosen in analogy with thermodynamics.) So $T_{\\mathrm{ent}}$ is a property of the ground state of a system. For the toy model, we see that $T_{\\mathrm{ent}}=1\/2$ since $\\Delta S=m\\log_{2}(2)=m$. In this case $T_{\\mathrm{ent}}$ is constant because there is a linear relationship between the entanglement extracted and the energy cost. For general systems, we would not expect $\\Delta E \\propto \\Delta S$ for the entire range of $\\Delta S$. Instead, we should think of the entanglement temperature as a function of the extracted entanglement. (This is also true in thermodynamics, where temperature can often be thought of as a function of other state functions, such as entropy or pressure.)\n\nIn the following sections, we give some physical and numerical arguments to find $\\Delta E$ as a function of $\\Delta S$ and hence find the entanglement temperature for some physical systems.\n\n\\section{The Energy Cost in General}\n\\label{sec:General Strategy}\nFor some quantum field theories or condensed matter systems, we can give a physical argument for the energy cost of entanglement extraction. In one dimensional systems, often the entanglement entropy (or, for example, the logarithmic negativity) of ground states can be calculated. This typically has the form $S(\\rho_I)=c_1\\log_2(N)+c_2$, where $c_i$ are constants and $\\rho_I$ is the state restricted to a contiguous region with $N$ sites \\cite{Korepin04,ECP10}. For a quantum field theory, regulated by a lattice with lattice spacing $a$, we have instead $S(\\rho_I)=c_1\\log_2(l\/a)+c_2$, where $l$ is the length of a region. Also, the entanglement entropy in the ground state of models close to the critical point is \\cite{CC04,ECP10}\n\\begin{equation}\n S(\\rho_I)=\\frac{c}{6}\\log_2(\\xi\/a),\n\\end{equation}\nwhere $\\xi\\gg a$ is the correlation length, $c$ is a constant and $I$ corresponds to the infinite half-line $(-\\infty,0]$. This is equivalent to a massive relativistic QFT with $1\/\\xi$ equal to the mass, e.g., for free bosons we have $c=1$.\n\nAs argued previously, the energy cost for extracting all of this entanglement is $\\Delta E = O(1\/a)$. At lattice spacing, $a=\\xi\/2^{6m\/c}$, we have $S(\\rho_I)=m$, meaning that the \\textit{most} entanglement we can extract is $m$ EPR pairs. (In some cases, exactly half of this entanglement is one-shot distillable \\cite{OLEC06}.) By probing higher energies, which corresponds to smaller values of $a$, we can extract more entanglement, and we have the energy cost $\\Delta E\\propto \\exp(Km)$, where $K=6\\ln(2)\/c$. This means that the energy cost of entanglement extraction increases \\textit{exponentially}:\\ there is infinite entanglement in the quantum field vacuum, but the cost of extraction grows quickly. (For gapless models the same argument goes through if Alice has access to a finite region and Bob has access to the rest. In contrast, if Alice's system is a halfline, there is infinite entanglement at any energy scale.)\n\nThe scaling is different for quantum fields in higher dimensional spaces. In many cases, the entanglement entropy of the ground state obeys an area law \\cite{RT06b}. Then in a region $A$ with area $\\partial A$, the leading contribution to the entropy is $S(\\rho_A)\\propto \\partial A\/a^{d-1}$. However, the energy cost of extracting all the entanglement scales like $\\Delta E \\propto \\partial A\/a^{d}$. Thus we get an idea for the energy cost of entanglement extraction:\\ $\\Delta E \\propto \\Delta S^{d\/(d-1)}$. And the entanglement temperature is then $T_{\\mathrm{ent}}\\propto 1\/a\\propto \\Delta E^{1\/d}$. The energy cost of entanglement extraction in QFT are plotted in figure \\ref{fig:qft}.\n\n\\begin{figure}\n\t\\setlength{\\figheight}{4cm}\n\t\\setlength{\\figwidth}{8cm}\n\t\\def.875{1}\n\n\n\t\\!\\!\\input{.\/QFT.tikz}\n\t\\footnotesize{\\caption{The energy cost of entanglement extraction from the quantum field vacuum depends heavily on the spatial dimension. Here we sketch the behaviour in dimensions $d=1,2,3$. When $d=1$, $\\Delta E\\propto \\exp(K\\Delta S)$, where $K$ is a constant, and for $d>1$, $\\Delta E \\propto \\Delta S^{d\/(d-1)}$.}\\label{fig:qft}}\n\\end{figure}\n\nFor more general systems, there is no clear way to proceed. Below we outline some potential methods to approach the problem. In the first two cases, we suppose that Alice and Bob use some LOCC protocol to extract $m$ EPR pairs into their ancillary systems, and in the third we consider the trade-off between entanglement change and energy cost numerically.\n\n\\subsection{Method I}\nAlice and Bob have the initial state $|\\Omega\\rangle_{AB}\\ket{00}_{A^{\\prime}B^{\\prime}}$, and then they apply some LOCC protocol to extract $m$ EPR pairs into the ancilla $A^{\\prime}B^{\\prime}$. We assume that the resulting state on the physical system after the protocol is also pure $\\ket{\\psi}_{AB}$. (It is possible that a protocol giving a mixed state on the physical system may be more efficient. In this case, we may use a superadditive entanglement measure, like the squashed entanglement \\cite{CW04}, to upper bound the entanglement left in the physical system.) Because they are using LOCC, the overall entanglement can only decrease:\n\\begin{equation}\n\\begin{split}\n \\text{Ent}\\left(|\\psi\\rangle_{AB}\\ket{\\Phi^+}^{\\otimes m}_{A^{\\prime}B^{\\prime}}\\right) &\\leq\\text{Ent}\\left(|\\Omega\\rangle_{AB}\\ket{00}_{A^{\\prime}B^{\\prime}}\\right)\\\\\n & = S_{\\mathrm{initial}},\n \\end{split}\n\\end{equation}\nwhere $S_{\\mathrm{initial}}$ is the initial entanglement entropy in the state $|\\Omega\\rangle_{AB}$ and $\\text{Ent}$ is an entanglement measure, which we take to be the entanglement entropy, since the states are all pure. Then we have that the entanglement entropy in the final state of physical system is $\\text{Ent}\\left(|\\psi\\rangle_{AB}\\right)\\leq S_{\\mathrm{initial}}-m$.\n\nSo what is the minimum energy cost of extracting this entanglement? We can get an idea by finding the state (or set of states) that have this final value of entanglement entropy while minimising the energy. In the appendix, we derive the equation\n\\begin{equation}\\label{eq:La}\n \\left[H - \\mu_1\\openone_A\\otimes\\log(\\rho_B) - \\mu_1 +\\mu_2\\right]\\ket{\\psi}_{AB}=0,\n\\end{equation}\nwhere $\\mu_i$ are Lagrange multipliers and $\\mathrm{tr}_A[\\ket{\\psi}\\bra{\\psi}]=\\rho_B$.\n\nThis is difficult to solve in general but may be simplified if we know something about the structure of $H$. This is the case for the toy model, where $H$ is a sum of commuting terms acting on different pairs of qubits $A_iB_i$.\n\nWith the ansatz $\\ket{\\psi}_{AB}=\\ket{\\psi_1}_{A_1B_1}\\otimes...\\otimes \\ket{\\psi_n}_{A_nB_n}$, we see from equation (\\ref{eq:La}) that each $\\ket{\\psi_i}_{A_iB_i}$ should have the same Schmidt vectors as $\\ket{\\Phi^{+}}_{A_iB_i}$. One possible solution is to take all qubit pairs to be in the same state:\\ $\\ket{\\psi}_{AB}=\\ket{\\phi}_{A_1B_1}\\otimes...\\otimes \\ket{\\phi}_{A_nB_n}$, where $\\ket{\\phi}=\\alpha\\ket{00}+\\beta\\ket{11}$. Then, since $S_{\\mathrm{initial}}=n$, one need only solve\n\\begin{equation}\\label{eq:ent}\n n-m= -n[\\alpha^2\\log_2(\\alpha^2)+\\beta^2\\log_2(\\beta^2)]\n\\end{equation}\nfor $\\alpha$ and $\\beta$. And the corresponding energy cost is $\\Delta E = n[1- (\\alpha+\\beta)^2\/2]$.\n\nFor example, with $m=n\/2$, one gets $\\Delta E\\simeq 0.38 m$. This is smaller than the optimal energy cost in the one-shot setting:\\ $\\Delta E= 0.5 m$. However, in the one-shot setting, Alice and Bob cannot prepare the state $\\ket{\\psi}_{AB}$ after extracting $m$ EPR pairs (because $\\ket{\\psi}_{AB}$ has maximal Schmidt rank). Interestingly, however, we get a nontrivial \\textit{upper} bound on the optimal energy cost of extracting entanglement in the asymptotic setting of many copies of this system. In the asymptotic setting, the criterion for deciding whether one bipartite entangled pure state can be transformed into another reversibly using LOCC is that the entanglement entropies are the same \\cite{NC00}. So we see that the energy cost of distilling $m$ EPR pairs (per copy of the physical system) in the asymptotic setting will be lower than in the one shot case.\n\n\\subsection{Method II}\nA second option is to maximise the overlap of the final state of the physical system (after the entanglement has been extracted) with its ground state. This gives a naive strategy at least. And for Hamiltonians of the form $H_{AB}=-\\ket{\\Omega}\\bra{\\Omega}$, we get an exact answer for the optimal energy cost.\n\nAs an example, take $\\ket{\\Omega}=(1\/\\sqrt{d})\\sum_{i=1}^{d}\\ket{i}_A\\ket{i}_B$, where $d=2^n$. Suppose that Alice and Bob extract $m$ EPR pairs using LOCC, leaving a pure state $\\ket{\\psi}$ in the physical system. Using the majorization criterion, this can be any state with Schmidt rank up to $K=2^{n-m}$. To minimise the energy cost, we need to find such a state having maximal overlap with $\\ket{\\Omega}$.\n\nWe may write the optimal $\\ket{\\psi}$ in its Schmidt basis as $\\sum_{i=1}^{K}\\alpha_i\\ket{a_i}_A\\ket{b_i}_B$. Next notice that \n\\begin{equation}\n\\left(\\sum_{i=1}^{d}\\bra{i}_A\\bra{i}_B\\right)\\ket{a_i}_A\\ket{b_i}_B\\leq 1.\n\\end{equation}\nThen we have\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle\\leq \\frac{1}{\\sqrt{d}}\\sum_{i=1}^K\\alpha_i\\leq \\sqrt{\\frac{K}{d}}=2^{-m\/2}.\n\\end{equation}\nTherefore, we see that the energy cost for extracting $m$ EPR pairs is $\\Delta E = 1-1\/2^m$.\n\n\\subsection{Method III}\nA third option is to consider the trade-off between entanglement and energy numerically. For a given Hamiltonian $H$, we consider a procedure in which the system starts in the ground state $\\ket{\\Omega}$, some entanglement is extracted, and the system is left in a final state $\\ket{\\psi}$. The energy cost of this procedure is \\mbox{$\\left\\langle \\psi \\middle|H\\middle|\\psi \\right\\rangle-\\left\\langle \\Omega \\middle|H\\middle|\\Omega \\right\\rangle$}, and the extracted entropy is upper bounded by \\mbox{$\\text{Ent}(\\ket{\\Omega})-\\text{Ent}(\\ket{\\psi})$}. In the asymptotic many-copy case, this is exactly the extracted entanglement entropy. We denote the entanglement temperature in that case by $T_{\\mathrm{ent}}^A$. We have that\n\\begin{align}\n\tT^{A}_{\\mathrm{ent}}\\leq \\frac{\\left\\langle \\psi \\middle|H\\middle|\\psi \\right\\rangle-\\left\\langle \\Omega \\middle|H\\middle|\\Omega \\right\\rangle}{\\text{Ent}(\\ket{\\Omega})-\\text{Ent}(\\ket{\\psi})}.\n\\end{align}\nNote that for a given amount of extracted entanglement $\\Delta S$, the one-shot entanglement temperature is lower bounded by the asymptotic-setting entanglement temperature $T_{\\mathrm{ent}}^A \\leq T_{\\mathrm{ent}}$.\n\nA given state does not necessarily give a tight bound on $T_{\\mathrm{ent}}^A$. For this we need to study the optimal trade-off between entanglement and energy, which is given by a \\emph{Pareto front}. By randomly generating states with low energy \\emph{and} low entanglement, we can numerically evaluate the above upper bound, and use this to compute the Pareto front. We describe a tensor network method for generating such samples in the appendix. In Figure~\\ref{fig:pareto} we present numerical results for two 1D spin models.\n\n\\begin{figure}\n\t\\setlength{\\figheight}{5cm}\n\t\\setlength{\\figwidth}{3.75cm}\n\t\\def.875{.875}\n\n\n\t\\! \\input{.\/HAF.tikz} \\!\\! \\input{.\/TIM.tikz} \\!\n\t\\footnotesize{\\caption{Asymptotic-case entanglement temperature ($T^A_{\\mathrm{ent}}= \\Delta E\/\\Delta S$) as a function of entropy change $\\Delta S$ for the critical Heisenberg anti-ferromagnet and the critical transverse-field Ising model, for multiple system sizes. Notice that near the ground state $T^A_{\\mathrm{ent}}\\propto \\Delta S$, which we prove is generic in the appendix. The one-shot entanglement temperature $T_{\\mathrm{ent}}$ is lower bounded by the asymptotic temperature $T_{\\mathrm{ent}}^A$.}\\label{fig:pareto}}\n\\end{figure}\n\n\\section{Outlook}\nWe introduced a framework to understand and quantify the energy cost of extracting entanglement from complex quantum systems. After looking at a toy model, which illustrated the key concepts, we defined the entanglement temperature. Then we analysed the energy cost of entanglement extraction in quantum field theories, and we saw that the energy cost of extracting entanglement depends on the spatial dimension. Finally, we looked at some general methods to approach the problem, including numerical methods for lattice models. \n\nQuantifying how much energy extracting $m$ EPR pairs costs in physical systems illuminates the entanglement structure of states, particularly ground states of, e.g., quantum fields. But it also can upper bound how efficient protocols such as entanglement harvesting can be. For general systems the optimal strategy for entanglement extraction may be hard to find. Still, it is heartening that, at least for quantum field theories, there is a relatively simple form of the entanglement temperature.\n\nIt would be interesting to combine the ideas here with those in \\cite{AMOP17}, where transformations between entangled states are considered using an additional resource:\\ an entanglement battery. This is a reservoir from which entanglement may be taken or deposited to facilitate state transformations, which may be impossible otherwise. One may then ask how this theory changes when there is also an energy cost associated with using the entanglement in the battery.\n\n\\section*{Acknowledgements}\nWe would like to thank David Reeb and Robin Harper for useful discussions.\n\nTF and TJO are supported by the DFG through SFB 1227 (DQ-mat) and the RTG 1991, the ERC grants QFTCMPS and SIQS, and the cluster of excellence EXC201 Quantum Engineering and Space-Time Research. CC acknowledges support from the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013, and from the AINST Postgraduate Scholarship (John Makepeace Bennett Gift). CB was supported by the research fund of Hanyang University (HY-2016-2237).\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection*{Introduction}\n Let $\\, A_1 := \\mathbb{C} \\langle x, \\, y \\rangle\/(xy-yx-1) \\,$\nbe the first Weyl algebra over $\\mathbb{C}$ with canonical generators $x$ and $y$. In his classic paper \\cite{D},\nDixmier described the group $\\, {\\rm{Aut}} \\, A_1 \\,$ of automorphisms of $ A_1 $: specifically, he proved that\n$\\, {\\rm{Aut}} \\, A_1 \\,$ is generated by the following transformations\n\\begin{equation}\n\\label{Phs}\n\\Phi_p\\,:\\,(x,\\,y) \\mapsto (x,\\,y + p(x))\\ ,\\qquad \\Psi_q\\,:\\, (x,\\,y) \\mapsto (x + q(y),\\,y)\\ ,\n\\end{equation}\nwhere $\\,p(x) \\in \\mathbb{C}[x]\\,$ and $\\,q(y) \\in \\mathbb{C}[y]\\,$. Using this result of Dixmier, Makar-Limanov (see \\cite{ML1, ML2})\nshowed that $\\, {\\rm{Aut}} \\, A_1 \\,$ is isomorphic to the group $\\, G_0 \\subset {\\rm{Aut}} \\,\\mathbb{C} \\langle x, \\, y \\rangle \\,$\nof `symplectic' (i.e. preserving $\\, \\omega = xy-yx$) automorphisms of the free algebra\n$\\,\\mathbb{C} \\langle x, \\, y \\rangle \\,$: the corresponding isomorphism\n\\begin{equation}\n\\label{isML}\nG_0 \\stackrel{\\sim}{\\to} {\\rm{Aut}} \\,A_1\n\\end{equation}\nis induced by the canonical projection $\\, \\mathbb{C} \\langle x, \\, y \\rangle \\to A_1\\,$. On the other hand, the results of\n\\cite{ML1} (see, e.g., \\cite{C}) also imply that $ G_0 $ is given by the amalgamated free product\n\\begin{equation}\n\\label{Aut}\nG_0 = A *_U B\\ ,\n\\end{equation}\nwhere $\\, A \\, $ is the subgroup of symplectic affine transformations\n\\begin{equation}\n\\label{A}\n(x,\\,y) \\mapsto (ax+by+e,\\,cx+dy+f)\\ , \\quad a,\\,b, \\ldots,\nf \\in \\mathbb{C}\\ ,\\quad ad-bc=1\\ ,\n\\end{equation}\n$\\, B \\, $ is the subgroup of triangular (Jonqui\\`eres) transformations\n\\begin{equation}\n\\label{B}\n(x,\\,y) \\mapsto (ax + q(y),\\,a^{-1}y+h)\\ , \\quad a \\in \\mathbb{C}^*,\\\nh \\in \\mathbb{C}\\ ,\\quad q(y) \\in \\mathbb{C}[y]\\ ,\n\\end{equation}\nand $\\, U\\,$ is the intersection of $ A $ and $ B $ in $ G_0 $:\n\\begin{equation}\n\\label{U}\n(x,\\,y) \\mapsto (ax+by+e,\\,a^{-1}y+h)\\ , \\quad a \\in \\mathbb{C}^*,\\\nb,\\,e,\\, h \\in \\mathbb{C}\\ .\n\\end{equation}\nCombining \\eqref{isML} and \\eqref{Aut}, we thus get decomposition $\\,{\\rm{Aut}}\\,A_1 \\cong A *_U B \\,$,\nwhich completely describes the structure of $\\,{\\rm{Aut}} \\,A_1\\,$ as a discrete group (cf. \\cite{A}).\n\nThe aim of the present paper is to generalize the above results to the case when\n$ A_1 $ is replaced by a noncommutative domain $D$, Morita equivalent to\n$A_1$ as a $\\mathbb{C}$-algebra. This question was originally posed by Stafford in \\cite{St}\n(see {\\it loc. cit.}, p. 636). To explain why it is natural, we recall that the algebras $\\,D\\,$\nare classified, up to isomorphism, by a single integer $\\, n \\ge 0 \\,$; the corresponding isomorphism classes\nare represented by the endomorphism rings $\\,D_n := \\End_{A_1} M_n \\,$ of certain distinguished right ideals\nof $ A_1 $ and can be realized geometrically as algebras of global differential operators on rational singular\ncurves (see \\cite{K, BW1} and \\cite{BW4} for a detailed exposition). Thus\nthe Dixmier group $\\,{\\rm{Aut}} \\,A_1 = {\\rm{Aut}} \\,D_0 \\,$ appears naturally as\nthe first member in the family $\\,\\{{\\rm{Aut}} \\,D_n\\, :\\, n \\ge 0 \\}\\,$. Our aim is to describe the `higher' groups\nin this family: in particular, to give a presentation of $\\,{\\rm{Aut}} \\,D_n\\,$ for arbitrary $ n \\ge 0 $ in terms\nof amalgamated products.\n\nThe groups $ {\\rm{Aut}} \\,D_n $ for $ n \\ge 1 $ can be naturally identified with subgroups of ${\\rm{Aut}} \\,D_0$.\nTo be precise, let $ {\\rm{Pic}}\\, D $ denote the (noncommutative) Picard group of a $\\mathbb{C}$-algebra $D$. By definition,\n$ {\\rm{Pic}}\\, D $ is the group of $\\mathbb{C}$-linear Morita equivalences of the category of $ D$-modules; its elements can be\nrepresented by the isomorphism classes of invertible $D$-bimodules $\\,[P]\\,$ (see, e.g., \\cite{B}). There is a\nnatural group homomorphism $\\,\\omega_D:\\, {\\rm{Aut}}\\,D \\to {\\rm{Pic}}\\,D \\,$, taking $\\, \\sigma \\in {\\rm{Aut}}\\,D \\,$ to the class of\nthe bimodule $ [{}_1 D_{\\sigma}] $, and if $\\, D' \\,$ is a ring Morita equivalent to $ D $, with\nprogenerator $ M $, then there is a group isomorphism $\\, \\alpha_M:\\, {\\rm{Pic}} \\,D' \\stackrel{\\sim}{\\to} {\\rm{Pic}} \\,D\\,$\ngiven by $\\,[P] \\mapsto [M^* \\otimes_D P \\otimes _D M]\\,$. Thus, in our situation, for each $ n \\ge 0 $\nwe have the following diagram\n\\begin{equation}\n\\la{D1}\n\\begin{diagram}[small, tight]\n{\\rm{Aut}} \\,D_n & \\rTo^{\\ \\omega_{D_n}\\ } & {\\rm{Pic}} \\,D_n \\\\\n\\dDotsto^{i_n} & & \\dTo_{\\alpha_{M_n}} \\\\\n{\\rm{Aut}} \\,D_0 & \\rTo^{\\ \\omega_{D_0}\\ } & {\\rm{Pic}} \\,D_0 \\\\\n\\end{diagram}\n\\end{equation}\nwhere the vertical map $ \\alpha_{M_n} $ is an isomorphism and the two horizontal maps are injective.\nMoreover, since $ D_0 = A_1 $, a theorem of Stafford (see \\cite{St}) implies that $ \\omega_{D_0} $ is actually\nan isomorphism. Inverting this isomorphism, we define the embedding\n$\\, i_n:\\,{\\rm{Aut}} \\,D_n \\,\\,\\hookrightarrow\\,\\, {\\rm{Aut}} \\,D_0 \\,$, which makes \\eqref{D1} a commutative diagram.\n\nRecall that we defined $ G_0 $ to be the automorphism group of the free algebra $\\,\\mathbb{C} \\langle x, \\, y \\rangle\\,$\npreserving $ [x,\\,y] $. Now, for $\\, n > 0 \\,$, we introduce the groups $ G_n $ geometrically,\nin terms of a natural action of $ G_0 $ on the {\\it Calogero-Moser spaces}\\, (see \\cite{W})\n\\begin{equation*}\n\\label{e1}\n{\\mathcal C}_n := \\{\\,(X,\\,Y) \\in \\mbox{\\tt Mat}_n(\\mathbb{C}) \\times\n\\mbox{\\tt Mat}_n(\\mathbb{C})\\ :\\ \\mbox{\\tt rk}([X,\\,Y] + I_n) = 1 \\,\\}\/\\,\\mbox{\\tt PGL}_n(\\mathbb{C})\\ ,\n\\end{equation*}\nwhere $ \\mbox{\\tt PGL}_n(\\mathbb{C}) $ operates on matrices $ (X,\\,Y) $ by simultaneous conjugation.\nThe action of $ G_0 $ on $ \\mathcal{C}_n $ is given by\n\\begin{equation}\n\\label{at}\n(X,\\,Y) \\mapsto (\\sigma^{-1}(X),\\, \\sigma^{-1}(Y))\\ , \\quad \\sigma \\in G_0\\ ,\n\\end{equation}\nwhere $ \\sigma^{-1}(X) $ and $ \\sigma^{-1}(Y) $ are the noncommutative polynomials\n$\\,\\sigma^{-1}(x) \\in \\mathbb{C} \\langle x, \\, y \\rangle \\,$ and $\\,\\sigma^{-1}(y) \\in \\mathbb{C} \\langle x, \\, y \\rangle \\,$\nevaluated at $(X,Y)$. It is known that $ {\\mathcal C}_n $ is a smooth affine algebraic variety\nof dimension $2n$, equipped with a natural symplectic structure, and it is easy to check that\n$G_0$ preserves that structure. Now, a theorem of Wilson and the first author (see \\cite{BW}) implies that\n\\eqref{at} is a transitive action for all $ n \\ge 0 $.\nWe define the groups $ G_n $ to be the stabilizers of points of $ \\mathcal{C}_n $\nunder this action: precisely, for each $ n \\ge 0 $, we fix a basepoint $ (X_0,\\,Y_0) \\in\n{\\mathcal C}_n $, with\n\\begin{equation*}\n\\label{base}\nX_0 = \\sum_{k=1}^{n-1} E_{k+1, k}\n\\quad , \\qquad\nY_0 = \\sum_{k=1}^{n-1} \\,(k-n)\\, E_{k, k+1}\\ ,\n\\end{equation*}\nwhere $ E_{i,j} $ stands for the elementary matrix with $(i,j)$-entry $1$, and let\n\\begin{equation*}\n\\label{gn}\nG_n := \\mbox{\\tt Stab}_{G_0}(X_0, Y_0)\\ ,\\quad n \\ge 0 \\ .\n\\end{equation*}\n\n\nThe following result can be viewed as a generalization of the above-mentioned theorem\nof Makar-Limanov; in a slightly different form, it has already appeared in\n\\cite{BW4} (cf. {\\it loc. cit.}, p.~120; see also \\cite{W2}).\n\\begin{theorem}\n\\label{T1}\nThere is a natural isomorphism of groups $\\,G_n \\stackrel{\\sim}{\\to} {\\rm{Aut}} \\, D_n\\,$.\n\\end{theorem}\n\\noindent\nSpecifically, we have group homomorphisms\n\\begin{equation*}\n\\label{ison}\nG_n \\,\\,\\hookrightarrow\\,\\, G_0 \\stackrel{\\sim}{\\to} {\\rm{Aut}} \\,A_1 \\stackrel{i_n}{\\hookleftarrow} {\\rm{Aut}} \\,\\mathcal{D}_n\\ ,\n\\end{equation*}\nwhere the first map is the canonical inclusion, the second is the Makar-Limanov isomorphism\n\\eqref{isML} and $ i_n $ is the embedding defined by \\eqref{D1}. We claim that the image\nof $ i_n $ coincides with the image of $ G_n $, which gives the required isomorphism.\n\nTheorem~\\ref{T1} is a simple consequence of the main results of \\cite{BW}: in fact, it is shown in \\cite{BW}\nthat there is a natural $G_0$-equivariant bijection (called the Calogero-Moser correspondence)\nbetween $\\, \\bigsqcup_{n \\ge 0} {\\mathcal C}_n \\,$ and the space of isomorphism classes of right\nideals of $ A_1 $. Under this bijection, the points $ (X_0, Y_0) \\in {\\mathcal C}_n $\ncorrespond precisely to the classes of the ideals $ M_n $.\n\nWe will use Theorem~\\ref{T1} to give a geometric presentation for the groups $ {\\rm{Aut}} \\,D_n $.\nTo this end, we associate to each space $ \\mathcal{C}_n $ a graph $ \\Gamma_n $ consisting of orbits of\ncertain subgroups of $ G_0 $ and identify $ G_n $ with the {\\it fundamental group}\n$\\, \\pi_1({\\mathbf \\Gamma}_n, \\ast)\\, $ of a graph of groups $ {\\mathbf \\Gamma}_n $ defined\nby the stabilizers of points of those orbits in $ \\Gamma_n $. The Bass-Serre theory of groups\nacting on graphs \\cite{Se} will give\nthen an explicit formula for $\\, \\pi_1({\\mathbf \\Gamma}_n, \\ast)\\, $ in terms of generalized\namalgamated products (see \\eqref{fgr2} below).\n\nTo define the graph $ \\Gamma_n $ we take the subgroups $A$, $B$ and $U$ of $G_0$\ndefined by the transformations \\eqref{A}, \\eqref{B} and \\eqref{U}. Restricting the action of $ G_0 $ on $ \\mathcal{C}_n $\nto these subgroups, we let $ \\Gamma_n $ be the oriented bipartite graph, with vertex and edge sets\n\\begin{equation}\n\\label{gamman}\n\\mbox{\\tt Vert}(\\Gamma_{n}) := (A \\backslash \\mathcal{C}_n)\\,\\bigsqcup\\, (B \\backslash \\mathcal{C}_n) \\ ,\\quad\n\\mbox{\\tt Edge}(\\Gamma_{n}) := U \\backslash \\mathcal{C}_n \\ ,\n\\end{equation}\nand the incidence maps $\\,\\mbox{\\tt Edge}(\\Gamma_{n}) \\to \\mbox{\\tt Vert}(\\Gamma_{n})\\,$ given by the canonical\nprojections $\\, i: U \\backslash \\mathcal{C}_n \\to A \\backslash \\mathcal{C}_n \\,$\nand $\\,\\tau: U \\backslash \\mathcal{C}_n \\to B \\backslash \\mathcal{C}_n \\,$. Since the elements of\n$A$ and $B$ generate $ G_0 $ and $ G_0 $ acts transitively on each $ \\mathcal{C}_n $, the graph $ \\Gamma_n $ is connected.\n\nNow, on each orbit in $ A \\backslash \\mathcal{C}_n $ and $\\,B \\backslash \\mathcal{C}_n\\,$\nwe choose a basepoint and elements $\\, \\sigma_A \\in G_0 \\,$ and $\\,\\sigma_B \\in G_0\\, $ moving these basepoints\nto the basepoint $ (X_0,\\,Y_0) $ of $ {\\mathcal C}_n $. Next, on each $U$-orbit $\\,\\mathcal{O}_U \\in\nU \\backslash \\mathcal{C}_n \\,$ we also choose a basepoint and an element $\\,\\sigma_U \\in G_0 \\,$ moving this\nbasepoint to $ (X_0,\\,Y_0) $ and such that $\\,\t\\sigma_U \\in \\sigma_A A \\,\\cap \\,\\sigma_B B\\,$,\nwhere $ \\sigma_A $ and $ \\sigma_B $ correspond to the (unique) $A$- and $B$-orbits containing $ \\mathcal{O}_U $.\nUsing a standard construction in the Bass-Serre theory (see \\cite{Se}, Sect.~5.4), we then assign to the\nvertices and edges of $ \\Gamma_n $ the stabilizers $\\,A_\\sigma = G_n \\cap \\sigma A \\sigma^{-1} \\,$,\n$\\,B_\\sigma = G_n \\cap \\sigma B \\sigma^{-1} \\,$, $\\,U_\\sigma = G_n \\cap \\sigma U \\sigma^{-1} \\,$ of the corresponding\nelements $ \\sigma $ in the graph of right cosets of $ G_0 $ under the action of $ G_n $.\nThese data together with natural group homomorphisms $\\, a_\\sigma: U_\\sigma \\hookrightarrow A_\\sigma \\,$\nand $\\, b_\\sigma:\\,U_\\sigma \\hookrightarrow B_\\sigma \\,$ define a graph of groups $ {\\mathbf \\Gamma}_n $ over\n$ \\Gamma_n $, and its fundamental group $\\, \\pi_1({\\mathbf \\Gamma}_n,\\,T)\\,$\nrelative to a maximal tree $\\,T \\subseteq \\Gamma_n \\,$ has canonical presentation (see \\cite{Se}, Sect.~5.1):\n\\begin{equation}\n\\label{fgr2}\n\\pi_1({\\mathbf \\Gamma}_n,\\,T) = \\frac{A_\\sigma \\ast_{\\,U_\\sigma} B_\\sigma \\ast \\, \\ldots\\, \\ast \\langle\\, \\mbox{\\tt Edge}(\\Gamma_n \\setminus T)\\,\\rangle}{(\\,e^{-1} a_\\sigma(g)\\, e = b_{\\sigma}(g)\\, :\\, \\forall\\,e \\in\n\\mbox{\\tt Edge}(\\Gamma_n \\setminus T),\\, \\forall\\, g \\in U_\\sigma\\,)}\\ .\n\\end{equation}\nIn \\eqref{fgr2}, the amalgams $\\, A_\\sigma \\ast_{\\,U_\\sigma} B_\\sigma \\ast \\, \\ldots \\,$ are taken along the stabilizers\nof edges of the tree $ T $, while $\\, \\langle\\, \\mbox{\\tt Edge}(\\Gamma_n \\setminus T)\\,\\rangle\\,$ denotes\nthe free group based on the set of edges of $ \\Gamma_n $ in the complement of $T$.\n\nOur main observation is the following.\n\\begin{theorem}\n\\label{T2}\nFor each $ n \\ge 0 $, the group $ G_n $ is isomorphic to $\\, \\pi_1({\\mathbf \\Gamma}_n,\\,T)\\,$. In particular,\n$ G_n $ has an explicit presentation of the form \\eqref{fgr2}.\n\\end{theorem}\n\\begin{proof}\nOne can prove Theorem~\\ref{T2} using the standard Bass-Serre theory (as exposed in \\cite{Se}, Ch~I, Sect.~5,\nor \\cite{DD}, Ch.~I, Sect.~9). However, it seems that the more economic and intuitively clearer proof is\nbased on topological arguments: namely, an abstract version of Van Kampen's Theorem, which we are now\ngoing to explain.\n\nLet $\\,\\mathfrak{G}_n := \\mathcal{C}_n \\rtimes G_0 \\,$ denote the (discrete) transformation groupoid\ncorresponding to the action of $ G_0 $ on $ \\mathcal{C}_n $. The canonical projection $\\,p:\\, \\mathfrak{G}_n \\to G_0 \\,$ is then\na connected covering of groupoids\\footnote{We refer to \\cite{M}, Ch.~3, for the theory of coverings of groupoids.},\nwhich maps identically the vertex group of $ \\mathfrak{G}_n $ at $\\, (X_0,\\,Y_0) \\in \\mathcal{C}_n \\,$ to the subgroup\n$ G_n \\subseteq G_0 $. Now, each of the subgroups $A$, $B$ and $U$ of $ G_0 $ can be lifted\nto $ \\mathfrak{G}_n\\,$: $\\,p^{-1}(A) = \\mathfrak{G}_n \\times_{G_0} A \\,$, $\\,p^{-1}(B) = \\mathfrak{G}_n \\times_{G_0} B \\,$\nand $\\,p^{-1}(U) = \\mathfrak{G}_n \\times_{G_0} U \\,$, and these fibred products\nare naturally isomorphic to the subgroupoids $\\, \\mathfrak{A}_n := \\mathcal{C}_n \\rtimes A $, $\\, \\mathfrak{B}_n := \\mathcal{C}_n \\rtimes B \\,$\nand $\\, \\mathfrak{U}_n := \\mathcal{C}_n \\rtimes U \\,$ of $ \\mathfrak{G}_n$, respectively. Since the coproducts in the category of\ngroups coincide with coproducts in the category of groupoids and the latter can be lifted through coverings\n(see \\cite{O}, Lemma~3.1.1), the decomposition \\eqref{Aut} implies\n\\begin{equation}\n\\la{vank}\n\\mathfrak{G}_n = \\mathfrak{A}_n *_{\\mathfrak{U}_n} \\mathfrak{B}_n\\ ,\\quad \\forall\\,n \\ge 0\\ .\n\\end{equation}\nNote that, unlike $ \\mathfrak{G}_n$, the groupoids $ \\mathfrak{A}_n $, $ \\mathfrak{B}_n $ and $ \\mathfrak{U}_n $ are not transitive\n(if $\\, n \\ge 1 $), so \\eqref{vank} can be viewed as an analogue of the Seifert-Van Kampen Theorem\nfor non-connected spaces (see, e.g., \\cite{Ge}, Ch.~6, Appendix). As in the topological situation, computing\nthe fundamental (vertex) group from \\eqref{vank} amounts to contracting the connected components (orbits) of\n$ \\mathfrak{A}_n $ and $ \\mathfrak{B}_n $ to points (vertices) and $ \\mathfrak{U}_n $ to edges. This defines a graph which is exactly\n$ \\Gamma_n $. Now, choosing basepoints in each of the contracted components and assigning the fundamental groups\nat these basepoints to the corresponding vertices and edges defines a graph of groups (see \\cite{HMM}, p.~46).\nBy {\\it loc.~cit.}, Theorem~3, this graph of groups is (conjugate) isomorphic to the graph $ {\\mathbf \\Gamma}_n $\ndescribed above, and our group $ G_n $ is isomorphic to $\\, \\pi_1({\\mathbf \\Gamma}_n,\\,T)\\,$.\n\\end{proof}\nTheorems~\\ref{T1} and~\\ref{T2} reduce the problem of describing the groups\n$ {\\rm{Aut}} \\,D_n $ to a purely geometric problem of describing the structure of the orbit spaces\nof $A$ and $B$ and $U$ on the Calogero-Moser varieties $ {\\mathcal C}_n$. Using the earlier results\nof \\cite{W} and \\cite{BW} and some basic invariant theory, one can obtain much\ninformation about these orbits (and thence about the groups $ G_n $). In particular, the graphs\n$ \\Gamma_n $ can be completely described for small $ n $; it turns that $ \\Gamma_n $ is a finite tree for\n$\\,n = 0,\\,1,\\, 2\\,$, but has infinitely many cycles for $ n \\ge 3 $ (see examples below).\n\nWe now explain the origin of $ \\Gamma_n $. It turns out that these graphs can be\nrealized as quotient graphs of a certain `universal' tree $ \\Gamma $ on which all\nthe groups $ {\\rm{Aut}} \\,D_n $ naturally act. Our construction of $ \\Gamma $ is motivated\nby algebraic geometry: specifically, a known application of the\nBass-Serre theory in the theory of surfaces (see, e.g., \\cite{GD}, \\cite{Wr}). In that approach,\nthe automorphism group of an affine surface $ S $ is described via its action on a tree whose\nvertices correspond to certain (admissible) projective compactifications of $ S $. Following\nthe standard (by now) philosophy in noncommutative geometry (see, e.g., \\cite{SV}), we may\nthink of our algebra $ D $ as the coordinate ring of a `noncommutative affine surface';\na `projective compactification' of $ D $ is then determined by a choice of filtration. Thus, we will define\n$ \\Gamma $ by taking as its vertices a certain class of filtrations on the algebra $D$.\nIt turns out that these filtrations can be naturally parametrized by an infinite-dimensional\n{\\it adelic Grassmannian} $\\,\\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ introduced in \\cite{W1} and studied in \\cite{W, BW, BW3}\n(in particular, we rely heavily on results of \\cite{BW3}). Our contruction is close in spirit to\nSerre's classic application of Bruhat-Tits trees for computing arithmetic subgroups of $ \\SL_2(\\mathbb{K}) $\nover the function fields of smooth curves (see \\cite{Se}, Chap.~II, \\S\\,2); however, at the moment,\nwe are not aware of any direct connection.\n\nWe begin by briefly recalling the definition of $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $. Let $ \\mathbb{C}[z] $ be the polynomial ring in one\nvariable $z$. For each $\\, \\lambda \\in \\mathbb{C} \\,$, we choose a {\\it $\\lambda$-primary} subspace in $ \\mathbb{C}[z] $,\nthat is, a $\\mathbb{C}$-linear subspace $\\, V_{\\lambda} \\subseteq \\mathbb{C}[z] \\,$ containing a power of the maximal ideal\n$\\,\\mathfrak{m}_\\lambda \\,$ at $\\, \\lambda $. We suppose that $\\, V_{\\lambda} = \\mathbb{C}[z] \\, $\nfor all but finitely many $\\, \\lambda$'s. Let $\\, V = \\bigcap_{\\lambda} V_{\\lambda} \\,$\n(such a subspace $\\, V \\,$ is called {\\it primary decomposable} in $ \\mathbb{C}[z] $) and, finally, let\n$$\nW = \\prod_{\\lambda}\\, (z - \\lambda)^{-n_{\\lambda}} \\, V \\subset\n\\mathbb{C}(z) \\, ,\n$$\nwhere $\\, n_{\\lambda} \\,$ is the codimension of $\\, V_{\\lambda} \\,$\nin $\\, \\mathbb{C}[z] \\,$. By definition, $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ consists of all subspaces\n$\\, W \\subset \\mathbb{C}(z) \\,$ obtained in this way. For each\n$\\, W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ we set\n\\begin{equation*}\n\\label{aw}\nA_W := \\{f \\in \\mathbb{C}[z] \\,:\\, f W \\subseteq W \\} \\,.\n\\end{equation*}\nTaking $\\,{\\rm{Spec}}\\,$ of $\\, A_W \\,$ gives then a rational curve $X$,\nthe inclusion $\\, A_W \\,\\,\\hookrightarrow\\,\\, \\mathbb{C}[z] \\,$ corresponds to normalization\n$\\, \\pi : \\mathbb{A}_{\\mathbb{C}}^1 \\to X \\,$ (which is set-theoretically a bijective map),\nand the $\\, A_W$-module $\\, W \\,$ defines\na rank 1 torsion-free coherent sheaf $\\, \\mathscr{L} \\,$ over $\\, X \\,$. In this way,\nthe points of $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ correspond bijectively to isomorphism classes of triples\n$\\, (\\pi, X, \\mathscr{L}) \\,$ (see \\cite{W1}).\n\nNow, following \\cite{BW}, for $\\, W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ we define\\footnote{In geometric terms, $ D(W) $\ncan be thought of as the ring $ D_{\\mathscr{L}}(X) $ of twisted differential operators on $X$ with\ncoefficients in $ \\mathscr{L} $.}\n\\begin{equation}\n\\label{dw}\nD(W) := \\{\\mathcal{D} \\in \\mathbb{C}(z)[\\partial_z] \\,:\\, \\mathcal{D} W \\subseteq W \\} \\, ,\n\\end{equation}\nwhere $\\, \\mathbb{C}(z)[\\partial_z] \\,$ is the ring of rational differential operators\nin the variable $z$. This last ring carries two natural filtrations:\nthe standard filtration, in which both generators $ z $ and $ \\partial_z $\nhave degree $1$, and the differential filtration, in which $\\,\\deg(z) = 0\\,$\nand $\\,\\deg(\\partial_z) = 1\\,$. These filtrations induce two different filtrations\non the algebra $ D(W) $, which we denote by $ \\{D_{\\bullet}^A(W)\\} $ and $ \\{D_{\\bullet}^B(W)\\} $\nrespectively.\n\nNow, let $D$ be a fixed domain Morita equivalent to $ A_1 $. Following \\cite{BW3}, we consider\nthe set\\footnote{More generally, we may think of $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ as a groupoid, in which the objects are\nthe $W$'s and the arrows are given by the algebra isomorphisms $ D(W) \\to D(W') $. For $ D = D(W) $,\nthe set $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}}(D) $ is then a costar in $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $, consisting of all arrows with target at $W$.\nIn \\cite{BW3}, this set was denoted by $\\,\\mbox{\\tt Grad}\\,D\\,$.} $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\,$\nof all algebra isomorphisms $\\,\\sigma_W: D(W) \\to D \\,$, where $ W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $\n(more precisely, $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\,$ is the set of all pairs $\\, (W, \\,\\sigma_W) \\,$, where\n$\\, W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} $ and $\\, \\sigma_W \\,$ is an isomorphism as above). Each\n$\\, \\sigma_W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) $ maps the two distinguished filtrations $ \\{D_{\\bullet}^A(W)\\} $ and\n$ \\{D_{\\bullet}^B(W)\\} $ into the algebra $ D $: we call their images the {\\it admissible}\nfiltrations on $D$ of type $A$ and type $B$, respectively. Let $ {\\mathbb P}_{A}(D) $ and $ {\\mathbb P}_{B}(D) $ denote\nthe sets of all such filtrations coming from various $\\,\\sigma_W \\in \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\\,$.\nBy definition, we have then two natural projections\n\\begin{equation}\n\\la{prc}\n{\\mathbb P}_{A}(D) \\stackrel{\\pi_A}{\\longleftarrow} \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\stackrel{\\pi_B}{\\longrightarrow} {\\mathbb P}_B(D)\\ .\n\\end{equation}\nWe say that $\\,(W, \\sigma_W) \\,$ and $\\,(W', \\sigma_W') \\,$ are {\\it equivalent} in $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D) \\,$\nif their images under $ \\pi_A $ and $ \\pi_B $ coincide. Writing $ \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\/\\!\\sim $ for the set of\nequivalence classes under this relation, we define an oriented graph $ \\Gamma $ by\n$$\n\\mbox{\\tt Vert}(\\Gamma) := {\\mathbb P}_A(D) \\,\\bigsqcup\\, {\\mathbb P}_B(D)\\ , \\quad \\mbox{\\tt Edge}(\\Gamma) := \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\/\\!\\sim\\ ,\n$$\nwith incidence maps $\\,\\mbox{\\tt Edge}(\\Gamma) \\to \\mbox{\\tt Vert}(\\Gamma)\\,$ induced by the\nprojections \\eqref{prc}. Observe that the group $ {\\rm{Aut}} \\,D $ acts naturally on the set $\\, \\mbox{\\rm{Gr}}^{\\mbox{\\scriptsize{\\rm{ad}}}} (D)\\,$\n(by composition), and this action induces an action of $ {\\rm{Aut}} \\,D $ on the graph $ \\Gamma $ via \\eqref{prc}.\nWe write $\\, {\\rm{Aut}} \\, D \\backslash \\Gamma\\,$ for the corresponding quotient graph.\n\n\\begin{theorem}\n\\la{T3}\n\n$(a)$ $ \\Gamma $ is a tree, which is independent of $ D $ (up to isomorphism).\n\n$(b)$ For each $ n \\ge 0 $, the graph $ {\\rm{Aut}} \\, D_n \\backslash \\Gamma $ is naturally isomorphic to $ \\Gamma_n $.\n\n\\end{theorem}\n\nTheorem~\\ref{T3} can be viewed as a generalization of the main results of \\cite{BW3}.\nIndeed, this last paper is concerned with a description of the maximal abelian ad-nilpotent ({\\it mad})\nsubalgebras of $ D_n \\,$: its main theorems (see {\\it loc. cit.}, Theorem~1.5 and Theorem~1.6) say\nthat the space $ \\mbox{\\rm Mad}(D_n) $ of all mad subalgebras of $ D_n $ is independent of $ D_n $\nand its quotient modulo the natural action of $ {\\rm{Aut}} \\,D_n $ is isomorphic to the orbit space\n$ B \\backslash \\mathcal{C}_n $. Now, it is easy to\nsee that every mad sublagebra defines an admissible filtration on $ D_n $ of type $B$, and conversely\nthe zero degree component of every filtration of type $B$ is a mad subalgebra of $D_n$. Thus, we have\na natural bijection $\\, {\\mathbb P}_B(D_n) \\cong \\mbox{\\rm Mad}(D_n) $, which is equivariant under the action\nof $ {\\rm{Aut}} \\,D_n $. This implies that $ {\\mathbb P}_B(D_n) $ does not depend on $ D_n $, which is part of\nTheorem~\\ref{T3}$(a)$, and\n$$\n{\\rm{Aut}} \\, D_n \\backslash {\\mathbb P}_B(D_n)\\,\\cong \\,{\\rm{Aut}} \\, D_n \\backslash \\mbox{\\rm Mad}(D_n) \\,\\cong\\,\nB \\backslash \\mathcal{C}_n\\ ,\n$$\nwhich is part of Theorem~\\ref{T3}$(b)$. In fact, the entire Theorem~\\ref{T3} can be proved using the techniques\nof \\cite{BW3}. We should also mention that for $ D = A_1(k) $ our construction of the tree $ \\Gamma $\nagrees with the one given in \\cite{A}.\n\n\\vspace{1ex}\n\nWe now look at examples of the graphs $ \\Gamma_n $ and groups $G_n$ for small $n$.\nFor $ n = 0 $, the space $ \\mathcal{C}_0 $ is just a point, and so are a fortiori\nits orbit spaces. The graph $ \\Gamma_0 $ is thus a segment, and the corresponding graph of\ngroups $ {\\mathbf \\Gamma}_0 $ is given by $\\,[\\,A \\stackrel{U}{\\longrightarrow} B\\,] \\,$.\nFormula \\eqref{fgr2} then says that $\\,G_0 = A \\ast_U B \\,$, which agrees, of course, with\nthe Makar-Limanov isomorphism \\eqref{Aut}.\n\nFor $ n =1 $, we have $\\, {\\mathcal C}_1 \\cong \\mathbb{C}^2 $, with $ (X_0, Y_0) $ corresponding\nto the origin. Since each of the groups\n$A$, $B$ and $U$ contains translations $\\,(x,y) \\mapsto (x+a, y+b) \\,$, $\\,a,b \\in \\mathbb{C} $,\nthey act transitively on $ {\\mathcal C}_1 $. So again $ \\Gamma_1 $ is just the segment, and\n$ {\\mathbf \\Gamma}_1 $ is given by $\\,[\\,A_1 \\stackrel{U_1}{\\longrightarrow} B_1\\,]\\,$,\nwhere $\\, A_1 := G_1 \\cap A \\,$, $\\, B_1 := G_1 \\cap B \\,$ and $\\, U_1 := G_1 \\cap U \\,$.\nSince, by definition, $ G_1 $ consists of all $\\, \\sigma \\in G_0 \\,$\npreserving $\\,(0, 0)\\,$, the groups $A_1$, $ B_1 $ and $U_1 $ are obvious:\n\\begin{eqnarray}\nA_1\\!\\!\\!\\!\\! &:& (x,\\,y) \\mapsto (ax + by,\\,cx+dy)\\ , \\quad a,\\,b,\\,c,\\,d \\in\n\\mathbb{C} \\ ,\\ ad-bc=1\\ ,\n\\nonumber \\\\\nB_1\\!\\!\\!\\!\\! &:& (x,\\,y) \\mapsto (ax + q(y),\\,a^{-1} y)\\ , \\quad a \\in \\mathbb{C}^*\\ ,\\\nq \\in \\mathbb{C}[y]\\ , \\ q(0) = 0\\ , \\nonumber \\\\\nU_1 \\!\\!\\!\\!\\! &:& (x,\\,y) \\mapsto (ax + by,\\,a^{-1} y)\\ , \\quad a \\in \\mathbb{C}^*\\ ,\\\nb\\in \\mathbb{C}\\ .\n\\nonumber\n\\end{eqnarray}\nIt follows from \\eqref{fgr2} that $\\, G_1 = A_1 \\ast_{U_1} B_1 \\,$.\n\nFor $ n = 2 $, the situation is already more interesting. A simple calculation shows that\n$ U $ has three orbits in $ {\\mathcal C}_2$: two closed orbits of dimension $3$ and one open orbit\nof dimension $4$. Moreover, the $B$-orbits coincide with the $U$-orbits. Combinatorially, this means\nthat the group $A$ acts transitively, and the graph $ \\Gamma_2 $ is a tree with one nonterminal and\nthree terminal vertices corresponding to the $A$-orbit and the $B$-orbits, respectively.\nIn this case, the graph of groups ${\\mathbf \\Gamma_2} $ is given by\n$$\n\\begin{diagram}\n& & & G_{2,y} \\rtimes \\mathbb{C}^* \\\\\n& & \\ruTo^{\\mathbb{C}^*} & \\\\\n&\\mathbb{C}^* & \\rTo^{\\mathbb{Z}_{2}\\quad} & G^{(1)}_{2, y} \\rtimes \\mathbb{Z}_{2} \\\\\n& & \\rdTo_{\\mathbb{C}^*} & \\\\\n& & & G_{2,x} \\rtimes \\mathbb{C}^* \\\\\n\\end{diagram}\n$$\nwhere $ G_{2,x} $ and $ G_{2,y} $ are the subgroups of $ G_0 $ consisting of all transformations\n$ \\Phi_p $ and $ \\Psi_q $ (see \\eqref{Phs}),\nwith $ p \\in \\mathbb{C}[x] $ and $ q \\in \\mathbb{C}[y] $ satisfying $ p(0) = p'(0) = 0 $ and\n$ q(0) = q'(0)= 0 $ respectively, and $\\, G^{(1)}_{2, y} := \\{\\,\\Phi_{-x}\\,\\Psi_q \\,\\Phi_{ x} \\in G_0 \\ :\\ q \\in \\mathbb{C}[y]\\ , \\\nq(\\pm 1) = 0\\,\\}\\,$. Formula \\eqref{fgr2} yields the presentation\n$$\n\\,G_2 = (G_{2,x} \\rtimes \\mathbb{C}^*) \\ast_{\\mathbb{C}^*}\n(G_{2,y} \\rtimes \\mathbb{C}^*) \\ast_{\\mathbb{Z}_2} (G^{(1)}_{2, y}\n\\rtimes \\mathbb{Z}_2) \\, .\n$$\nIn particular, $ G_2 $ is generated by its subgroups $\\, G_{2,x} $, $\\, G_{2,y} $, $\\, G^{(1)}_{2, y}$\nand $ \\mathbb{C}^* $.\n\nNow, for $ n = 3 $, the structure of the graph $ \\Gamma_3 $ and the group $ G_3 $ is much more complicated.\nThe graph $ \\Gamma_3 $ is {\\it not}\na tree: in fact, it has infinitely many circuits. Nevertheless, the group $ G_3 $ can still be described\nexplicitly:\n$$\nG_3 = \\frac{\\pi_1(T_3,\\,G_3) \\,\\star\\,\n\\langle E_+(\\Gamma_3 \\setminus\nT_3)\\rangle}{(\\,e^{-1} \\alpha_e(g)\\, e = \\alpha_{e^*}(g)\\, :\\, \\forall\\,e \\in\nE_+(\\Gamma_3 \\setminus T_3),\\, \\forall\\, g \\in G_e\\,)} \\ ,\n$$\nwhere $ T_3 $ is a (maximal) tree in $ \\Gamma_3 $ given in Figure \\ref{T3}, $\\,\\pi_1(T_3,\\,G_3)\\,$ is the\ncorresponding tree product of stabilizer groups, and the complement graph $\\,\\Gamma_3 \\setminus T_3 \\,$\nis shown in Figure \\ref{graph4}.\n\nWe would like to end this paper with some questions and conjectures.\n\n\\medskip\n\n{1.} By \\cite{ML1}, it is known that $ G_0 $ is isomorphic to the group $\\,\\mbox{\\rm SAut}\\,\\mathbb{A}_{\\mathbb{C}}^2 \\,$ of\n{\\it symplectic} automorphisms of the affine plane $\\,\\mathbb{A}_{\\mathbb{C}}^2\\,$ (as in the case of the Weyl algebra,\nthe isomorphism $\\,G_0 \\cong \\mbox{\\rm SAut}\\, \\mathbb{A}_{\\mathbb{C}}^2 \\,$ is induced by the canonical projection\n$\\, \\mathbb{C}\\langle x,\\,y \\rangle \\to \\mathbb{C}[x,\\,y] $). Thus, the groups $ G_n $ can be naturally identified\nwith subgroups of $\\,{\\rm{Aut}}\\,\\mathbb{A}_{\\mathbb{C}}^2 \\,$. Do these last subgroups have a geometric interpretation?\n\n{2.}\nIn this paper, we have described the structure of $ G_n $ and $ {\\rm{Aut}} \\,D_n $ as discrete groups.\nHowever, these two groups carry natural {\\it algebraic} structures and can be viewed as\ninfinite-dimensional algebraic groups (in the sense of Shafarevich \\cite{Sh}). Despite being isomorphic\nto each other as discrete groups, they are not isomorphic as algebraic groups\n(for $n=0$, this phenomenon was observed in \\cite{BW}.) A natural question is to explicitly describe\nthe algebraic structures on $ G_n $ and $ {\\rm{Aut}} \\,D_n $; in particular, to compute the corresponding\n(infinite-dimensional) Lie algebras. The last question was an original motivation for our work. For\n$ G_0 $, the answer is known (see \\cite{G}).\n\n{3.}\nCompute the homology of the groups $ G_n $ for all $n$. Again, for $ n = 0 \\,$, the answer is known\n(see \\cite{Al}): $\\,H_*(G_0,\\,\\mathbb{Z}) \\cong H_*(\\SL_2(\\mathbb{C}),\\,\\mathbb{Z})\\,$. One may wonder whether the groups\n$ H_*(G_n,\\,\\mathbb{Z}) $ are strong enough invariants to distinguish the algebras $ D_n $\nup to isomorphism. Unfortunately, the answer is `no': in fact, it follows from our description of $ G_1 $ that\n$\\,H_*(G_1,\\,\\mathbb{Z}) \\cong H_*(\\SL_2(\\mathbb{C}),\\,\\mathbb{Z})\\,$. However, for $ n \\ge 2 $, it seems that the groups $ H_*(G_n,\\,\\mathbb{Z}) $\nare neither isomorphic to $\\,H_*(\\SL_2(\\mathbb{C}),\\,\\mathbb{Z})\\,$ nor to each other, so they may provide\ninteresting invariants.\n\n{4.}\nFinally, we would like to propose an extension of the well-known {\\it Dixmier Conjecture}\nfor $ A_1 \\,$ (see \\cite{D}, Probl\\`eme~11.1) to the class of Morita equivalent algebras.\nWe recall that if $ D $ is a domain Morita equivalent to $ A_1 $, then there is a unique\ninteger $ n \\ge 0 $ such that $\\,D \\cong D_n \\,$, where $\\,D_n \\,$ is the endomorphism ring of\nthe right ideal $\\, M_n = x^{n} A_1 + (y+n x^{-1})\\,A_1 $. For two unital $\\mathbb{C}$-algebras $A$ and $B$,\nwe denote by $\\,\\Hom\\, (A,\\,B)\\,$ the set of all unital $\\mathbb{C}$-algebra homomorphisms $\\,A \\to B \\,$.\n\\begin{conjecture}\n\\la{DC}\nFor all $\\, n, m \\ge 0 \\,$, we have\n\\begin{equation*}\n\\Hom\\, (D_n,\\,D_m) = \\left\\{\n\\begin{array}{lll}\n\\emptyset & \\mbox{if}\\quad n \\ne m \\\\*[1ex]\n{\\rm{Aut}} \\,D_n & \\mbox{if} \\quad n = m\n\\end{array}\n\\right.\n\\end{equation*}\n\\end{conjecture}\n\\noindent\nFormally, Conjecture~\\ref{DC} is a strengthening of the Dixmier Conjecture for $ A_1 $: in fact,\nin our notation, the latter says that $\\,\\Hom\\, (D_0,\\,D_0) = {\\rm{Aut}} \\,D_0\\,$. Does actually the Dixmier\nConjecture imply Conjecture~\\ref{DC}?\n\n\n\\subsection*{Acknowledgments}\nWe are grateful to J.~Alev, V.~Bavula, O.~Chalykh, K.~Vogtmann, D.~Wright, G.~Wilson and E.~Zelmanov for interesting discussions,\nquestions and comments. We would also like to thank D.~Wright for providing us with reference \\cite{Wr},\nwhich turned out to be very useful, and G.~Wilson for sending us a copy of Quillen's private notes on trees and amalgams.\nThis work was partially supported by NSF grant DMS 09-01570.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLocal volatility model was introduced by \\cite{Dupire:94} and \\cite{derman\/kani:94} as a natural extension of the celebrating Black-Scholes model to take into account an existence of option smile. It is able to exactly replicate the local volatility function $\\sigma(T,K)$ where $K,T$ are the option strike and time to maturity, at given pairs $(T,K)$ where the European options prices or their implied volatilities are known. This process is called calibration of the local volatility (or, alternatively, implied volatility) surface. Various approaches to solving this important problem were proposed, see, eg, survey in \\cite{ItkinLipton2017} and references therein.\n\nAs mentioned in \\cite{ItkinLipton2017}, there are two main approaches to solving the calibration problem. The first approach relies on some parametric or non-parametric regression to construct a continuous implied volatility (IV) surface matching the given market quotes. Then the corresponding local volatility surface can be found via the well-known Dupire's formula, see, e.g., \\cite{ItkinSigmoid2015} and references therein.\n\nThe second approach relies on the direct solution of the Dupire equation using either analytical or numerical methods. The advantage of this approach is that it guarantees no-arbitrage \\footnote{But only if an analytical or numerical method in use does preserve no-arbitrage. This includes various interpolations, etc.}. However, the problem of the direct solution can be ill-posed, \\cite{Coleman2001}, and is rather computationally expensive. For instance, in \\cite{ItkinLipton2017} the Dupire equation (a partial differential equation (PDE) of the parabolic type) is solved by i) first using the Laplace-Carson transform, and ii) then applying various transformations to obtain a closed form solution of the transformed equation in terms of Kummer hypergeometric functions. Still, it requires an inverse Laplace transform to obtain the final solution.\n\nWith the second approach in use one also has to make an assumption about the behavior of the local\/implied volatility surface at strikes and maturities where the market quotes are not known. Usually, by a tractability argument the corresponding local variance is seen either piecewise constant, \\cite{LiptonSepp2011}, or piecewise linear \\cite{ItkinLipton2017} in the log-strike space, and piecewise constant in the time to maturity space \\footnote{See, however, comments in \\cite{ItkinLipton2017} about their assumptions.}.\n\nTo improve computational efficiency of calibration, an important step is made in \\cite{CarrNadtochiy2017} where Local Variance Gamma (LVG) model has been introduced (the first version refers to 2014 and can be found in \\cite{CarrLVGOrig2014}). This model assumes that the risk-neutral process for the underlying futures price is a pure jump Markov martingale, and that European option prices are given at a continuum of strikes and at one or more maturities. The authors construct a time-homogeneous process which meets a single smile and a piecewise time-homogeneous process, which can meet multiple smiles. However, in contrast to eg, \\cite{ItkinLipton2017}, their construction leads not to a PDE, but to a partial differential difference equation (PDDE), which permits both explicit calibration and fast numerical valuation. In particular, it does not require application of any optimization methods, rather just a root solver. In \\cite{CarrNadtochiy2017} this model is used to calibrate the local volatility surface assuming its piecewise constant structure in the strike space.\n\nOne of the potential criticism of this calibration method is the fact that the resulting local volatility function has a finite number of discontinuities. So it would be advantaged to relax the piecewise constant behavior of the surface. This is similar to how \\cite{ItkinLipton2017} was developed to overcome the same problem as compared with \\cite{LiptonSepp2011}.\n\nOn this way, recently \\cite{FalckDeryabin2017} applied the LVG model to the FX options market where usually option prices are quoted only at five strikes. They assumed that the local volatility function is continuous, piecewise linear in the four inner strike subintervals and constant in the outer subintervals. A closed form solution of the PDDE derived in \\cite{CarrLVGOrig2014}) is obtained with this parametrization, and calibration of some volatility smiles is provided. Still, to calibrate the model the authors rely on a residual minimization by using a least-square approach. So, despite an improved version of the LVG model is used, computational efficiency of this method is not perfect.\n\nAnother remark of \\cite{CarrNadtochiy2017} is about the limitation that the risk-neutral price process of the underlying is assumed to be a martingale, i.e. the main driving process in \\eqref{D} doesn't have a drift. However, the drift may not be negligible. If the drift is deterministic, e.g when the interest rate and dividends are deterministic, and the drift is a deterministic function of them, the calibration problem can be reduced to the driftless case by discounting, but this assumption might be inconsistent with the market. Therefore, an expansion of the proposed model that allows for a non-zero and stochastic drift is very desirable. In particular, it would be interesting to expand the LVG model to a risk-neutral price process obtained by stochastic time change of a drifted diffusion. In this way, similar to local Variance Gamma model, \\cite{MadCarrChang}, we introduce both stochastic volatility and stochastic drift.\n\nWith this in mind, our ultimate goals in this paper are as follows. First, we propose an expanded version of the LVG model by adding drift to the governing underlying process. It turns out that to proceed we need to re-derive and re-think every step in construction proposed in \\cite{CarrNadtochiy2017}. We show that still it is possible to find an ordinary differential equation (ODE) for the option price which plays a role of Dupire's equation for the standard local volatility model, and how calibration of multiple smiles (the whole local volatility surface) can be done in such a case.\n\nFurther, assuming the local variance to be a piecewise linear function of strike and piecewise constant function of time we solve this ODE in closed form in terms of Confluent hypergeometric functions. Calibration of the model to market smiles does not require solving any optimization problem. In contrast, it can be done term-by-term by solving a system of non-linear algebraic equations for each maturity, and thus is much faster.\n\nThe rest of the paper is organized as follows. In Section~\\ref{Process} the Expanded Local Variance Gamma model is formulated. In Section~\\ref{ForwardEq} we derive a forward equation (which is an ordinary differential equation (ODE)) for Put option prices using a homogeneous Bochner subordination approach. Section~\\ref{LVpwconst} generalizes this approach by considering the local variance being piece-wise constant in time. In Section~\\ref{SolutionODE} a closed form solution of the derived ODE is given in terms of Confluent hypergeometric functions. The next Section discusses computation of a source term of this ODE which requires a no-arbitrage interpolation. Using the idea of \\cite{ItkinLipton2017}), we show how to construct non-linear interpolation which provides both no-arbitrage, and a nice tractable representation of the source term, so that all integrals in the source term can be computed in closed form. In Section~\\ref{calib} calibration of multiple smiles in our model is discussed in detail. To calibrate a single smile we derive a system of nonlinear algebraic equations for the model parameters, and explain how to obtain a smart guess for their initial values. In Section~\\ref{asympt}\nasymptotic solutions of our ODE at extreme values of the model parameters are derived which improve computational accuracy and speed of the numerical solution. Section~\\ref{numExp} presents the results of some numerical experiments where calibration of the model to the given market smiles is done term-by-term. The last Section concludes.\n\n\n\\section{Process} \\label{Process}\n\nBelow where possible we follow the notation of \\cite{CarrNadtochiy2017}.\n\nLet $W_t$ be a $\\mathbb{Q}$ standard Brownian motion with time index $t \\ge 0$. Consider a stochastic process $D_t$ to be a time-homogeneous diffusion\n\\begin{equation} \\label{D}\n d D_t = \\mu D_t d t + \\sigma(D_t) d W_t,\n\\end{equation}\n\\noindent where the volatility function $\\sigma$ is local and time-homogeneous, and $\\mu$ is deterministic.\n\nA unique solution to \\eqref{D} exists if $\\sigma(D) : \\mathbb{R} \\to \\mathbb{R}$ is\nLipschitz continuous in $D$ and satisfies growth conditions at infinity. According to \\eqref{D} we have $D_t \\in (-\\infty,\\infty)$ while $t \\in [0,\\infty)$. Since $D$ is a time-homogeneous Markov process, its infinitesimal generator ${\\cal A}$ is given by\n\\begin{equation} \\label{gen}\n{\\cal A} \\phi(D) \\equiv \\left[\\mu D {\\nabla}_D + \\frac{1}{2}\\sigma^2(D) {\\nabla}^2_D \\right] \\phi(D)\n\\end{equation}\n\\noindent for all twice differentiable functions $\\phi$. Here ${\\nabla}_x$ is a first order\ndifferential operator on $x$. The semigroup of the $D$ process is\n\\begin{equation} \\label{semig}\n{\\cal T}^D_t \\phi(D_t) = e^{t {\\cal A}} \\phi(D_t) = \\EQ[\\phi(D_t)|D_0 = D], \\quad \\forall t \\ge 0.\n\\end{equation}\n\nIn the spirit of Variance Gamma model, \\cite{MadanSeneta:90,MadCarrChang} and similar to \\cite{CarrNadtochiy2017}, introduce a new process $D_{\\Gamma_t}$ which is $D_t$ subordinated by the unbiased Gamma clock $\\Gamma_t$. The density of the unbiased Gamma clock $\\Gamma_t$ at time $t \\ge 0$ is\n\\begin{equation} \\label{gammaDen}\n\\mathbb{Q}\\{\\Gamma_t \\in d\\nu\\} = \\dfrac{\\nu^{m-1} e^{-\\nu m \/t}}{(t^*)^m \\Gamma(m)} d\\nu, \\quad \\nu > 0, \\quad m \\equiv t\/t^*.\n\\end{equation}\nHere $t^* > 0$ is a free parameter of the process, $\\Gamma(x)$ is the Gamma function. It is easy to check that\n\\begin{equation} \\label{gammaExp}\n\\EQ[\\Gamma_t] = t.\n\\end{equation}\n\\noindent Thus, on average the stochastic gamma clock $\\Gamma_t$ runs synchronously with the calendar time $t$.\n\nAs applied to the option pricing problem, we introduce a more complex construction.\nNamely, consider options written on the underlying process $S_t$. Without loss of generality and for the sake of clearness let us treat below $S_t$ as the stock price process. Here, in contrast to \\cite{CarrNadtochiy2017}, we don't ignore interest rates $r$ and continuous dividends $q$ assuming them to be deterministic (below for simplicity of presentation we treat them as constants, but this can be easily relaxed). Then, let us define $S_t$ as\n\\begin{equation} \\label{sub}\nS_t = D_{\\Gamma_{X(t)}}, \\qquad X(t) = \\dfrac{1 - e^{-(r-q) t}}{r-q}.\n\\end{equation}\n\\noindent It is clear that in the limit $ r \\to 0, \\ q \\to 0$ we have $X(t) = t$, i.e., in this limit our construction coincides with that in \\cite{CarrNadtochiy2017} who considered a driftless diffusion and assumed $S_t = D_{\\Gamma_t}$. Also based on \\eqref{gammaExp}\n\\begin{equation} \\label{gammaXexp}\n\\EQ[\\Gamma_{X(t)}] = X(t).\n\\end{equation}\nFunction $X(t)$ starts at zero, ie, $X(0) = 0$, and is a continuous increasing function of time $t$. Indeed, if $r - q > 0$, then $X(t)$ is increasing in $t$ on $t \\in [0,\\infty)$, and at $t \\to \\infty$ it tends to constant. The infinite time horizon is not practically important, but for any finite time $t$ function $X(t)$ can be treated as an increasing function in $t$. If $r - q < 0$, function $X(t)$ is strictly increasing $\\forall t \\in [0,\\infty)$. Thus, $X(t)$ has all properties of a good clock. Accordingly, $\\Gamma_{X(t)}$ has all properties of a random time.\n\nUnder a risk-neutral measure $\\mathbb{Q}$, the total gain process, including the underlying price appreciation and dividends, after discounting at the risk free rate should be a martingale, see, eg, \\cite{Shreve:1992}. This process obeys the following stochastic differential equation\n\\begin{align} \\label{mart}\nd \\left( e^{-r t} S_t e^{q t} \\right) &= e^{(q-r)t} \\left[ (q-r) S_t dt + d S_t\\right]. \\end{align}\nTaking an expectation of both parts we obtain\n\\begin{align} \\label{martCond}\n\\EQ[d \\left( e^{(q-r)t} S_t\\right)] &=\ne^{(q-r)t} \\left\\{ (q-r) \\EQ[S_t] d t + d \\EQ[S_t] \\right\\}.\n\\end{align}\nObserve, that from \\eqref{sub}, \\eqref{D}\n\\begin{align} \\label{e1}\n\\EQ[d S_t] &= \\EQ[d {D_{\\Gamma_{X(t)}}}] = \\mu \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}] + \\EQ[\\sigma(D_{\\Gamma_{X(t)}}) d W_{\\Gamma_{X(t)}}] = \\mu \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}],\n\\end{align}\n\\noindent because the process $W_{\\Gamma_t}$ is a local martingale, see \\cite{RevuzYor1999}, chapter 6. Accordingly, the process $W_{\\Gamma_{X(t)}}$ inherits this property from $W_{\\Gamma_t}$, hence $\\EQ[\\sigma(D_{\\Gamma_{X(t)}}) d W_{\\Gamma_{X(t)}}] = 0$.\n\nFurther assume that the Gamma process $\\Gamma_t$ is independent of $W_t$ (and, accordingly, ${\\Gamma_{X(t)}}$ is independent of $W_{\\Gamma_{X(t)}}$). Then the expectation in the RHS of \\eqref{e1} can be computed, by first conditioning on ${\\Gamma_{X(t)}}$, and then integrating over the distribution of ${\\Gamma_{X(t)}}$ which can be obtained from \\eqref{gammaDen} by replacing $t$ with $X(t)$, i.e.\n\\begin{align} \\label{e2}\n\\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}} | S_s] &= \\int_0^\\infty \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}| {\\Gamma_{X(t)}} = \\nu] \\dfrac{\\nu^{m-1} e^{-\\nu m \/X(t)}}{(t^*)^m \\Gamma(m)} \\\\\n&= \\int_0^\\infty \\EQ [D_\\nu] \\dfrac{\\nu^{m-1} e^{-\\nu m \/X(t)}}{(t^*)^m \\Gamma(m)} d \\nu, \\quad\n\\nu > 0, \\quad m \\equiv X(t)\/t^*. \\nonumber\n\\end{align}\nThe find $\\EQ [D_\\nu]$ we take into account \\eqref{D} to obtain\n\\begin{align} \\label{e3}\nd \\EQ [D_\\nu] = \\EQ [d D_\\nu]\n= \\EQ [\\mu D_\\nu d\\nu + \\sigma(D_\\nu) D_\\nu d W_\\nu] = \\mu \\EQ [D_\\nu] d\\nu.\n\\end{align}\nSolving this equation with respect to $y(\\nu) = \\EQ [D_\\nu | D_s]$, we obtain $\\EQ [D_\\nu | D_s] = D_s e^{\\mu (\\nu-s)}$. Since we condition on time $s$, it means that $D_s = D_{\\Gamma_{X(s)}} = S_s$, and thus\n$\\EQ [D_\\nu | D_s] = S_s e^{\\mu (\\nu-s)}$.\n\nFurther, we substitute this into \\eqref{e2}, set the parameter of the Gamma distribution $t^*$ to be $t^* = X(t)$ (so $m = 1$) and integrate to obtain\n\\begin{equation} \\label{intRight}\nd \\EQ[S_t | S_s] = \\EQ[d S_t | S_s] = \\mu \\EQ [{D_{\\Gamma_{X(t)}}} d {\\Gamma_{X(t)}}] = S_s e^{- s \\mu} \\frac{\\mu}{1 - \\mu X(t)}.\n\\end{equation}\nSetting now $m = r-q$ and solving this equation we find\n\\begin{equation} \\label{ESsol}\n\\EQ[S_t | S_s] = S_s (r-q) e^{(q-r)(s-t)}.\n\\end{equation}\nSubstituting \\cref{ESsol} and \\cref{intRight} into \\cref{martCond} yields $d \\left( e^{-r t} S_t e^{q t} \\right) = 0$. Thus, if we chose $\\mu = r-q$, the right hands part of \\eqref{mart} vanishes, and our discounted stock process with allowance for non-zero interest rates and continuous dividends becomes a martingale. So the proposed construction can be used for option pricing.\n\nThis setting can be easily generalized for time-dependent interest rates $r(t)$ and continuous dividends $q(t)$. We leave it for the reader.\n\nThe next step is to consider connection between the original and time-changed processes. It is known from \\cite{BochnerPDE1949} that the process $G_{\\Gamma_t}$ defined as\n\\[ d G_t = \\sigma^2(G) d W_t \\]\n\\noindent is a time-homogeneous Markov process. As the deterministic process $\\mu t$ is also time-homogeneous, the whole process $D_t$ defined in \\eqref{D} is also a time-homogeneous Markov process. Accordingly, the semigroups $T^S_t$ of $S_t$ and $T^D_t$ of $D_{\\Gamma_{X(t)}}$ are connected by the Bochner integral\n\\begin{equation} \\label{BI}\n{\\cal T}^S_t U(S) = \\int_0^\\infty {\\cal T}^D_\\nu U(S) \\mathbb{Q}\\{{\\Gamma_{X(t)}} \\in d\\nu\\}, \\quad \\forall t \\ge 0,\n\\end{equation}\n\\noindent where $U(S)$ is a function in the domain of both ${\\cal T}^D_t$ and ${\\cal T}^S_t$.\nIt can be derived by exploiting the time homogeneity of the $D$ process, conditioning on the gamma time first, and taking into account the independence of $\\Gamma_t$ and $W_t$ (or $\\Gamma_{\\Gamma_{X(t)}}$ and $W_{\\Gamma_{X(t)}}$ in our case).\n\nWe set parameter $t^*$ of the gamma clock to $t^* = X(t)$. Then \\eqref{BI} and \\eqref{gammaDen} imply\n\\begin{equation} \\label{BI1}\n{\\cal T}^S_{t} U(S) = \\int_0^\\infty {\\cal T}^D_\\nu U(S) \\dfrac{e^{-\\nu\/X(t)}}{X(t)} d\\nu.\n\\end{equation}\nIn what follows for the sake of brevity we will call this model as an Expanded Local Variance Gamma model, or ELVG.\n\n\\section{Forward equation for Put option prices} \\label{ForwardEq}\n\nFollowing \\cite{CarrNadtochiy2017} we interpret the index $t$ of the semigroup ${\\cal T}^S_t$ as the maturity date $T$ of a European claim with the valuation time $t = 0$. Also let the test function $U(S)$ be the payoff of this European claim, ie,\n\\begin{equation} \\label{payoff}\nU(S_T) = e^{-r T}(K - S_T)^+.\n\\end{equation}\nThen define\n\\begin{equation} \\label{P0}\nP(S_0,T,K) = {\\cal T}^S_T U(S_0)\n\\end{equation}\n\\noindent as the European Put value with maturity $T$ at time $t=0$ in the ELVG model. Similarly\n\\begin{equation} \\label{P0D}\nP^D(S_0,\\nu,K) = {\\cal T}^D_\\nu U(S_0)\n\\end{equation}\n\\noindent would be the European Put value with maturity $\\nu$ at time $t=0$ in the model of \\eqref{D}\\footnote{Below for simplicity of notation we drop the subscript '0' in $S_0$.}. Then the Bochner integral in \\eqref{BI1} takes the form\n\\begin{equation} \\label{Bochner2}\nP(S,T,K) = \\int_0^\\infty P^D(S,\\nu,K) p e^{- p \\nu} d \\nu, \\quad p \\equiv 1\/X(T).\n\\end{equation}\nThus, $P(S,X(T),K)$ is represented by a Laplace-Carson transform of $P^D(S,\\nu,K)$ with $p$ being a parameter of the transform. Note that\n\\begin{equation} \\label{init}\nP(S,0,K) = P^D(S,0,K) = U(S).\n\\end{equation}\nTo proceed, we need an analog of the Dupire forward PDE for $P^D(S,\\nu,K)$.\n\n\\subsection{Derivation of the Dupire forward PDE \\label{dupFWPDE}}\n\nDespite this can be done in many different ways, below for the sake of compatibility we do it in the spirit of \\cite{CarrNadtochiy2017}.\n\nFirst, differentiating \\eqref{P0D} by $\\nu$ with allowance for \\eqref{semig} yields\n\\begin{align} \\label{Dt}\n{\\nabla}_\\nu P^D(S,\\nu,K) &= e^{-r \\nu} e^{\\nu {\\cal A}}\\left[ {\\cal A} - r\\right] U(S)\n= e^{-r \\nu} \\EQ \\left[ {\\cal A} - r\\right] U(S).\n\\end{align}\nWe take into account the definition of the generator ${\\cal A}$ in \\eqref{gen}, and also remind that at $t=0$ we have $D_0 = S_0$. Then \\eqref{Dt} transforms to\n\\begin{equation} \\label{Dt1}\n{\\nabla}_\\nu P^D(S,\\nu,K) = -r P^D(S,\\nu,K) + (r-q) S {\\nabla}_S P^D(S,\\nu,K) + e^{-r \\nu}\\frac{1}{2} \\EQ \\left[ \\sigma^2(S) {\\nabla}_S^2 U(S) \\right].\n\\end{equation}\nHowever, we need to express the forward equation using a pair of independent variables $(\\nu,K)$ while \\eqref{Dt} is derived in terms of $(\\nu,S)$. To do this, observe that\n\\begin{align} \\label{sir}\ne^{-r \\nu} \\EQ \\left[\\sigma^2(S) {\\nabla}_S^2 U(S) \\right] &= e^{-r \\nu}\\EQ \\left[\\sigma^2(S) \\delta(K-S)\\right] = e^{-r \\nu} \\EQ \\left[\\sigma^2(K) \\delta(K-S)\\right] \\\\\n&= e^{-r \\nu} \\EQ \\left[\\sigma^2(K) {\\nabla}_K^2 U(S)\\right] = \\sigma^2(K) {\\nabla}_K^2 P^D(S,\\nu,K). \\nonumber\n\\end{align}\n\\noindent where the sifting property of the Dirac delta function $\\delta(S-K)$ has been used. Also\n\\begin{align} \\label{term2}\n-r & P^D(S,\\nu,K) + (r-q) S \\nabla_S P^D(S,\\nu,K) \\\\\n&= e^{-r \\nu} \\EQ\\left[ -r (K-S)^+ + (r-q)S \\fp{(K-S)^+}{S} \\right] \\nonumber \\\\\n&= e^{-r \\nu} \\EQ\\left[ -r (K-S)^+ - (r-q)(K-S) \\fp{(K-S)^+}{S} + (r-q)K \\fp{(K-S)^+}{S}\\right] \\nonumber \\\\\n&= e^{-r \\nu} \\EQ\\left[ -r (K-S)^+ + (r-q)(K-S)^+ - (r-q)K \\fp{(K-S)^+}{K}\\right] \\nonumber \\\\\n&= -q P^D(S,\\nu,K) - (r-q) K \\nabla_K P^D(S,\\nu,K). \\nonumber\n\\end{align}\nTherefore, using \\eqref{sir} and \\eqref{term2}, \\eqref{Dt} could be transformed to\n\\begin{align} \\label{Dup1}\n\\nabla_\\nu P^D(S,\\nu,K) &= -q P^D(S,\\nu,K) - (r-q) K \\nabla_K P^D(S,\\nu,K) + \\frac{1}{2} \\sigma^2(K) K^2 \\nabla_K^2 P^D(S,\\nu,K) \\nonumber \\\\\n&\\equiv {{\\cal A}}^K P^D(S,\\nu,K), \\\\\n{{\\cal A}}^K &= -q - (r-q) K \\nabla_K + \\frac{1}{2} \\sigma^2(K) K^2 \\nabla_K^2. \\nonumber\n\\end{align}\nThis equation looks exactly like the Dupire equation with non-zero interest rates and continuous dividends, see, eg, \\cite{Tysk2012} and references therein. Note, that ${\\cal A}^K$ is also a time-homogeneous generator.\n\n\\subsection{Forward partial divided-difference equation} \\label{FPDDder}\n\nOur final step is to apply the linear differential operator ${{\\cal A}}^K$ defined in \\eqref{Dup1} to both parts of \\eqref{Bochner2}. Using time-homogeneity of $D_t$ and, again, the Dupire equation \\eqref{Dup1}, we obtain\n\\begin{align} \\label{b2}\n-q &P(S,T,K) - (r-q) K {\\nabla}_K P(S,T,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2\nP(S,T,K) \\\\\n&= \\int_0^\\infty p e^{- p \\nu} \\left[-q P^D(S,\\nu,K) - (r-q) K {\\nabla}_K P^D(S,\\nu,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 P^D(S,\\nu,K)\\right] d \\nu \\nonumber \\\\\n&= \\int_0^\\infty p e^{- p \\nu} {\\nabla}_\\nu P^D(S,\\nu,K) d \\nu = - p P^D(S,0,K) + p \\int_0^\\infty P^D(S,\\nu,K) p e^{- p \\nu} d \\nu \\nonumber \\\\\n&= p\\left[P(S,T,K) - P^D(S,0,K) \\right] = p\\left[P(S,T,K) - P(S,0,K) \\right], \\nonumber\n\\end{align}\n\\noindent where in the last line \\eqref{init} was taken into account.\n\nThus, finally $P(S,T,K)$ solves the following problem\n\\begin{align} \\label{finDup}\n-q P(S,T,K) &- (r-q) K {\\nabla}_K P(S,T,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2\nP(S,T,K) \\\\\n&= \\dfrac{P(S,T,K) - P(S,0,K)}{X(T)}, \\qquad P(S,0,K) = (K-S)^+. \\nonumber\n\\end{align}\nAt $ r = q = 0$ this equation translates to the corresponding equation in \\cite{CarrNadtochiy2017}. In contrast to the Dupire equation which belongs to the class of PDE, \\eqref{finDup} is an ODE, or, more precisely, a partial divided-difference equation (PDDE), since the derivative in time in the right hands part is now replaced by a divided difference. In the form of an ODE it reads\n\\begin{equation} \\label{finDupPut}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 - (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{X(T)}\\right) \\right] P(S,T,K) =\n- \\dfrac{P(S,0,K)}{X(T)}.\n\\end{equation}\nThis equation could be solved analytically for some particular forms of the local volatility function $\\sigma(K)$ which are considered later in this paper. Also in the same way a similar equation could be derived for the Call option price $C_0(S,T,K)$ which reads\n\\begin{align} \\label{finDupCall}\n\\Big[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 + (r-q) K {\\nabla}_K &- \\left(q + \\dfrac{1}{X(T)}\\right) \\Big] C_0(S,T,K) = - \\dfrac{C_0(S,0,K)}{X(T)}, \\nonumber \\\\\nC_0(S,0,K) &= (S-K)^+.\n\\end{align}\n\nSolving \\eqref{finDupPut} or \\eqref{finDupCall} provides the way to determine $\\sigma(K)$ given market quotes of Call and Put options with maturity $T$. However, this allows calibration of just a single term. Calibration of the whole local volatility surface, in principle, could be done term-by-term (because of the time-homogeneity assumption) if \\eqref{finDupPut}, \\eqref{finDupCall} could be generalized to this case. We consider this in the following Section.\n\n\\section{Local variance piece-wise constant in time} \\label{LVpwconst}\n\nTo address calibration of multiple smiles, we need to relax some assumptions about time-homogeneity of the process $D_t$ defined in \\eqref{D}. This includes several steps which are described below in more detail.\n\n\\subsection{Local variance}\n\nHere we assume that the local variance $\\sigma(D_t)$ is no more time-homogeneous, but a piece-wise constant function of time $\\sigma(D_t,t)$.\n\nLet $T_1, T_2, \\ldots, T_M$ be the time points at which the variance rate $\\sigma^2(D_t)$ jumps deterministically. In other words, at the interval $t \\in [T_0,T_1)$, the variance rate is $\\sigma^2_0(D_t)$, at $t \\in [T_1, T_2)$ it is $\\sigma^2_1(D_t)$, etc. This can be also represented as\n\\begin{align} \\label{sigmaPW}\n\\sigma^2(D_t,t) &= \\sum_{i=0}^M \\sigma^2_i(D_t) w_i(t), \\\\\nw_i(t) &\\equiv {\\mathbf 1}_{t - T_i} - {\\mathbf 1}_{t - T_{i+1}}, \\ i=0,\\ldots,M,\n\\quad T_0 = 0, \\ T_{M+1} = \\infty, \\quad {\\mathbf 1}_x =\n\\begin{cases}\n1, & x \\ge 0 \\\\\n0, & x < 0.\n\\end{cases}\n\\nonumber\n\\end{align}\n\nNote, that\n\\[ \\sum_{i=0}^M w_i(t) = {\\mathbf 1}_t - {\\mathbf 1}_{t-\\infty} = 1, \\quad \\forall t \\ge 0.\\]\n\\noindent Therefore, in case when all $\\sigma^2_i(D_t)$ are equal, ie, independent on index $i$, \\eqref{sigmaPW} reduces to the case considered in the previous Sections.\n\nIt is important to notice that our construction implies that the volatility $\\sigma(D_t)$ jumps as a function of time at the calendar times $T_0, T_1,\\ldots,T_M$, and not at the business times $\\nu$ determined by the gamma clock. Otherwise, the volatility function would change at random (business) times which means it is stochastic. But this definitely lies out of scope of our model. Therefore, we need to change \\eqref{sigmaPW} to\n\\begin{align} \\label{sigmaPW_exp}\n\\sigma^2({D_{\\Gamma_{X(t)}}}, {\\Gamma_{X(t)}}) &= \\sum_{i=0}^M \\sigma^2_i(D_t) {\\bar w}_i(\\EQ({\\Gamma_{X(t)}})), \\\\\n{\\bar w}_i(t) &= {\\mathbf 1}_{X^{-1}(t) - T_i} - {\\mathbf 1}_{X^{-1}(t) - T_{i+1}}, \\ i=0,\\ldots,M, \\nonumber \\\\\nX^{-1}(t) &= \\dfrac{1}{q-r} \\log \\left[1 - (r-q)t \\right].\n\\end{align}\nHence, when using \\eqref{sub} we have\n\\begin{align} \\label{sigmaPW_exp1}\n\\sigma^2(D_t, t)\\Big|_{t = \\Gamma_{X(t)}} &= \\sum_{i=0}^M \\sigma^2_i(D_t) \\bar{w}_i(X(t)) = \\sum_{i=0}^M \\sigma^2_i(D_t) w_i(t).\n\\end{align}\n\nAccordingly, if the calendar time $t$ belongs to the interval $T_0 \\le t < T_1$, the infinitesimal generator ${\\cal A}$ of the semigroup ${\\cal T}^D_\\nu$ is a function of $\\sigma(D_t)$ (and not on $\\sigma(D_\\nu)$). As at $T_0 \\le t < T_1$ we assume $\\sigma(D) = \\sigma_0(D)$, i.e., is constant in time, it doesn't depend of $\\nu$. Thus, ${\\cal A}$ (which for this interval of time we will denote as ${\\cal A}_0$) is still time-homogeneous.\n\nSimilarly, one can see, that for $T_1 \\le t < T_2$ the infinitesimal generator ${\\cal A}_1$ of the semigroup ${\\cal T}^D_\\nu$ is also time-homogeneous and depends on $\\sigma_1(D)$, etc.\n\n\\subsection{Bochner subordination}\n\nWe start with re-definition of \\eqref{P0}, \\eqref{P0D}. We now define the European Put value with maturity $T$ at the evaluation time $t=X(T_1)$ in the ELVG model\n\\begin{equation} \\label{P02}\nP(S_0,T_1 + T,K) = {\\cal T}^S_T [e^{-r T} P(S_0,T_1,K)].\n\\end{equation}\nAnd, clearly we are interesting in the value of $T$ to be $T = T_2 - T_1$.\n\nSimilarly, we define the European Put value with maturity $\\nu$ at the evaluation time $t=T_1$ in the model given by \\eqref{D} as\n\\begin{equation} \\label{P0D2}\nP^D(S_0,T_1 + \\nu,K) = {\\cal T}^D_\\nu [e^{-r \\nu} P(S_0,T_1,K)].\n\\end{equation}\nBy these definitions\n\\[ P(S_0,T_1 + T,K)\\Big|_{T=0} = P^D(S_0,T_1 + \\nu,K)\\Big|_{\\nu=0} = P(S_0,T_1,K)]. \\]\n\nIn contrast to \\eqref{Bochner2}, in case of multiple smiles at $t > T_1$ we need to change the definition of $t$ in \\eqref{BI1} from $t \\mapsto X(t)$ to\n\\begin{equation} \\label{newt*}\nt \\mapsto X(T_1+t) - X(T_1) \\equiv \\Delta x(T_1,t).\n\\end{equation}\nThis definition implies two observations.\n\nFirst, function $\\Delta x(T_1,t)$ starts at zero at $t=0$ and is an increasing function of time. Also, in case $r=q=0$ we have $\\Delta x(T_1,t) = t$. Therefore, $\\Delta x(T_1,t)$ can be used as a good clock. Accordingly, similar to \\eqref{gammaExp} we have\n\\begin{equation} \\label{gammaXexp1}\n\\EQ[\\Gamma_{\\Delta x(T_1,t)}] = \\Delta x(T_1,t).\n\\end{equation}\nSecond, a proof that in our model the discounted stock price is a martingale\ngiven in Section~\\ref{Process} could be repeated for times $t: \\ T_1 < t \\le T_2$.\nWhen doing so, at $t > T_1$ we reset the definition of $S_t$ to\n\\[ S_{T_1+t} = D_{\\Gamma_{\\Delta x(T_1,t) }}, \\quad t \\ge 0. \\]\nThen instead of \\eqref{e1} we now have\n\\begin{align} \\label{e11}\n\\EQ[d S_{T_1+t}] &= \\EQ[d D_{\\Gamma_{\\Delta x(T_1,t)}}] = \\mu \\EQ [D_{\\Gamma_{\\Delta x(T_1,t)}} d \\Gamma_{\\Delta x(T_1,t)}] + \\EQ[\\sigma(D_{\\Gamma_{\\Delta x(T_1,t)}}) d W_{\\Gamma_{\\Delta x(T_1,t) }}] \\\\\n&= \\mu \\EQ [D_{\\Gamma_{\\Delta x(T_1,t)}}] d \\Delta x(T_1,t) =\n\\mu \\EQ [D_{\\Gamma_{\\Delta x(T_1,t)}}] d X(T_1+t). \\nonumber\n\\end{align}\nOn the other hand,\n\\begin{align} \\label{martCond1}\n\\EQ[d \\left( e^{(q-r)(T_1+t)} S_{T_1+t} \\right)] &=\ne^{(q-r)(T_1+t)} \\left\\{ (q-r) \\EQ[S_{T_1+t}] d t + d \\EQ[S_{T_1+t}] \\right\\} \\\\\n&= e^{(q-r)t}[\\mu + (q-r)S_{T_1}e^{-(r-q)T_1}] dt \\nonumber\n\\end{align}\n\nOne can check, that with $\\mu = r-q$ the RHS of \\cref{martCond1} vanishes, therefore this construction can be used for option pricing.\n\nThe definition in \\eqref{newt*} implies that parameter $t$ of the Gamma random clock is reset at the point $T_1$, i.e., at $0 \\le t \\le T_1$ it is $t \\mapsto X(t) = X(t) - X(0)$, while at $T_1 < t \\le T_2$ it is $t \\mapsto X(T_1+t) - X(T_1)$. Using the definition of $w_i(t)$ in \\eqref{sigmaPW}, this could be written as\n\\begin{equation} \\label{t*2}\nt \\mapsto \\sum_{i=0}^M w_i(T_i+t)[X(T_i+t) - X(T_i)]\n\\end{equation}\nResetting $t$ was also first proposed in \\cite{CarrNadtochiy2017} but in a different form.\n\nThen, the Bochner integral in \\eqref{BI1} transforms to\n\\begin{equation} \\label{BI3}\n{\\cal T}^S_{T} P(S,T_1,K) = \\int_{0}^\\infty {\\cal T}^D_\\nu P(S,T_1 + \\nu,K) \\dfrac{\\nu^{m-1} e^{-\\nu m \/\\Delta X(T_1,T)}}{(t^*)^m \\Gamma(m)} d\\nu.\n\\end{equation}\nSince for a tractability reason we still want to have $m \\equiv \\Delta X(T_1,T)\/t^* = 1$. we need to redefine $t^*$ in accordance with \\eqref{t*2}. Based on that, the Bochner integral in \\eqref{Bochner2} now finally reads\n\\begin{align} \\label{Bochner22}\nP(S,T_1 + T,K) &= \\int_0^\\infty P^D(S,T_1+\\nu,K) p e^{- p \\nu} d \\nu, \\quad\np \\equiv 1\/\\Delta X(T_1, T).\n\\end{align}\n\n\\subsection{Forward partial divided-difference equation for the second term} \\label{FPDDder2}\n\nNow we need to derive a Forward partial divided-difference equation for the second term $T_2$ similar to how this is done in Section~\\ref{FPDDder}. Obviously, the Put price $P^D(S_0,T_1 + \\nu,K)$ solves the same Dupire equation \\eqref{Dup1}. Therefore, proceeding in the same way as in Section~\\ref{FPDDder}, we apply linear differential operator $\\cal L$ defined in \\eqref{Dup1} to both parts of \\eqref{Bochner22}. Using time-homogeneity of $D_t$ at the interval $[T_1, T_2)$ and again the Dupire equation \\eqref{Dup1}, we obtain\n\\begin{align} \\label{b21}\n-q &P(S,T_1 + T,K) - (r-q) K {\\nabla}_K P(S,T_1 + T,K) + \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 P(S,T_1 + T,K) \\nonumber \\\\\n&= \\int_0^\\infty p e^{- p \\nu} \\Big[-q P^D(S,T_1 + \\nu,K) - (r-q) K {\\nabla}_K P^D(S,T_1 + \\nu,K) \\\\\n&+ \\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 P^D(S,T_1 + \\nu,K)\\Big] d \\nu\n= \\int_0^\\infty p e^{- p \\nu} {\\nabla}_\\nu P^D(S,T_1 + \\nu,K) d \\nu \\nonumber \\\\\n&= - p P^D(S,T_1,K) + p \\int_0^\\infty P^D(S,T_1+\\nu,K) p e^{- p \\nu} d\\nu \\nonumber \\\\\n&= p\\left[P(S,T_1 + T,K) - P^D(S,T_1,K)\\right] = p\\left[P(S,T_1 + T,K) - P(S,T_1,K) \\right]. \\nonumber\n\\end{align}\nFinally, taking $T = T_2 - T_1$ we obtain an ODE for the Put price $P(S,T_2,K)$.\n\\begin{equation} \\label{finDupPut2}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 - (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{\nX(T_2) - X(T_1)}\\right) \\right] P(S,T_2,K) = - \\dfrac{P(S,T_1,K)}{X(T_2) - X(T_1)}.\n\\end{equation}\nHere the local variance function $\\sigma^2(K) = \\sigma^2_1(K)$ as it corresponds to the interval $(T_1,T_2]$ where the above ODE is solved.\n\nWe continue in the same way to derive an ODE for the Put price $P(S,T_i,K), \\ i=1,\\ldots,M$, which finally reads\n\\begin{equation} \\label{finDupPutI}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 - (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{\nX(T_i) - X(T_{i-1})}\\right) \\right] P(S,T_i,K) = - \\dfrac{P(S,T_{i-1},K)}{X(T_i) - X(T_{i-1})}.\n\\end{equation}\nThis is a recurrent equation that can be solved for all $i=1,\\ldots,M$ sequentially starting with $i=1$ subject to some boundary conditions. The natural boundary conditions for the Put option price are, \\cite{hull:97}\n\\begin{equation} \\label{bc}\n\\begin{array}{lll}\nP(S,T_i,K) = 0, & & K \\to 0, \\\\\nP(S,T_i,K) = {\\cal D}_i K - {\\cal Q}_i S \\approx {\\cal D}_i K, & & K \\to \\infty,\n\\end{array}\n\\end{equation}\n\\noindent where ${\\cal D}_i = e^{-r T_i}$ is the discount factor, and ${\\cal Q}_i = e^{-q T_i}$.\n\nA similar equation can be obtained for the Call option prices, which reads\n\\begin{equation} \\label{finDupCallI}\n\\left[\\frac{1}{2} \\sigma^2(K) {\\nabla}_K^2 + (r-q) K {\\nabla}_K - \\left(q + \\dfrac{1}{\nX(T_i) - X(T_{i-1})}\\right) \\right] C(S,T_i,K) = - \\dfrac{C(S,T_{i-1},K)}{X(T_i) - X(T_{i-1})},\n\\end{equation}\n\\noindent subject to the boundary conditions\n\\begin{equation} \\label{bcCall}\n\\begin{array}{lll}\nC(S,T_i,K) = {\\cal Q}_i S, & & K \\to 0, \\\\\nC(S,T_i,K) = 0, & & K \\to \\infty.\n\\end{array}\n\\end{equation}\n\n\n\n\n\\section{Solution of the ODE \\eqref{finDupPutI}} \\label{SolutionODE}\n\nBelow we use the approach similar to \\cite{ItkinLipton2017} by assuming the local variance to be a piecewise linear continuous function of strike. In contrast to \\cite{ItkinLipton2017}, instead of a standard local volatility model in this paper we use the ELVG model. As the result, instead of a partial differential (Dupire) equation, we face a problem of solving the ODE in \\eqref{finDupPutI}.\n\nFirst, it is useful to change the dependent variable from $P(S,T_j,K)$ to\n\\[ V(S,T_j,K) = P(S,T_j,K) - {\\cal D}_j K, \\]\n\\noindent which is known as a {\\it covered Put}. The advantage of the covered Put is that according to \\eqref{bc} its price obeys homogeneous boundary conditions.\n\nUsing this definition we now re-write \\eqref{finDupPutI} in a more convenient form (while with some loose of notation)\n\\begin{align} \\label{finODE}\n& - v(x) V_{x,x}(x) + b_1 x V_x(x) + b_{0,j} V(x) = c_j, \\\\\nb_1 &= (r - q) p_j, \\quad b_{0,j} = q p_j + 1, \\quad c_j = V(T_{j-1},x) + \\beta x, \\nonumber \\\\\np_j &= X(T_j) - X(T_{j-1}) > 0, \\quad x = \\frac{K}{S}, \\quad V(x) = V(S,T_j,x), \\quad v(x) = p_j\\frac{\\sigma^2(x)}{2 S^2}. \\nonumber \\\\\n\\beta &= - S[{\\cal D}_j(1 + p_j r) - {\\cal D}_{j-1}]. \\nonumber\n\\end{align}\nIn \\eqref{finODE} $x$ is the inverse moneyness. In what follows we also assume that $r > q > 0$, but this assumption could be easily relaxed.\n\nFurther, suppose that for each maturity $T_j, \\ j \\in [1,M]$ the market quotes are provided at a set of strikes $K_i, \\ i=1,\\ldots,n_j$ where the strikes are assumed to be sorted in the increasing order. Then the corresponding continuous piecewise linear local variance function $v_j(x)$ on the interval $[x_{i},x_{i+1}]$ reads\n\\begin{equation} \\label{vDef}\nv_{j,i}(x) = v^0_{j,i} + v^1_{j,i} x,\n\\end{equation}\n\\noindent where we use the super-index $0$ to denote a level $v^0$, and the super-index $1$ to denote a slope $v^1$. Subindex $i=0$ in $v^0_{j,0}, v^1_{j,0}$ corresponds to the interval $(0, x_1]$. Since $v_j(x)$ is continuous, we have\n\\begin{equation} \\label{cont}\nv^0_{j,i} + v^1_{j,i} x_{i+1} = v^0_{j,i+1} + v^1_{j,i+1} x_{i+1}, \\quad i=0,\\ldots,n_j-1.\n\\end{equation}\nThe first derivative of $v_j(x)$ experiences a jump at points $x_{i}, \\irg{i}{1}{n_j}$. We also assume that $v(x,T)$ is a piecewise constant function of time, i.e., $v^0_{j,i}, v^1_{j,i}$ don't depend on $T$ on the intervals $[T_j, T_{j+1}), \\ j \\in [0,M-1]$, and jump to new values at the points $T_j, \\irg{j}{1}{M}$.\n\nWith the above assumptions in mind, \\eqref{finODE} can be solved by induction. One starts with $T_0 = 0$, and on each time interval $[T_{j-1},T_j], \\ \\irg{j}{1}{M}$ solves the problem \\eqref{finODE} for $V(x) \\mapsto P(S,T_j,x) - d_j S x$.\n\nSince $v(x)$ is a piecewise linear function, the solution of \\eqref{finODE} can also be constructed separately for each interval $[x_{i-1},x_i]$. By taking into account the explicit representation of $v(x)$ in \\eqref{vDef}, from \\eqref{finODE} for the $i$-th spatial interval we obtain\n\\begin{align} \\label{Laplace2}\n-(b_2 + a_2 x) V_{xx}(x) &+ b_1 x V_x(x) + b_0 V(x) = c \\\\\nb_2 &= v^0_{j,i}, \\ a_2 = v^1_{j,i}. \\nonumber\n\\end{align}\nWe proceed by introducing a new independent variable $z = (b_2 + a_2 x)b_1\/a_2^2, \\ z \\in \\mathbb{R}^+$, so that \\eqref{Laplace2} transforms to\n\\begin{align} \\label{Laplace3}\n-z V_{zz}(z) &+ (z-q_2)V_z(z) + q_1 V(z) = \\chi \\\\\nq_1 &= b_0\/b_1, \\ q_2 = b_2 b_1\/a_2^2, \\ \\chi = c\/b_1. \\nonumber\n\\end{align}\n\nThe \\eqref{Laplace3} is an {\\it inhomogeneous} Laplace equation, \\cite{PolyaninSaitsevODE2003}, page 155. It is well known that if $y_1=y_1(z)$, $y_2=y_2(z)$ are two fundamental solutions of the corresponding {\\it homogeneous} equation, then the general solution of \\eqref{Laplace3} can be represented as\n\\begin{align} \\label{solInhom}\nV(z) &= C_1 y_1(z) + C_2 y_2(z) + \\frac{1}{b_1} I_{12}(z) \\\\\nI_{12}(z) &= -y_2(z) \\int \\dfrac{ y_1(z) f(z)} {W z}d z + y_1(z) \\int \\dfrac{ y_2(z) f(z)}{W z}d z \\equiv I_1 + I_2, \\nonumber \\\\\nf(z) &= V(T_{j-1},z) - k_1 - k_2 z, \\quad\nk_1 = \\beta \\frac{b_2}{a_2}, \\quad k_2 = - \\beta \\frac{a_2}{b_1}, \\nonumber\n\\end{align}\n\\noindent where $W = y_1 (y_2)_z - y_2 (y_1)_z$ is the so-called Wronskian, and $\\beta$ is defined in \\eqref{finODE}. Then the problem is to determine suitable fundamental solutions of the homogeneous Laplace equations. Based on \\cite{PolyaninSaitsevODE2003}, if $a_2 \\ne 0$, they read\n\\begin{equation} \\label{homog}\ny_i(z) = {\\mathcal V}_i (q_1, q_2, z), \\quad i=1,2\n\\end{equation}\nHere ${\\mathcal V}_i(a,b,z)$ is an arbitrary solution of the degenerate hypergeometric equation, i.e., Kummer's function, \\cite{as64}. Two types of Kummer's functions are known, namely $M(a,b,z)$ and $U(a,b,z)$, which are Kummer's functions of the first and second kind.\n\nIt is known, that there exist several pairs of such independent solutions. Therefore, for every spatial interval in $z$ among all possible fundamental pairs we have to determine just one which is numerically satisfactory at this interval (see \\cite{Olver1997} for the detailed definition of satisfactory solutions and the corresponding discussion). Since our boundary conditions are set at zero and positive infinity, we need a numerically satisfactory solution for the positive half of the real line.\n\nSimilar to \\cite{ItkinLipton2017}, in the vicinity of the origin we choose the numerically satisfactory pair as, \\cite{Olver1997}\n\\begin{align} \\label{Kummer0}\ny_1(\\chi) &= M\\left(q_1, q_2, z\\right) = e^z M\\left(q_2-q_1, q_2, -z\\right), \\\\\ny_2(\\chi) &= z^{1-q_2} M\\left(q_1 - q_2 + 1, 2-q_2, z\\right) =\nz^{1-q_2} e^z M\\left(1-q_1, 2-q_2, -z\\right), \\nonumber \\\\\nW &= \\sin(\\pi q_2) z^{-q_2} e^{z}\/\\pi. \\nonumber\n\\end{align}\nHowever, in the vicinity of infinity the numerically satisfactory pair is, \\cite{Olver1997}\n\\begin{align} \\label{Kummer1}\ny_1(\\chi) &= U\\left(q_1, q_2, z\\right) = z^{1-q_2} U\\left(q_1-q_2+1, 2-q_2, z\\right), \\\\\ny_2(\\chi) &= e^z U\\left(q_2 - q_1, q_2, -z\\right) = e^z z^{1-q_2} U\\left(1-q_1, 2-q_2, -z\\right), \\nonumber \\\\\nW &= (-1)^{q_1 - q_2} e^{z} z^{-q_2}. \\nonumber\n\\end{align}\n\nAs two solutions $J_1(q_1,q_2, z), J_2(q_1,q_2, z)$ are independent, \\eqref{solInhom} is a general solution of \\eqref{Laplace3}. Two constants $C_1,C_2$ should be determined based on the boundary conditions for the function $V(z)$.\n\nThe boundary conditions for the ODE \\eqref{Laplace2} in a strike $K$ space (or in $x$ space) should be set at zero and infinity. Based on the usual shape of the local variance curve and its positivity, for $x \\to 0$, we expect that $v^1_{j,i} < 0$. Similarly, for $x \\to \\infty$ we expect that $v^1_{j,i} > 0$. In between these two limits the local variance curve for a given maturity $T_j$ is assumed to be continuous, but the slope of the curve could be both positive and negative, see, e.g., \\cite{ItkinSigmoid2015} and references therein. Also, by definition $z = v_{j,i}$, and $\\mathrm{Dom}(z) = \\mathbb{R}^+$. Thus, at high strikes $a_2 = v^1_{j,i} > 0$. Therefore, the boundary conditions for \\eqref{Laplace3} should be set at $z = b_2$ (which corresponds to the boundary $K=0$) and at $z \\to \\infty$. These are the boundary conditions given in \\eqref{bc}.\n\n\\section{Computation of the source term} \\label{SolutionInt}\n\nComputation of the source term $p I_{12}$ in \\eqref{solInhom} could be achieved in several ways. The most straightforward one is to use numerical integration as the Put price $P(x,T_{i-1})$ as a function of $x$ is already known when we solve \\eqref{finODE} for $T=T_i$. However, as this is discussed in detail in \\cite{ItkinLipton2017}, function $P(x,T_{i-1})$ is known only for a discrete set of points in $x$. Therefore, some kind of interpolation is necessary to find its values at the other points.\n\n\\subsection{No-arbitrage interpolation}\n\nAs shown in \\cite{ItkinLipton2017}, this interpolation must preserve no-arbitrage. So, for instance, a standard linear interpolation is not a good candidate, since its violates no-arbitrage conditions. Indeed, given three Put option prices $P(K_1), P(K_2), P(K_3)$ for three strikes $K_1 < K_2 < K_3$, the necessary and sufficient conditions for an arbitrage-free system read, \\cite{CoxRubinstein1985}\n\\begin{align} \\label{noarb}\nP(K_3) &> 0, \\qquad P(K_2) < P(K_3), \\\\\nBs = (K_3 - K_2)P(K_1) &- (K_3 - K_1)P(K_2) + (K_2 - K_1)P(K_3) > 0. \\nonumber\n\\end{align}\nSuppose that we want to use linear interpolation in the strike space on the interval $[K_1,K_3]$ to find the unknown Put option price $P(K_2)$ given the values of $P(K_1), P(K_3)$,\n\\begin{equation*}\nP(K_2) \\equiv P_l(K_2) = \\dfrac{P(K_1) K_3 - P(K_3) K_1}{K_3 - K_1} + \\dfrac{P(K_3) - P(K_1)}{K_3 - K_1} K_2.\n\\end{equation*}\nWhen plugging this expression into the second line of \\eqref{noarb}, the left hands side of the latter vanishes, so the third no-arbitrage condition is violated.\n\nIn \\cite{ItkinLipton2017} it is shown that this problem could be resolved if we use linear interpolation with a modified independent variable (further on we denote it as $P_F(K)$),\n\\begin{align} \\label{lin2}\nP(K_2) &\\equiv P_F(K_2) \\\\\n&= \\dfrac{P(K_1) f(K_3) - P(K_3) f(K_1)}{f(K_3) - f(K_1)} + \\dfrac{P(K_3) - P(K_1)}{f(K_3) - f(K_1)} f(K_2), \\nonumber\n\\end{align}\n\\noindent where $f(K)$ is a convex and increasing function in $[K_1,K_3]$. Indeed, if $f(K)$ is convex, then $P(K_2) = P_F(K_2) = P_l(K_2) - \\varepsilon, \\ \\varepsilon > 0$ (see Fig.~2 in \\cite{ItkinLipton2017}). Substitution of this expression into the second line of \\eqref{noarb} gives $(K_3 - K_1) \\varepsilon > 0$, which is true. The second condition in \\eqref{noarb} now reads\n\\[ (P(K_1) - P(K_3))(f(K_3) - f(K_2))(f(K_1) - f(K_3)) > 0, \\]\n\\noindent which is also true since $f(K)$ is an increasing function of $K$.\n\nAlternatively, one can use non-linear interpolation. In \\cite{ItkinLipton2017}) both approaches were combined, and it was proved that the new interpolation scheme preserves no-arbitrage. Moreover, the final representation of the modified Put price (which is a dependent variable in their approach) obtains a nice tractable representation, so the integral $I_{12}$ can be computed in closed form. Here we want to exploit the same idea, thus significantly improving performance of our model as compared with the numerical integration.\n\nTherefore, here we propose the following interpolation scheme\n\\begin{align} \\label{linNew}\nP(x) &\\equiv P_F(x) = \\gamma_1 + \\gamma_2 x^2, \\quad x_1 \\le x \\le x_3, \\\\\n\\gamma_1 &= \\dfrac{P(x_3) x_1^2 - P(x_1) x^2_3}{x_1^2 - x_3^2}, \\qquad\n\\gamma_2 = \\dfrac{P(x_1) - P(x_3)}{x_1^2 - x_3^2}. \\nonumber\n\\end{align}\nThen Proposition similar to that in \\cite{ItkinLipton2017} can be proved.\n\\begin{proposition} \\label{prop1}\nThe interpolation scheme in \\eqref{linNew} is arbitrage free at the interval $[K_1,K_3]$.\n\\end{proposition}\n\\begin{proof}\nObserve, that the no-arbitrage conditions in \\eqref{noarb} are discrete versions of the conditions\n\\[ P > 0, \\quad P_K > 0, \\quad P_{K,K} > 0, \\]\nThey, in turn, correspond to the conditions\n\\[ P > 0, \\quad P_x > 0, \\quad P_{x,x} > 0, \\]\n\\noindent as $x'(K) = 1\/S > 0$. By differentiating the first line of \\eqref{linNew} one can check that the proposed interpolation obeys these conditions provided that $P$ is an increasing function of $K$ (or $x$) given the values of all other parameters to be constant. For instance, this is the case for the Black-Scholes Puts.\n\\hfill $\\blacksquare$\n\\end{proof}\n\nAs by definition $z$ is a linear function of $x$, a similar interpolation scheme can be used in the $z$ space, with a similar proof of no-arbitrage.\n\n\\subsection{No-arbitrage at consecutive intervals}\n\nProposition~\\ref{prop1} guarantees that the proposed interpolation doesn't introduce an arbitrage into the solution if any three strikes belong to the same interval $[K_1,K_3]$. However, what if we consider strikes $K_2, K_3, K_4$ as this is schematically depicted in Fig.~\\ref{str2interv}.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[line\/.style={<->},thick, framed, scale=1.4,\nredline\/.style={shape=rectangle, draw=red, line width=2},\nblueline\/.style={shape=rectangle, draw=blue, line width=2}\n]\n\n\\draw[->] (-3.0,0) -- (5,0) node[right] {$K$};\n\\draw[->] (-3.0,-0.2) -- (-3.0,4.5) node[right] {$P(K)$};\n\\draw[red,ultra thick] (-2,1) parabola (4.5,3.5);\n\\node at (-3.0,-0.3) {$0$};\n\n\\node at (-2,-0.3) {$K_1$};\n\\node at (-2,1.3) {$P_1$};\n\\node at (-2,1.) {$\\bullet$};\n\\draw[red, dashed] (-2,0) -- (-2,1.);\n\n\\node at (-0,-0.3) {$K_2$};\n\\node at (0.2,0.6) {$P_2$};\n\\node at (-0,0.8) {$\\bullet$};\n\\draw[red, dashed] (-0,0) -- (-0,0.8);\n\n\\node at (1.5,-0.3) {$K_3$};\n\\node at (1.5,2.) {$P_3$};\n\\node at (1.5,1.7) {$\\bullet$};\n\\draw[red, dashed] (1.5,0) -- (1.5,1.7);\n\n\\node at (2.5,-0.3) {$K_4$};\n\\node at (2.7,1.5) {$P_4$};\n\\node at (2.5,1.65) {$\\bullet$};\n\\draw[red, dashed] (2.5,0) -- (2.5,1.65);\n\n\\node at (4.5,-0.3) {$K_5$};\n\\node at (4.5,3.8) {$P_5$};\n\\node at (4.5,3.5) {$\\bullet$};\n\\draw[red, dashed] (4.5,0) -- (4.5,3.5);\n\n\\draw[blue,ultra thick] (-2,1.) parabola bend (-0.2, 0.8) (1.5,1.7);\n\\draw[blue,ultra thick] (1.5,1.7) parabola bend (2., 1.6) (4.5,3.5);\n\n\\matrix [draw,below left] at (1.5, 4.5) {\n \\node [redline,label=right:Exact solution] {}; \\\\\n \\node [blueline,label=right:Interpolation] {}; \\\\\n};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Three strikes $K_2, K_3, K_4$ which belong to the consecutive intervals.}\n\\label{str2interv}\n\\end{figure}\n\n\nHere at the interval $[K_1,K_3]$ the exact solution is depicted by the red line, while our quadratic interpolation is in blue. Accordingly, the Put prices $P_1,P_3$ are the market quotes, so they assumed to be the exact prices with no market arbitrage. By our construction, these prices also don't have a model arbitrage. At the consecutive interval $[K_3,K_5]$ a similar construction applies.\n\nWe have to emphasize that this graph is pure illustrative, and no-arbitrage interpolation guarantees that $P'(K) > 0$, while the blue line in Fig.~\\ref{str2interv} doesn't support this. However, if we draw an accurate picture by using the above formulae, it would be almost impossible to distinguish the red and blue lines. Therefore, we changed convexity and skew of the blue line to make the difference visible.\n\nBy Proposition~\\ref{prop1} given a set of strikes $K_1, K_2, K_3$ the price $P_2$ obtained by interpolation preserves no-arbitrage. The same is true for $P_5$ given the Put prices $P_3, P_5$ at strikes $K_3, K_5$.\nNow assume that given $K_1, K_3, K_5$ and $P_1, P_3, P_5$ we want to check the no-arbitrage conditions for the set of strikes $K_2, K_3, K_4$. The Proposition~\\ref{prop1} doesn't help in this situation, so we need a special consideration of this case.\n\nObviously, the first and second conditions in \\eqref{noarb} are still satisfied in this case, so we need to check that the butterfly spread is positive. Unfortunately, at the moment we don't have a general analytical solution of this problem, while some particular cases can be addressed. Thus this remains an open question. However, we checked this condition numerically. In doing so we used the Black-Scholes Put prices $P_1, P_3, P_5$\\footnote{This is done to preserve upper bounds on the Put price that $P(S, K,T,r) \\le K e^{-rT}$, \\cite{Levy1985}.} and built a 2D plot of $Bs$ which is the left-hands side of the third line in \\eqref{noarb}. The results for two cases presented in Table~\\ref{tabCases} are presented in Fig.~\\ref{intCase1}, \\ref{intCase2}.\n\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{|r|r|r|r|r|r|r|r|}\n\\hline\nTest & $S$ & $r$ & $\\sigma_{BS}$ & $T$ & $K_1$ & $K_3$ & $K_5$ \\\\\n\\hline\n1 & 100 & 0.01 & 0.5 & 2 & 80 & 100 & 130 \\\\\n\\hline\n2 & 100 & 0.1 & 0.1 & 0.1 & 90 & 100 & 105 \\\\\n\\hline\n\\end{tabular}\n\\caption{Parameters of the test for non-negativity of the Butterfly spread. $\\sigma_{BS}$ is the Black-Scholes implied volatility.}\n\\label{tabCases}\n\\end{center}\n\\end{table}\n\nOverall, we ran a lot of tests and didn't find any case where the butterfly spread would become negative. This partly supports our no-arbitrage interpolation. More sophisticated cases where, e.g., instead of strike $K_3$ in the butterfly spread at strikes $K_2, K_3, K_4$ we use another strike $K_6$ such that $K_1 < K_2 < K_3 < K_4 < K_6 < K_5$, could be treated in a similar way. Again, our numerical tests didn't reveal any case where a butterfly spread would become negative.\n\n\\begin{figure}[h!]\n\\begin{minipage}{0.46\\linewidth}\n\\begin{center}\n\\fbox{\\includegraphics[width=\\linewidth]{test1.png}}\n\\caption{Butterfly spread $Bs$ for a set of strikes $K_2, K_3, K_4$ computed in Test 1 in Table~\\ref{tabCases}.}\n\\label{intCase1}\n\\end{center}\n\\end{minipage}\n\\hspace{0.04\\linewidth}\n\\begin{minipage}{0.46\\linewidth}\n\\begin{center}\n\\fbox{\\includegraphics[width=\\linewidth]{test2.png}}\n\\caption{Butterfly spread $Bs$ for a set of strikes $K_2, K_3, K_4$ computed in Test 2 in Table~\\ref{tabCases}.}\n\\label{intCase2}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Computing the integrals in \\eqref{solInhom} far from $z = 0$}\n\nUsing the interpolation scheme proposed in above, consider the first integral in \\eqref{solInhom}. To remind, we compute it at some interval $z \\in [z_i,z_{i+1}], \\ \\irg{i}{1}{n_j}$. Picking together the solutions in \\eqref{Kummer0} with the interpolation scheme for $P(z,T_{j-1})$ and Wronskians in \\eqref{Kummer0}, and substituting them into the first integral in \\eqref{solInhom} we obtain\n\\begin{align} \\label{Int1}\n\\int & \\dfrac{ y_2(z) f(z,T_{j-1})}{W z}d z = A \\Big[-B_0\n+ B_1 M(-2-q_1, -1-q_2, -z)\n+ B_2 M(-1-q_1, -q_2, -z) \\\\\n&+ B_3 M(-q_1, 1-q_2, -z)\\Big], \\nonumber \\\\\nA &= \\dfrac{1}{b_1^2 q_1} \\pi(1-q_2) \\csc(\\pi q_2), \\nonumber \\\\\nB_0 &= \\frac{1}{a_2^2 (q_1+1) (q_1+2)} \\Big[\na_2 b_1 (q_1+2) \\left(a_2^2 \\beta q_2 - b_1 (q_1+1) (\\beta b_2-a_2 \\gamma_1)\n\\right) \\nonumber \\\\\n&+ \\gamma_2 \\left(2 a_2^4 q_2 (q_2+1)-2 a_2^2 b_1 b_2 (q_1+2)q_2 + b_1^2 b_2^2 (q_1+1) (q_1+2) \\right) \\Big],\n\\nonumber \\\\\nB_1 &= 2 a_2^2 \\gamma_2 \\frac{q_2 (q_2+1)}{(1+q_1)(2+q_1)}, \\nonumber \\\\\nB_2 &= (a_2 b_1 \\beta - 2 b_1 b_2 \\gamma_2 + 2 a_2^2 \\gamma_2 z) \\frac{q_2}{1+q_1}, \\nonumber \\\\\nB_3 &= \\frac{1}{a_2^2}\n\\left[a_2 b_1\\left(a_2^2 \\beta z + a_2 b_1 \\gamma_1 - \\beta b_1 b_2 \\right)\n + \\gamma_2 \\left(b_1 b_2 - a_2^2 z\\right)^2\\right]. \\nonumber\n\\end{align}\nSimilarly\n\\begin{align} \\label{Int2}\n\\int & \\dfrac{ y_1(z) f(z,T_{j-1})}{W z}d z = \\bar{A} \\Big[\n\\bar{B}_1 M(q_2-q_1, 1+q_2, -z)\n+ \\bar{B}_2 \\ \\pFq{2}{2}{q_2-q_1, 1+q_2; q_2,2+q_2; -z} \\nonumber \\\\\n&+ \\bar{B}_3 \\ \\pFq{2}{2}{q_2-q_1, 2+q_2; q_2,3+q_2; -z} \\Big], \\\\\nA &= \\pi z^{q_2} \\csc(\\pi q_2) \\Gamma(q_2), \\nonumber \\\\\nB_1 &=\n\\frac{a_2^2 \\gamma_1 - a_2 \\beta b_2 + b_2^2 \\gamma_2}{a_2^2 \\Gamma(1+q_2)}, \\nonumber \\\\\nB_2 &=\\frac{\\Gamma(q_2+1) }{\\Gamma(q_2) \\Gamma(2+q_2) b_1} (a_2 \\beta -2 b_2 \\gamma_2) z, \\nonumber \\\\\nB_3 &= \\frac{\\Gamma(q_2+1) }{\\Gamma(q_2) \\Gamma(3+q_2) b_1^2}\na_2^2(1+q_2)\\gamma_2 z^2, \\nonumber\n\\end{align}\n\\noindent where $\\pFq{p}{q}{{a_1,...,a_p}; {b_1,...,b_q}; z}$ is the generalized hypergeometric function, \\cite{Olver1997}.\n\n\n\\subsection{Computing the integrals in \\eqref{solInhom} far from $z = \\pm \\infty$}\n\nHere we proceed in the same way as in the previous section. Again, we pick together the solutions in \\eqref{Kummer1} with the interpolation scheme for $P(z,T_{j-1})$ and Wronskians in \\eqref{Kummer1}, and substitute them into the first integral in \\eqref{solInhom} we obtain\n\\begin{align} \\label{Int11}\n\\int & \\dfrac{ y_2(z) f(z,T_{j-1})}{W z}d z = (-1)^{q_2 - q_1} [C_0 J_0 + C_1 J_1 + C_2 J_2], \\\\\nJ_i &= \\int z^i U(1-q_1, 2-q_2, -z) dz, \\nonumber \\\\\nC_0 &= \\frac{b_2^2 \\gamma_2}{a_2^2} - \\frac{\\beta b_2}{a_2} + \\gamma_1, \\quad\nC_1 = \\frac{a_2 \\beta - 2 b_2 \\gamma_2}{b_1}, \\quad\nC_2 = \\frac{a_2^2 \\gamma_2}{b_1^2}. \\nonumber\n\\end{align}\nIt is known, \\cite{as64}, that\n\\begin{equation*}\nJ_0 = - \\frac{1}{q_1}U(-q_1, 1-q_2, -z).\n\\end{equation*}\nThen, $J_1, J_2$ can be found using integration by parts to yield\n\\begin{align*}\nJ_1 &= z J_0 + \\frac{1}{q_1(1+q_1)}U(-1-q_1, -q_2, -z), \\\\\nJ_2 &= z J_1 + \\frac{1}{q_1(2+3 q_1 + q_1^2)}U(-2-q_1, -1-q_2, -z)\n - z\\frac{1}{q_1(1 + q_1)}U(-1-q_1,-q_2,-z).\n\\end{align*}\nSimilarly\n\\begin{align} \\label{Int21}\n\\int & \\dfrac{ y_1(z) f(z,T_{j-1})}{W z}d z = (-1)^{q_2 - q_1} [C_0\n{\\mathcal J}_0 + C_1 {\\mathcal J}_1 + C_2 {\\mathcal J}_2], \\\\\n{\\cal J}_i &= \\int z^i e^{-z} U(1+q_1-q_2, 2-q_2, z) dz. \\nonumber\n\\end{align}\nThe integrals ${\\mathcal J}_i $ have been considered in \\cite{ItkinLipton2017} using the approach of \\cite{kummerInt1970}. Borrowing from there the result\n\\begin{equation*}\n{\\mathcal J}_0 = \\int e^{-z} U(1+q_1-q_2,2-q_2,z) d z = -e^{-z} U(q_1-q_2,1-q_2,z),\n\\end{equation*}\n\\noindent and using integration by parts, we obtain\n\\begin{align*}\n{\\mathcal J}_1 &= z {\\mathcal J}_0 + e^{-z}U(q_1-q_2-1, -1-q_2, z), \\\\\n{\\mathcal J}_2 &= z {\\mathcal J}_1 - \\int {\\mathcal J}_1 dz = (z-1) {\\mathcal J}_1 - \\int e^{-z}U(q_1-q_2-1, -1-q_2, z) dz \\\\\n&= (z-1) {\\mathcal J}_1 + e^{-z}U(q_1-q_2-2, -2-q_2, z).\n\\end{align*}\n\n\n\\subsection{Some additional notes}\n\nBased on the no-arbitrage interpolation and some analytics proposed in this Section, we managed to find the solution \\eqref{solInhom} of the forward equation \\eqref{finODE} in closed form. This solution by construction is arbitrage free at any interval where the local variance function defined in \\eqref{vDef} is linear. In other words we proved, that if we consider, say 3 strikes $0 < K_1 < K_2 < K_3 < \\infty$ such that, e.g., $x_1 = K_1\/S \\in [x_i, x_{i+1}]$, $x_2 = K_2\/S \\in [x_i, x_{i+1}]$, $x_3 = K_3\/S \\in [x_{i}, x_{i+1}]$, then the solution at these 3 points obeys no-arbitrage conditions.\n\n\n\\section{Calibration of smile for a given term $T_i$} \\label{calib}\n\nCalibration problem for the local volatility model can be formulated as follows: given market quotes of Call and\/or Put options corresponding to a set of $N$ strikes $\\{K\\}:= K_j, \\ j \\in [1,N]$ and same maturity $T_i$, find the local variance function $\\sigma(K)$ such that these quotes solve equations in \\eqref{finDupPutI}, \\eqref{finDupCallI}.\n\nAs mentioned in \\cite{ItkinLipton2017}, there are two main approaches to solving this problem. The first approach attempts to construct a continuous implied volatility (IV) surface matching the market quotes by using either some parametric or non-parametric regression, and then generates the corresponding LV surface via the well-known relationship between the local and implied variances also known as the Dupire formula, see, e.g., \\cite{ItkinSigmoid2015} and references therein. To be practically useful, this construction should guarantee no arbitrage for all strikes and maturities, which is a serious challenge for any model based on interpolation. If the no-arbitrage condition is satisfied, then the LV surface can be calculated using the Dupire formula. The second approach relies on the direct solution of the corresponding forward equation (which is the Dupire equation in the Black-Scholes world, or \\eqref{finDupCallI}, \\eqref{finDupPutI} in our model) using either analytical or numerical methods. The advantage of this approach is that it guarantees no-arbitrage. However, the problem of the direct solution can be ill-posed, \\cite{Coleman2001}, and is rather computationally intensive.\n\nIn this Section we show that the second approach could be significantly simplified when using the ELVG model, so calibration of the smile could be done very fast and accurate.\n\nFurther, for the sake of certainty, suppose that all known market quotes are Puts, despite this can be easily relaxed. Also, suppose that the shape of a local variance is given by some function $\\sigma_j(K) = f_j(K, p_1,\\ldots,p_L)$, where $p_1,\\ldots,p_L$ is a set of the model parameters to be determined. For instance, in \\cite{LiptonSepp2011, CarrNadtochiy2017} the local variance is assumed to be a piecewise constant function of strike, while in \\cite{ItkinLipton2017} this is a piecewise linear function of strike.\n\nIn this paper we also assume the local variance to be a piecewise linear function of strike. Moreover, for our model we obtained a closed form representation of the Put option prices via parameters of the model given in Sections~\\ref{SolutionODE},\\ref{SolutionInt}. Therefore, calibration of the model to the given set of smiles could be provided as follows. First, using the above-mentioned closed form solution for a fixed interval in $x$ where parameters of the model are constant, we construct the combined solution valid for all $x \\in \\mathbb{R}^+$. At the second step, the parameters of the local variance function $v^0_{j,i}, v^1_{j,i}$ can be found together with the integration constants $C_1, C_2$ in \\eqref{solInhom} by solving a system of non-linear algebraic equations.\n\n\n\\subsection{The combined solution in $x \\in \\mathbb{R}^+$ \\label{wholeSol}}\n\nSuppose that the Put prices for $T=T_j$ are known for $n_j$ ordered strikes. The location of these strikes on the $x$ line is schematically depicted in Fig.~\\ref{Fig3}.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tikzpicture}[line\/.style={<->},thick, framed, scale=1.4]\n\n\\draw[->] (-3.0,0) -- (5,0) node[right] {$x$};\n\\draw[->] (-3.0,-0.2) -- (-3.0,3.5) node[right] {$v(x)$};\n\\draw[red,ultra thick] (0.5,0.5) parabola (4.5,1.5);\n\\draw[red,ultra thick] (0.5,0.5) parabola (-2,3);\n\\node at (-3.0,-0.3) {$0$};\n\\node at (-2,-0.3) {$x_1$};\n\\node at (-2,3.) {$\\bullet$};\n\\draw[red, dashed] (-2,0) -- (-2,3.);\n\\node at (-2.5,0.5) {$B_1$};\n\\draw[blue, dashed] (-2,3.) -- (-2.8,3.2);\n\\node at (-1,-0.3) {$x_2$};\n\\node at (-1,1.4) {$\\bullet$};\n\\draw[red, dashed] (-1,0) -- (-1,1.4);\n\\node at (-1.5,0.5) {$B_{12}$};\n\\draw[blue, dashed] (-2,3.) -- (-1,1.4);\n\\node at (1.5,-0.3) {$x_3$};\n\\node at (1.5,0.55) {$\\bullet$};\n\\draw[red, dashed] (1.5,0) -- (1.5,0.55);\n\\node at (-0.5,0.5) {$B_{23}$};\n\\draw[blue, dashed] (-1,1.4) -- (1.5,0.55);\n\\node at (2.5,-0.3) {$\\ldots$};\n\\node at (3.5,-0.3) {$x_{n_j}$};\n\\node at (3.5,1.05) {$\\bullet$};\n\\draw[red, dashed] (3.5,0) -- (3.5,1.05);\n\\draw[blue, dashed] (1.5,0.55) -- (3.5,1.05);\n\\draw[blue, dashed] (3.5,1.05) -- (4.5,2.);\n\\node at (4,0.5) {$B_{n_j}$};\n\\node at (0.5,1.3) {$2$};\n\\node at (0.7,0.3) {$1$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Construction of the combined solution in $x \\in \\mathbb{R}^+$: 1 (red solid line - the real (unknown) local variance curve, 2 (dashed blue line) - a piecewise linear solution.}\n\\label{Fig3}\n\\end{figure}\n\nRecall, that the Put prices are given by \\eqref{solInhom}, which in a more convenient form at the interval $x_{i-1} \\le x \\le x_i$ and at $T=T_j$ can be represented as\n\\begin{align} \\label{combPut}\nP_i(x) &= \\CO{i}{j} {\\cal J}_1(q_1, q_2, z)\n+ \\CT{i}{j} {\\cal J}_2(q_1, q_2, z) + \\frac{1}{b_1}I_{12}(z) + {\\cal D}_j K, \\\\\nz &\\equiv (b_2 + a_2 x)b_1\/a_2^2 = (v_{j,i}^0 + v_{j,i}^1 x)b_1\/a_2^2. \\nonumber\n\\end{align}\nHere, for consistency we change notation of two integration constants which belong to the $i$-th interval in $x$ and $j$-th maturity to $\\CO{i}{j},\\CT{i}{j}$.\n\nFor the open interval $B_1$ in Fig.~\\ref{Fig3}, since function $K_\\nu(z)$ diverges when $z \\to 0$, we have to put $\\CO{1}{j} = 0$ as the boundary condition\\footnote{\nActually, since $x \\to 0$ implies $z = v \\to b_2$, so $b_2$ should be non-negative, $b_2 \\ge 0$. Therefore, the only case when $z \\to 0$ at $x \\to 0$ is when $b_2 = 0$.}. Therefore, \\eqref{combPut} contains just one yet unknown constant $\\CT{1}{j}$. For the closed intervals $x \\in [x_{i-1}, x_{i}], \\ i \\in [2,n_j]$ the solutions in \\eqref{combPut} have two yet unknown constants $\\CO{i}{j}, \\CT{i}{j}$, since $x$ is finite on the corresponding intervals, and both solutions $y_1(x), y_2(x)$ are well-behaved. Finally, for the interval $x \\in [x_{n_j}, \\infty)$, according to the boundary conditions in \\eqref{bc} we must set $C^{(2)}_{2,n_j+1} = 0$.\n\nRigorously speaking, we also have to show that in the limits $x \\to 0$ and $x \\to \\infty$ the source term $I_{12}(z)$ in \\eqref{solInhom} also vanishes. This could be done similar to Proposition~2 in \\cite{ItkinLipton2017}.\n\nThus, we have $2 n_j$ unknown constants to be determined. Since the local volatility function $v_i$ is continuous at the points $x_i, \\ i=1,\\ldots,n_j$, so should be the Put options prices $P(x,T_j)$. Therefore, we require that at the points $x_i, \\ i = 1,...,n_j$ the solution for Puts and its first derivative in $x$ should be a continuous function of $x$. Thus, if the local variance function is known, the above constants solve a system of $2 n_j$ algebraic equations. This system has a block-diagonal structure where each block is a 2x2 matrix. Therefore, it can be easily solved with the linear complexity $O(n_j)$.\n\nWhen computing the first derivatives, we take into account that, \\cite{as64}\n\\begin{align} \\label{der}\n\\fp{M(a,b,z)}{z} &= \\dfrac{a}{b}M(a+1,b+1,z), \\quad\n\\fp{U(a,b,z)}{z} = -a U(a+1,b+1,z), \\\\\n\\partial_z I_{12}(z) &= \\left[\\frac{y'_{1}(z)}{y_{1}(z)} I_1 + \\frac{y'_{2}(z)}{y_{2}(z)} I_2 \\right] a_2. \\nonumber\n\\end{align}\nTherefore, computing the derivatives of the solution doesn't cause any new technical problem.\n\n\\subsection{Additional equations for calibration}\n\nAs we have already mentioned above, the standard way of doing calibration of the local volatility model would be that described, e.g., in \\cite{ItkinLipton2017}. Namely, given the maturity $T_j$ and some initial guess of the local variance parameters $v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$, the following steps represented in Panel~\\ref{Algo} have to be achieved, e.g., in the standard least-square method,\n\\begin{center}\n\\begin{algorithm}[H]\n \\KwIn{Strikes $z_i, \\ i \\in [1,n_j]$, Put prices $V^{market}_i, \\ i \\in [1,n_j]$}\n \\KwOut{$v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$}\n \\emph{{\\bf Initialization}: The initial guess of $v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$, the tolerance $\\epsilon$} \\;\n\\While{1}{\n 1. Solve the system for $\\CO{i}{j}, \\CT{i}{j}$ \\;\n 2. Compute Put option prices $V(x)$\\;\n 3. Compute the total error $\\Delta = \\sum_{i=1}^{n_j} [V(x_i) - V^{market}(x_i)]^2$\\;\n \\eIf{ $\\Delta > \\epsilon$}{\n New guess for $v_{j,i}^0, v_{j,i}^1, \\ \\forall i \\in [1,n_j]$\\;\n }{\n break\\;\n }\n }\n\\caption{Calibration of the local volatility model using a least-square method.}\n\\label{Algo}\n\\end{algorithm}\n\\end{center}\n\\vspace{0.2in}\nHere $V^{market}(z_i)$ are the market Put quotes at the given strikes and maturity. Obviously, when the number of calibration parameters (strikes) is high, this algorithm is slow even if the closed form solution is known and can be used at Step~2. Things become even worse when a numerical solution at Step~2 has to be used if the closed form solution is not available.\n\nHowever, in our case this tedious algorithm can be fully eliminated. Indeed, at every point $i$ in strike space, $i \\in [1,n_j]$ we have four unknown variables $v_{j,i}^0, v_{j,i}^1, \\CO{i}{j}, \\CT{i}{j}$. We also have four equations which contain these variables, namely\n\\begin{align} \\label{finSystem}\nP_i(x)|_{x=x_i} &= P_{i+1}(x)|_{x = x_i}, \\\\\nP_i(x)|_{x=x_i} &= P_{market}(x_i), \\nonumber \\\\\n\\fp{P_{i+1}(x)}{x}\\Big|_{x = x_i} &= \\fp{P_{i}(x)}{x}\\Big|_{x = x_i}, \\nonumber \\\\\nv^0_{j,i} + v^1_{j,i} x_{i} &= v^0_{j,i+1} + v^1_{j,i+1} x_{i}, \\quad i=1,\\ldots,n_j. \\nonumber\n\\end{align}\nAlso, based on \\eqref{cont}, the last line in \\eqref{finSystem} could be re-written as a recurrent expression\n\\begin{equation} \\label{recurr}\nv^0_{j,i} = v^0_{j,n_j} + \\sum_{k=i+1}^{n_j} x_{k}(v^1_{j,k} - v^1_{j,k-1}) , \\quad i=0,\\ldots,n_j-1.\n\\end{equation}\n\nThe \\eqref{finSystem} is a system of $4 n_j$ nonlinear equations with respect to $4 (n_j+1)$ variables $v_{j,i}^0$, $v_{j,i}^1$, $\\CO{i}{j}$, $\\CT{i}{j}$. We remind that according to the boundary conditions $\\CO{1}{j} = \\CT{n_j}{j} = 0$. Therefore, we need two additional conditions to provide a unique solution. For instance, often traders have an intuition about the asymptotic behavior of the volatility surface at infinity, which, according to our construction, is determined by $v^1_{j,n_j}$ and $v^1_{j,0}$.\n\nOverall, solving the nonlinear system of equations \\eqref{finSystem} provides the final solution of our problem. This can be done by using standard methods, and, thus, no any optimization procedure is necessary. However, a good initial guess still would be helpful for a better (and faster) convergence.\n\n\\subsection{Smart initial guess}\n\nThe initial guess of the solution of \\eqref{combPut} can be constructed, for instance, as follows. We take advantage of the fact that according to \\eqref{finODE} the local variance function $v(x)$ could be explicitly expressed as\n\\begin{equation} \\label{localVarappr}\nv(x) = \\dfrac{b_1 x V_x(x) + b_0 V(x) - c}{V_{x,x}(x)}.\n\\end{equation}\n\nGiven maturity $T_j$ and approximating derivatives by central finite differences with the second order of approximation in step $h$ in the strike space (see, e.g. \\cite{ItkinBook}), \\eqref{localVarappr} can be represented in the form\n\\begin{align} \\label{localVarappr1}\nv^0_{j,i} + v^1_{j,i} x_{i} &= \\dfrac{b_1 x V_x(x_i) + b_{0,j} V(x_i) - c_j}{V_{x,x}(x_i)}, \\\\\nV_{x}(x_i) &= \\alpha_{-1} V(x_{i-1}) + \\alpha_{0} V(x_{i}) + \\alpha_{1} V(x_{i+1}), , \\nonumber \\\\\nV_{x,x}(x_i) &= \\delta_{-1} V(x_{i-1}) + \\delta_{0} V(x_{i}) + \\delta_{1} V(x_{i+1}), \\nonumber \\\\\n\\alpha_{-1} &= - \\dfrac{h_{i+1}}{h_{i}(h_{i+1} + h_{i})}, \\quad\n\\alpha_{0} = \\dfrac{h_{i+1}-h_{i}}{h_{i+1}h_{i}}, \\quad\n\\alpha_{1} = \\dfrac{h_{i}}{h_{i+1}(h_{i+1} + h_{i})}. \\nonumber \\\\\n\\delta_{-1} &= \\dfrac{2}{h_{i}(h_{i+1} + h_{i})}, \\quad\n\\delta_{0} = -\\dfrac{2}{h_{i+1}h_{i}}, \\quad\n\\delta_{1} = \\dfrac{2}{h_{i+1}(h_{i+1} + h_{i})}. \\nonumber \\\\\nh_i &= x_i-x_{i-1}, \\quad i \\in [1,n_j]. \\nonumber\n\\end{align}\nFurther, associating Put prices $P(S,T_j,x_{i})$ with the given market quotes, the right hands side of the first line in \\eqref{localVarappr1} can be found explicitly. This then can be combined with the last line of \\eqref{finSystem} to produce a system of $2(n_j-1)$ equations for $v^1_{j,i}$ and $v^1_{j,i}, \\ i \\in [1,n_j]$. Finally, we take into account the asymptotic behavior of the volatility surface in $x$ at zero and infinity, which, according to our construction, is determined by $v^1_{j,n_j}$ and $v^1_{j,0}$ and is assumed to be known. Thus, we obtain a closed system of $2(n_j-1)$ linear equations with a banded matrix which can be easily solved with a linear complexity. This provides an explicit representation of the local variance function over the whole set of intervals in the strike space determined according to our approximation where the continuous derivatives are replace by finite differences.\n\nNote, that at the first and last strike intervals the approximation of the first and second derivatives by central finite differences should be replaced by one-sided approximations, in more detail see \\cite{ItkinBook}, chapter 2.\n\nIt could also happen that at some strikes this solution (the smart guess) gives rise to a negative local variance. In such a case we do another step which is a kind of smoothing. Namely, we exclude from the initial guess all values where the local variance is negative and using the remaining points create a spline. Then the negative values in the initial guess are replaced by those given by the constructed spline.\n\nThe final step utilizes the exact representation \\eqref{combPut} of the Put price in the ELVG model. As the variance function is already known from the previous step, this equation contains two yet unknown constants $\\CO{i}{j},\\CT{i}{j}$. Accordingly, they can be found by solving the system of 2 linear equations represented by the first and third lines of \\eqref{finSystem}. Then, after this last step is complete, all unknown variables are determined, and thus found solution could be used as an educated initial guess for solving \\eqref{finSystem} numerically.\n\n\\section{Asymptotic solutions} \\label{asympt}\n\nIn many practical situations either some coefficients $a_2 = v^1_{j,i}$, or both\n$b_2 = v^0_{j,i}, \\ a_2 = v^1_{j,i}$ in \\eqref{Laplace2} are small. Of course, in that case the general solution \\eqref{combPut} remains valid. However, in this case when computing the values of Kummer functions numerically, numerical errors significantly grow. This is especially pronounced when computing the integral $I_{12}$. The main point is that either the Kummer function takes a very small value, and then the constants $\\CO{i}{j},\\CT{i}{j}$ should be big to compensate, or vice versa. Resolution of this requires a high-precision arithmetics, and, which is more important, taking many terms in a series representation of the Kummer functions, which significantly slows down the total performance of the method.\n\nOn the other hand, to eliminate these problems we can look for asymptotic solutions of \\eqref{Laplace2} taking into account the existence of small parameters from the very beginning. This approach was successfully elaborated on in \\cite{ItkinLipton2017}, and below we proceed in a similar spirit.\n\n\\subsection{Small $a_2$}\n\nWe can build the solution of \\eqref{Laplace3} directly using an independent variable $x$ (so not switching to the variable $z$). We represent it as a series on the small parameter $a_2$, i.e.\n\\begin{equation} \\label{ser}\nV(x) = \\sum_{i=0}^\\infty a_2^i V_i(x).\n\\end{equation}\nIn the zero-order approximation by plugging \\eqref{ser} into \\eqref{Laplace3} and neglecting by terms proportional to $a_2 \\ll 1$ we obtain the following equation for $V_0(x)$\n\\begin{equation} \\label{asymA1}\n-b_2 V_{xx}(x) + b_1 x V_x(x) + b_0 V(x) = c.\n\\end{equation}\nThis equation is simpler than \\eqref{combPut}. Still, its solution is given by a general formula\n\\begin{equation*}\nV(x) = C_1 y_1(x) + C_2 y_2(x) + I_{12}(x),\n\\end{equation*}\n\\noindent but the fundamental solutions $y_1(x), y_2(x)$ now read\n\\begin{equation*}\ny_1(x) = {\\mathcal H} \\left(-\\frac{b_0}{b_1}, \\sqrt{\\frac{b_1}{2 b_2}} x\\right), \\qquad\ny_2(x) = M\\left(\\frac{b_0}{2 b_1}, \\frac{1}{2}, \\frac{b_1}{2 b_2}x^2\\right), \\end{equation*}\n\\noindent where ${\\mathcal H}(a, x), \\ a,x \\in \\mathbb{R}$ is the generalized Hermite polynomial $H_a(x)$, \\cite{as64}.\n\n\\subsection{Small $|z|$}\n\nBased on the definition of $z = (b_2 + a_2 x)b_1\/a_2^2$, this could occur in two cases: either at some finite interval in the strike space $|a_2| \\gg |b_1 x|, \\ |a_2| \\gg |b_2|$, or just $z$ is small, so $b_2$ and $a_2$ have the opposite signs. In any case we have a small parameter under the high-order derivative. This equation belongs to the class of singularly perturbed differential equations, \\cite{Wasow1987}. It can be solved by using either the method of matching asymptotic expansions, \\cite{Nayfeh2000}, or the method of boundary functions, \\cite{VasBut1995}. The latter was used in \\cite{ItkinLipton2017} in a similar situation, so for further details we refer a reader to that paper.\n\nHowever, we can partly eliminate this by constructing solutions of \\eqref{Laplace2} using the original variable $x$. Then we have to consider various cases where instead of a small parameter $z$ some other combinations of parameters could be small or large. But if so, a general solution as a function of the original independent variable $x$ could be represented as regular series on the new small parameter. Then, truncating the series, one gets a simplified solution.\n\nTo make it more transparent let us represent the general solution of \\eqref{Laplace2} expressed in variable $x$, rather than in $z$, as this was done in \\eqref{solInhom}\n\\begin{align} \\label{combPutX}\nV(x) &= \\CO{i}{j} y_1(x) + \\CT{i}{j} y_2 (x) + I_{12}(x), \\\\\ny_i(x) &= a_2^k (b_2+a_2 x)^k {\\mathcal V}_i\\left(\n-1 - \\frac{b_0}{b_1} + \\frac{b_1 b_2}{a_2^2}, 2 - \\frac{b_1 b_2}{a_2^2},\n\\frac{b_1}{a_2^2}(b_2 + a_2 x)\\right), \\quad i=1,2.\\nonumber \\\\\nk &= 1 - \\frac{b_1 b_2}{a_2^2}. \\nonumber\n\\end{align}\nObserve, that based on the definition of $b_1$ in \\cref{finODE}, $b_1 \\approx (r-q)\\Delta T$, so usually small. Therefore, small $z$ doesn't mean that\n$w$ is necessarily small. Below we consider two cases.\n\n\\subsubsection{$w \\ll 1$}\n\nAs $|z| \\ll 1$ and $w \\ll 1$ we have $w \\ll |a_2^2\/b_1|$. So $a_2 \\ge \\sqrt{b_1}$.\nIn this case $w \\ll 1$ is an actual small argument. Therefore, the general solution \\eqref{combPutX} can be expanded into series on small $w$. The condition $0 < w \\ll 1$ implies that $a_2$ and $b_2$ have the opposite signs. If $a_2 > 0$ (and so $b_2 < 0$), then in the zero-order approximation we obtain\n\\begin{align} \\label{asymZ}\ny_1(w) &= (a_2 w)^{k-1} \\left[\\dfrac{\\Gamma(-k)}{\\Gamma(b_0\/b_1)} a_2 w + O(w^2) \\right] - \\left(\\frac{b_1}{a_2^3}\\right)^{1-k} \\frac{\\Gamma(k-1)}{b_1\\Gamma(k + b_0\/b1)}(a_2 b_1 b_2 - b_0 a_2 w) + O(w^2) , \\nonumber \\\\\ny_2(w) &= (a_2 w)^{k-1} \\left[a_2 w + O(w^2)\\right].\n\\end{align}\nAs $b_1 > 0$ we have $k-1 > 0$.\n\nIf $a_2 < 0$ and $b_2 > 0$, then both RHS in \\eqref{asymZ} should be multiplied by a factor\n$\\exp(-2 i \\pi b_1 b_2\/a_2^2)$.\n\n\\subsubsection{$a^2_2 \\gg |b_1 w|$}\n\nIn this case we can also expand the solution in \\eqref{combPutX} into series on small $z$ to obtain\n\\begin{align} \\label{asym1}\ny_1(z) &= \\frac{1}{\\Gamma(1 + q_1 - q_2)}\\left[\\Gamma(1 - q_2) - q_1 \\Gamma(- q_2) z\\right] + z^{-q_2} \\left[\\frac{\\Gamma(q_2-1)}{\\Gamma(q_1)}z + O(z^2)\\right] + O(z^2), \\\\\ny_2(w) &= 1 + \\frac{q_1}{q_2}z + O(z^2). \\nonumber\n\\end{align}\nNote, that based on the definition $q_2 = b_2 b_1\/a_2^2$, at large $a_2$ the coefficient $q_2$ could also be small. But $z\/q_2 = 1 + a_2 x\/b_2 = w\/b_2 = O(1)$.\n\n\n\\section{Numerical experiments} \\label{numExp}\n\nIn our numerical test we use the same data set as in \\cite{ItkinSigmoid2015, ItkinLipton2017}. This is done first, to compare performance and a quality of the fit for all those models. Also, we already know that these smiles are difficult to fit precisely, see discussions in \\cite{ItkinSigmoid2015, ItkinLipton2017}.\n\nTo remind, we take data from \\url{http:\/\/www.optionseducation.org} on XLF traded at NYSEArca on March 25, 2014. The spot price of the index is $S = 22.64$, and $r = 0.0148,\\ q=0.01$. The option implied volatilities ($I_{call}, I_{put}$) are given in Tables~\\ref{TabOptC},\\ref{TabOptP}. We take all OTM quotes and some ITM quotes which are\nvery close to the at-the-money (ATM). When strikes for Calls and Puts coincide, we take an average of $I_{call}$ and $I_{put}$ with weights proportional to $1 - |\\Delta|_c$ and $1-|\\Delta|_p$ respectively, where $\\Delta_c, \\Delta_p$ are option Call and Put deltas~\\footnote{By doing so we do take into account effects reported in \\cite{callPutIV}, who pointed out that the IVs calculated from Call and Put option prices corresponding to the same strike do not coincide, although they should be equal in theory. Our weights are chosen according to a pure empirical rule of thumb, and a more detailed investigation of this effect is required.}.\n\\begin{sidewaystable*}[!htb]\n\\begin{center}\n\\footnotesize\n\n\\caption{XLF implied volatilities for the Put options.}\n\\label{TabOptC}\n\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n \\multicolumn{1}{|c|}{\\multirow{2}[4]{*}{T}} & \\multicolumn{17}{c|}{K,Put} \\\\\n\\cline{2-18} & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 19.5 & 20 & 20.5 & 21 & 21.5 & 22 & 23 \\\\\n\\hline\n 4\/4\/2014 & & & & & & & & & & & & 39.53 & & 23.77 & 19.73 & 16.67 & \\\\\n\\hline\n 4\/11\/2014 & & & & & & & & & & & 35.89 & 30.33 & 26.62 & 22.06 & 18.49 & 16.11 & \\\\\n\\hline\n 4\/19\/2014 & & & & & & & & & & 32.90 & & 26.79 & & 20.14 & & 15.19 & 12.93 \\\\\n\\hline\n 5\/17\/2014 & & & & & & & & 37.66 & 33.27 & 26.88 & & 23.08 & & 18.94 & & 16.12 & 13.86 \\\\\n\\hline\n 6\/21\/2014 & & & & & & 40.51 & 37.21 & 31.41 & 27.84 & 23.90 & & 21.07 & & 18.88 & & 16.95 & 15.82 \\\\\n\\hline\n 7\/19\/2014 & & & & & & 36.71 & 33.35 & 29.96 & 26.09 & 22.81 & & 20.29 & & 18.13 & & 16.30 & 14.93 \\\\\n\\hline\n 12\/20\/2014 & & & & & 31.98 & 29.38 & 27.21 & 25.30 & 23.75 & 22.09 & & 20.67 & & 19.44 & & 18.36 & 17.60 \\\\\n\\hline\n 1\/17\/2015 & 42.75 & 38.79 & 35.60 & 33.26 & 30.94 & 28.82 & 26.52 & 24.96 & 23.12 & 21.67 & & 20.29 & & 19.10 & & 17.90 & 18.07 \\\\\n\\hline\n\\end{tabular}%\n\n\\bigskip\n\\bigskip\n\n\\caption{XLF implied volatilities for the Put options.}\n\\label{TabOptP}\n\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n \\multicolumn{1}{|c|}{\\multirow{2}[4]{*}{T}} & \\multicolumn{13}{c|}{K,Call} \\\\\n\\cline{2-14} & 21 & 21.5 & 22 & 22.5 & 23 & 23.5 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\\\\n\\hline\n 4\/4\/2014 & & 16.60 & 14.69 & 14.40 & 14.86 & & & & & & & & \\\\\n\\hline\n 4\/11\/2014 & & 16.89 & 14.96 & 14.52 & 14.77 & 14.98 & & & & & & & \\\\\n\\hline\n 4\/19\/2014 & & & 15.79 & & 13.38 & & 15.39 & & & & & & \\\\\n\\hline\n 5\/17\/2014 & 16.71 & & 14.48 & & & & 13.75 & & & & & & \\\\\n\\hline\n 6\/21\/2014 & 16.31 & & 14.78 & & & & 13.92 & 14.28 & 16.58 & & & & \\\\\n\\hline\n 7\/19\/2014 & 16.82 & & 15.24 & & & & 14.36 & 14.19 & 15.20 & & & & \\\\\n\\hline\n 12\/20\/2014 & 17.63 & & 16.61 & & & & 15.86 & 15.47 & 15.12 & 15.18 & 15.03 & & \\\\\n\\hline\n 1\/17\/2015 & & & & & 16.95 & & 17.25 & 16.23 & 15.73 & 15.50 & 15.58 & 15.86 & 16.47 \\\\\n\\hline\n\\end{tabular}%\n\n\\end{center}\n\\end{sidewaystable*}\n\nWe have already mentioned that in our model for each term the slopes of the smile at plus and minus infinity, $v^1_{j,n_j}$ and $v^1_{j,0}$, are free parameters. So often traders have an intuition about these values. However, in our numerical experiments we take for them just some plausible values. In more detail, for a normalized variance $v(x)$ defined in \\eqref{finODE}, for all smiles we use $v^1_{j,0} = -0.1$, and $v^1_{j,n_j} = 0.1$. Accordingly, for the instantaneous variance $\\sigma^2(x) = 2 S^2 v(x)\/p_j$ the slopes at both zero and plus infinity are time-dependent and can be computed by using the above formula.\n\nWhen calibrating the model to market data, we use the standard Matlab {\\it fsolve} function, and utilize a \"trust-region-dogleg\" algorithm (see Matlab documentation on {\\it fsolve}). Parameter \"TypicalX\" has to be chosen carefully to speedup calculations.\n\nThe results of this calibration which is done term-by term, are given in Fig.~\\ref{FigPut}. Here each subplot corresponds to a single maturity $T$ (marked in the legend) and shows market data (discrete points) and computed values (solid line). It can be seen that this simple local calibration algorithm provides a very accurate fit for all terms\\footnote{Note, that in \\cite{ItkinLipton2017} in the last subplot the fit is not perfect in the vicinity of $X = -0.5$, where $X = \\log K\/F$ and $F = S e^{(r-q)T}$.}.\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width = 0.8\\textwidth]{fit.png}}\n\\caption{Term-by-term fitting of market Put prices constructed using the whole set of data in Tab.~\\ref{TabOptC},\\ref{TabOptP}.}\n\\label{FigPut}\n\\end{center}\n\\end{figure}\n\nWe constructed the calibration algorithm to be smart enough in a sense that based on the values of parameters at each iteration it decides itself which particular solution (full or asymptotic) should be used at this iteration. We also observed that all full and asymptotic solutions are utilized by the algorithm when calibrating these market smiles.\n\nTable~\\ref{Perf} presents some performance measures of our algorithm. It can be seen that the elapsed time depends on the number of iterations and function evaluations necessary to converge to the given tolerance (we use a relative tolerance $\\varepsilon = 10^{-4}$). This, in turn, depends on the number of evaluated Kummer functions (for the full solution), or number of exponential and Gamma functions (for the asymptotic solutions). Of course, the asymptotic solutions are much faster to evaluate, therefore an average time to calibrate a typical term is less than a second. For the last term 8 in Tab.~\\ref{Perf} calibration is slow for two reasons: i) full solution is used based on the values of parameters, and 2) the number of strikes is higher than for the other terms. But the main reason is that the market data for this term is quite irregular. In any case, performance of this model is much better than that reported in both \\cite{ItkinSigmoid2015} and \\cite{ItkinLipton2017}.\n\\begin{table}[H]\n\\begin{center}\n\\small\n\\begin{tabular}{|r|r|r|r|r|r|}\n\\hline\nTerm & $T$, years & Elapsed time, sec & iterations & function evaluations & strikes \\\\\n\\hline\n1 & 0.0274 & 0.86 & 97 & 1202 & 6 \\\\\n\\hline\n2 & 0.0466 & 2.83 & 97 & 1808 & 9 \\\\\n\\hline\n3 & 0.0685 & 1.43 & 95 & 1200 & 6 \\\\\n\\hline\n4 & 0.1452 & 0.64 & 48 & 433 & 8 \\\\\n\\hline\n5 & 0.2411 & 0.90 & 37 & 470 & 12 \\\\\n\\hline\n6 & 0.3178 & 2.98 & 82 & 1523 & 12 \\\\\n\\hline\n7 & 0.7397 & 6.60 & 106 & 3017 & 15 \\\\\n\\hline\n8 & 0.8164 & 149.67 & 56 & 1317 & 21 \\\\\n\\hline\n\\end{tabular}\n\\caption{Performance characteristics of the algorithm in the described experiment.}\n\\label{Perf}\n\\end{center}\n\\end{table}\n\nThe local variance curves obtained as a result of this fitting are given term-by-term in Fig.~\\ref{termXLF}. The corresponding local variance surface is represented in Fig.~\\ref{lvXLF}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth]{local_var.png}}\n\\caption{Term-by-term fitting of the instantaneous local variance $\\sigma^2(x,T)$.}\n\\label{termXLF}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth, height=3in]{surf.png}}\n\\caption{The instantaneous local variance surface $\\sigma^2(x,T)$ constructed by using the proposed approach.}\n\\label{lvXLF}\n\\end{center}\n\\end{figure}\n\nBy comparing the surface with that given in \\cite{ItkinLipton2017}, one can notice that the shape is quite different while for calibration we use the same market smiles. This is because in \\cite{ItkinLipton2017} the standard local volatility model is used, where the underlying price follows a Geometric Brownian motion equipped with an instantaneous local volatility function, while in this paper the model is quite different.\n\n\nTo look at a more regular surface, we proceed with another example which is taken from \\cite{Balaraman2016}. In that paper an implied volatility surface of S\\&P500 is presented, and the local volatility surface is constructed using the Dupire formula. In our test we take data for the first 12 maturities and all strikes as they are given in \\cite{Balaraman2016}, and apply our model to calibrate the local variance surface as this is described in above. When doing so we set $v^1_{j,0} = -0.3$, and $v^1_{j,n_j} = 0.1$ for all smiles.\n\nThe results of this calibration are presented in Fig.~\\ref{fitSP},\\ref{termSP},\\ref{lvSP}. By construction, our surface preserves no-arbitrage, while for the approach in \\cite{Balaraman2016} they have to solve some additional problems\\footnote{As this is mentioned in \\cite{Balaraman2016}, the correct pricing of local volatility surface requires an arbitrage free implied volatility surface. If the input implied volatility surface is not arbitrage free, this can lead to negative transition probabilities and\/or negative local volatilities and can give rise to mispricing.}.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width = 0.8\\textwidth]{fit_sp500.png}}\n\\caption{Term-by-term fitting of market S\\&P500 Put prices constructed using data of \\cite{Balaraman2016}.}\n\\label{fitSP}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth]{local_var_sp500.png}}\n\\caption{Term-by-term fitting of the instantaneous local variance $\\sigma^2(x,T)$ for S\\&P500.}\n\\label{termSP}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\fbox{\\includegraphics[width=0.7\\textwidth, height=3in]{surf_sp500.png}}\n\\caption{The instantaneous local variance surface $\\sigma^2(x,T)$ for S\\&P500 constructed by using the proposed approach.}\n\\label{lvSP}\n\\end{center}\n\\end{figure}\n\n\nIn Table~\\ref{PerfSP500} we present the performance of our algorithm in this experiment.\nIt can be seen that here the elapsed time is similar or shorter as compared with the previous test presented in Table~\\ref{Perf}.\n\n\\begin{table}[H]\n\\begin{center}\n\\small\n\\begin{tabular}{|r|r|r|r|r|r|}\n\\hline\nTerm & $T$, years & Elapsed time, sec & iterations & function evaluations & strikes \\\\\n\\hline\n 1 & 0.0822 & 1.09 & 99 & 1604 & 8 \\\\ \\hline\n 2 & 0.1671 & 0.56 & 40 & 377 & 8 \\\\ \\hline\n 3 & 0.2521 & 2.32 & 94 & 1615 & 8 \\\\ \\hline\n 4 & 0.3315 & 1.70 & 97 & 1186 & 8 \\\\ \\hline\n 5 & 0.4164 & 0.10 & 15 & 64 & 8 \\\\ \\hline\n 6 & 0.4986 & 2.35 & 111 & 1600 & 8 \\\\ \\hline\n 7 & 0.5836 & 2.40 & 111 & 1584 & 8 \\\\ \\hline\n 8 & 0.6658 & 2.25 & 131 & 1604 & 8 \\\\ \\hline\n 9 & 0.7507 & 1.51 & 95 & 1072 & 8 \\\\ \\hline\n 10 & 0.8356 & 2.30 & 98 & 1603 & 8 \\\\ \\hline\n 11 & 0.9178 & 0.07 & 13 & 46 & 8 \\\\ \\hline\n 12 & 1.0027 & 72.80 & 74 & 795 & 8 \\\\ \\hline\n\\end{tabular}\n\\caption{Performance characteristics of the algorithm for calibration of a S\\&P500 surface.}\n\\label{PerfSP500}\n\\end{center}\n\\end{table}\n\n\n\\clearpage\n\\section{Conclusions}\n\nIn this paper we propose an expanded version of the Local Variance Gamma model of\n\\cite{CarrNadtochiy2017} which we refer as an Expanded Local Variance Gamma model, or ELVG. Two main improvements are introduced as compared with the LVG model. First, we add drift to the governing underlying process. It turns out that this a relatively minor (at the first glance) improvement requires a interesting trick to preserve tractability of the model, which is a non-trivial time-change. We show that still in this new model it is possible to derive an ordinary differential equation for the option price which plays a role of Dupire's equation for the standard local volatility model.\n\nThe second novelty of the paper as compared with the LVG model is that in contrast to \\cite{CarrNadtochiy2017} we consider a local variance to be a piecewise linear function of strike, while in \\cite{CarrNadtochiy2017} it was piecewise constant. We proceed in the spirit of \\cite{ItkinLipton2017} by describing a no-arbitrage interpolation, and then construct a closed-form solution of our ODE in terms of hypergeometric and generalized hypergeometric functions. An important advantage of this approach is that calibration of the model to market smiles does not require solving any optimization problem, and can be done term-by-term by solving a system of non-linear algebraic equations for each maturity, which, in general, is significantly faster, especially since we provide an algorithm for constructing a smart initial guess. We also provide various asymptotic solutions which allow a significant acceleration of the numerical solution and improvement of its accuracy in the corresponding cases (i.e, when parameters of the model at some iteration obey the conditions to apply the corresponding asymptotic).\n\nIn principle, somebody could claim that solving a system of nonlinear equations with a generic solver is not much different from solving a nonlinear optimization problem. Obviously, when our ODE is used as an alternative to the Dupire equation, the difference comes from the fact that calibration based on the Dupire equation requires solving this PDE at every iteration by either numerically, or semi-analytically by using a Laplace transform, which is obviously slower. As was mentioned in Introduction there exist many other calibration algorithms which reduce to a nonlinear optimization problem (e.g., taking a sufficiently large parametric family of local volatility functions and choosing the parameters that provide the best fit of observed prices). For the latter computation of the objective function is fast, but optimization must be constrained to preserve no-arbitrage, and, thus, slow.\n\nIn our numerical test we use same market data as in \\cite{ItkinSigmoid2015, ItkinLipton2017}. The results of the test demonstrate robustness of the proposed approach from both the speed and accuracy point of view, especially in cases where the above referred papers experienced some difficulties with achieving a perfect fit. An additional test performed for the S\\&P500 data taken from \\cite{Balaraman2016} gives rise to the same conclusion.\n\n\n\n\\clearpage\n\n\\section*{References}\n\n\\newcommand{\\noopsort}[1]{} \\newcommand{\\printfirst}[2]{#1}\n \\newcommand{\\singleletter}[1]{#1} \\newcommand{\\switchargs}[2]{#2#1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe motion of a compressible viscous, heat-conductive, and Newtonian polytropic fluid occupying a spatial domain $\\Omega \\subset \\mathbb{R}^{3}$ is governed by the following full compressible Navier-Stokes system:\n\\be \\la{a0}\n\\begin{cases}\n \t\\n_t+{\\rm div} (\\n u)=0,\\\\\n \t(\\n u)_t+{\\rm div}(\\n u\\otimes u)+\\na P=\\div \\mathbb{S},\\\\\n \t(\\n E)_t+\\div (\\n Eu+Pu)=\\div (\\kappa \\na \\te)+\\div (\\mathbb{S}u),\n\\end{cases}\\ee\nwhere $\\mathbb{S}$ and $E$ are respectively the viscous stress tensor and the total energy given by\n$$\\mathbb{S}=2\\mu\\mathfrak{D}(u) +\\lambda \\div u \\mathbb{I}_3,~~E=e+\\frac{1}{2}|u|^2,$$\nwith $\\mathfrak{D}(u) = (\\nabla u + (\\nabla u)^{\\rm tr})\/2$ and $\\mathbb{I}_3$ denoting the deformation tensor and the $3\\times3$ identity matrix respectively.\nHere, $t\\ge 0$ is time, $x\\in \\Omega$ is the spatial coordinate, and $\\n$, $u=\\left(u^1,u^2,u^3\\right)^{\\rm tr},$ $e$, $P,$ and $\\te $ represent respectively the fluid density, velocity, specific internal energy, pressure, and absolute temperature.\nThe viscosity coefficients $\\mu$ and $\\lambda$ are constants satisfying the physical restrictions:\n\\be\\la{h3} \\mu>0,\\quad 2 \\mu + 3\\lambda\\ge 0.\\ee\nThe heat-conductivity coefficient $\\ka$ is a positive constant.\nWe consider the ideal polytropic fluids so that $P$ and $e$ are given by the state equations:\n\\be \\la{pg}\nP(\\n,e)=(\\ga-1)\\n e=R\\n \\te, \\quad e=\\frac{R\\theta}{\\ga-1},\n\\ee\nwhere $\\ga>1$ is the adiabatic constant and $ R$ is a positive constant.\n\n\nLet $\\Omega \\subset \\r^3 $ be a simply connected bounded domain.\nNote that for the classical solutions, the system \\eqref{a0} can be rewritten as\n\\be \\la{a1}\n\\begin{cases}\n\t\\n_t+{\\rm div} (\\n u)=0,\\\\\n\t\\n (u_t+ u\\cdot \\na u )=\\mu\\Delta u+(\\mu+\\lambda)\\na({\\rm div}u)-\\na P,\\\\\n\t\\frac{R}{\\ga-1}\\n ( \\te_t+u\\cdot\\na \\te)=\\ka\\Delta\\te-P\\div u+\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2.\n\\end{cases}\n\\ee\nWe consider the system \\eqref{a1} subjected to the given initial data\n\\be \\la{h2}\n(\\rho,\\n u,\\n \\te)(x,{t=0})=(\\rho_0,\\n_0 u_0, \\n_0\\te_0)(x), \\quad x\\in \\Omega,\n\\ee\nand boundary conditions\n\\be \\la{h1}\nu\\cdot n=0,~~\\mbox{curl} u\\times n=0,~~\\na \\theta\\cdot n=0~~~~\\text{on}~\\p\\Omega \\times(0,T),\n\\ee\nwhere $n =(n^1, n^2, n^3)^{\\rm{tr}}$ is the unit outward normal vector on $\\partial\\Omega$.\n\n\nThere is a lot of literature on the global existence and large time\nbehavior of solutions to (\\ref{a0}). The one-dimensional problem\nwith strictly positive initial density and temperature has been\nstudied extensively by many people (see \\cite{kazh01,Kaz,akm} and\nthe references therein). For the multi-dimensional case, the local existence and uniqueness of classical solutions are known in \\cite{Na,se1} in the absence of vacuum. \nThe global classical solutions were\nfirst obtained by Matsumura-Nishida \\cite{M1} for initial data close\nto a non-vacuum equilibrium in some Sobolev space $H^s.$\nLater, Hoff \\cite{Hof1} studied the\nglobal weak solutions with strictly positive initial density and\ntemperature for discontinuous initial data. On the other hand, in\nthe presence of vacuum, this issue becomes much more complicated.\nConcerning viscous compressible fluids in a barotropic regime, where\nthe state of these fluids at each instant $t>0 $ is completely\ndetermined by the density $\\n=\\n(x,t)$ and the velocity $u=u(x,t),$\nthe pressure $P$ being an explicit function of the density, the\nmajor breakthrough is due to Lions \\cite{L1} (see also Feireisl\n\\cite{F1, feireisl1}), where he obtained global existence of weak\nsolutions, defined as solutions with finite energy, when the\npressure $P$ satisfies $P(\\n)=a\\n^\\ga (a>0,\\ga>1)$ with suitably large $\\ga.$\nThe main restriction on initial data is that the initial energy is\nfinite, so that the density vanishes at far fields, or even has\ncompact support. Recently, Huang-Li-Xin \\cite{hulx} and Li-Xin \\cite{2dlx} established the global well-posedness\nof classical solutions to the Cauchy problem for the 3D and 2D barotropic compressible Navier-Stokes equations in whole space with smooth initial data that are of small energy but\npossibly large oscillations, in particular, the initial density is allowed to vanish.\nMore recently, for slip boundary condition in bounded domains,\nCai-Li \\cite{C-L} obtained the global classical solutions with initial vacuum, provided that the initial energy is suitably small.\n\n\n\nCompared with the barotropic flows, it seems much more difficult and complicated to study the global well-posedness of solutions to full compressible Navier-Stokes system (\\ref{a0}) with vacuum, where some additional difficulties arise, such as the degeneracy of both momentum and energy equations, the strong coupling between the velocity and temperature, et al. For specific pressure laws excluding the perfect gas\nequation of state, the question of existence of so-called\n``variational\" solutions in dimension $d\\ge 2$ has been recently\n addressed\nin \\cite{feireisl,feireisl1}, where the temperature equation is\nsatisfied only as an inequality which justifies the notion of\nvariational solutions.\n Moreover, for a very particular form of the viscosity\ncoefficients depending on the density, Bresch-Desjardins\n\\cite{bd} obtained global stability of weak solutions.\nFor the global well-posedness of classical solutions to the full compressible Navier-Stokes system \\eqref{a0}, it is shown in {Xin} \\cite{X1} that there is no solution in $C^{1}\\left([0, \\infty), H^{s}\\left(\\mathbb{R}^{d}\\right)\\right)$ for large $s$ to the Cauchy problem for the full compressible Navier-Stokes\nsystem without heat conduction provided that the initial density has compact support. See also the recent generalizations to the\ncase for non-compact but rapidly decreasing at far field initial densities (\\cite{R}).\nRecently, Huang-Li \\cite{H-L} established the global existence and uniqueness for the classical solutions to the 3D Cauchy problem with interior vacuum provided the initial energy is small enough. Later, Wen-Zhu \\cite{W-C} obtained the global existence and uniqueness of the classical solutions for vanishing far-field density under the assumption that the initial mass is sufficiently small or both viscosity and heat-conductivity coefficients are large enough. It should be mentioned here that the results of \\cite{H-L,W-C} hold only for the Cauchy problem. However,\n the global existence of classical solutions or even weak ones with vacuum to multi-dimensional full compressible Navier-Stokes system \\eqref{a0} in general bounded domains remains completely open except for spherically or cylindrically symmetric\ninitial data (see \\cite{wenzhu1,wenzhu2}). In fact, one of the aims of this paper is to study the global well-posedness of classical solutions to full compressible Navier-Stokes system \\eqref{a0} in general bounded domains.\n\n\n\n\n\n\n\n\n\n\n\n\n\nBefore stating the main results, we explain the notations and conventions used throughout this paper. We denote\n\\bnn\\int fdx\\triangleq\\int_{\\Omega}fdx,\\enn\nand \\bnn \\overline{f}\\triangleq\\frac{1}{|\\O|}\\int_{\\Omega}fdx,\\enn\nwhich is the average of a function $f$ over $\\Omega$.\nFor $1\\le p\\le \\infty $ and integer $k\\ge 0,$ we adopt the simplified notations for Sobolev spaces as\nfollows:\n\\be\\ba\\notag \\begin {cases}\nL^p=L^p(\\Omega),\\quad W^{k,p}=W^{k,p}(\\Omega),\\quad H^k=W^{k,2}(\\O),\\\\\nH_\\omega^i= \\left.\\left\\{f\\in H^i \\right | f\\cdot n=0,\\,\\curl f\\times n=0 \\rm \\,\\,{on}\\,\\, \\p\\O\\right\\}\\,(i=1,2).\n\\end{cases}\\ea\\ee\n\nWithout loss of generality, we assume that\n\\be\\la{m}\\overline{\\rho_0}=\\frac{1}{|\\O|}\\int \\rho_0 dx=1.\\ee\nWe then define the initial energy $C_0$ as follows:\n\\be\\la{e}\\ba C_0\\triangleq &\\frac{1}{2}\\int\\n_0\n|u_0|^2dx+R \\int \\left( 1+\\n_0\\log {\\n_0}-\\n_0 \\right)dx\\\\\n&+\\frac{R}{\\ga-1}\\int \\n_0\\left(\\te_0- \\log {\\te_0} -1\n\\right) dx .\\ea\\ee\n\nThe first main result in this paper can be stated as follows:\n\n\n\n\n\\begin{theorem}\\la{th1}\nLet $\\O\\subset\\r^3$ be a simply connected bounded smooth domain, whose boundary $\\p\\O$ has a finite number of 2-dimensional connected components. For given numbers $M>0$ (not necessarily\nsmall), $q\\in (3,6),$\n $\\on> 2,$ and $\\bt>1,$\nsuppose that the initial data $(\\n_0,u_0,\\te_0)$ satisfies \\be\\ba\n\\la{co3} \\n_0\\in W^{2,q}, \\quad u_0 \\in H^2_\\omega,\\quad\n\\te_0\\in\\left.\\left\\{f\\in H^1 \\right| \\na f\\cdot n=0 \\,\\,\\rm {on}\\,\\, \\p\\O\\right\\}, \\ea\\ee\n\\be \\la{co4} 0\\le\\inf\\rho_0\\le\\sup\\rho_0< \\hat{\\rho},\\quad 0 \\le\\inf\\te_0\\le\\sup\\te_0\\le \\bt, \\quad \\|\\na u_0\\|_{L^2} \\le M,\n \\ee\n and the compatibility condition\n\\be\n\\la{co2}-\\mu \\Delta u_0-(\\mu+\\lambda)\\na\\div u_0+R\\na (\\n_0\\te_0)\n=\\sqrt{\\n_0} g,\\ee\nwith $g\\in L^2. $ Then there exists a positive constant $\\ve$\ndepending only\n on $\\mu,$ $\\lambda,$ $\\ka,$ $ R,$ $ \\ga,$ $\\on,$ $\\bt$, $\\O$, and $M$ such that if\n \\be\n \\la{co14} C_0\\le\\ve,\n \\ee the problem (\\ref{a1})--(\\ref{h1})\nadmits a unique global classical solution $(\\rho,u,\\te)$ in\n $\\Omega\\times(0,\\infty)$ satisfying\n \\be\\la{h8}\n 0\\le\\rho(x,t)\\le 2\\hat{\\rho},\\quad \\te(x,t)\\ge 0,\\quad x\\in \\Omega,\\,~~ t\\ge 0,\n \\ee\n and \\be\n \\la{h9}\\begin{cases}\n \\rho\\in C([0,T];W^{2,q}),\\\\\n u \\in C([0,T];W^{1,\\tilde p})\\cap L^\\infty(0,T;H^2)\\cap L^\\infty(\\tau,T; W^{3,q}), \\\\\n \\te\\in L^\\infty(\\tau,T;H^4)\\cap C([\\tau,T]; W^{3,\\tilde p}), \\\\\n u_t \\in L^2(0,T;H^1)\\cap\nL^{\\infty}(\\tau,T;H^2)\\cap H^1(\\tau,T;H^1), \\\\\n \\te_t \\in\nL^{\\infty}(\\tau,T;H^2)\\cap H^1(\\tau,T;H^1),\\end{cases} \\ee for any $0<\\tau0,\\ee\n\\be\\la{hq4}\\rho\\in [0,2\\hat{\\rho}] \\quad \\mbox{ \\rm a.e.}, \\quad \\te\\ge 0 \\quad\\mbox{ \\rm a.e.},\\ee\n and the exponential decay property (\\ref{h11}) with $p\\in [1 ,\\infty)$, $\\tilde p\\in [1,6)$, and $r\\in[1,6]$. In addition, there exists some positive constant $C$ depending only on $ \\mu,$ $\\lambda,$ $ \\ka,$ $ R,$ $ \\ga,$ $\\on$, $\\bt,$ $\\O$, and $M $ such that, for $\\si(t)\\triangleq\\min\\{1,t\\},$ the following estimates hold\n \\be\\la{hq5}\\ba \\sup_{t\\in (0,\\infty)} \\| u \\|_{H^1}^2 +\\int_0^\\infty\\int\\left| (\\n u)_t+{\\rm div}(\\n u\\otimes u)\\right|^2dxdt\\le C,\\ea\\ee\n \\be\\la{hq7}\\ba &\\sup_{t\\in (0,\\infty)} \\int\\left((\\n-1)^2+\\n |u|^2+\\n(R\\te-\\overline P)^2\\right)dx \\\\&\\quad +\\int_0^\\infty\\left( \\|\\na u\\|^2_{L^2}+ \\|\\na \\te\\|_{L^2}^2 \\right)dt\\le CC_0^{1\/4}, \\ea\\ee\n\\be\\la{hq8}\\ba\n&\\sup_{t\\in (0,\\infty)}\\left(\\si\\|\\na u \\|^2_{L^6}+\\si^2\\|\\te\\|^2_{H^2}\\right) \\\\\n&+\\int_0^\\infty\\left( \\si\\|u_t\\|^2_{L^2}+\\si^2\\|\\na \\dot u\\|_{L^2}^2+\\si^2\\|\\te_t\\|^2_{H^1}\\right)dt\\le C. \\ea\\ee\nMoreover, $(\\n,u,\\te)$ satisfies (\\ref{a1})$_3$ in the weak form, that is, for any test function $\\psi\\in\n\\mathcal{D}(\\O\\times(-\\infty,\\infty)),$\n\\be\\la{vu019}\\ba &\\frac{R}{\\ga-1} \\int\\n_0 \\te_0\\psi(\\cdot,0) dx+\\frac{R}{\\ga-1}\\int_0^\\infty\\int\\n \\te \\left(\\psi_t+u \\cdot\\na\\psi\n\\right)dxdt\\\\ & = \\ka \\int_0^\\infty\\int \\na\\te \\cdot\\na\\psi dxdt+R\\int_0^\\infty\\int \\n\\te \\div u \\psi dxdt\\\\&\\quad -\\int_0^\\infty\\int \\left(\\lambda(\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2 \\right) \\psi dxdt.\\ea\\ee\n\n\\end{theorem}\n\n\nNext, as a direct application of \\eqref{h11}, the following Corollary \\ref{th3}, whose proof is similar to that of \\cite[Theorem 1.2]{C-L}, shows that the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially.\n\n\\begin{cor} \\la{th3}\nIn addition to the conditions of Theorem \\ref{th1}, assume further that there exists some point $x_0\\in\\Omega$ such that $\\rho_0(x_0)=0.$ Then for any $\\hat p>3$, there exists some positive constant $C$ depending on $\\mu$, $\\lambda$, $\\ka,$ $ R,$ $ \\ga,$ $\\on,$ $\\bt$, $\\O$, $\\hat p$ and $M$ such that the unique global classical solution $(\\rho,u,\\te)$ to the problem (\\ref{a1})--(\\ref{h1}) obtained in Theorem \\ref{th1} satisfies that for any $t\\geq 1$,\n\\be\\notag\\|\\nabla \\rho(\\cdot,t)\\|_{L^{\\hat p} }\\geq Ce^{Ct}.\\ee\n\\end{cor}\n\n\n\nA few remarks are in order:\n\n\n\n\n\n\n\\begin{remark}\nIt is easy to deduce from (\\ref{h9}) and the Sobolev imbedding theorem that for any $0<\\tau0, \\quad \\inf\\limits_{x\\in\\Omega}\\te_0(x)> 0,\\\\\nu_0\\cdot n=0,~~\\curl u_0\\times n=0,~~\\na\\te_0\\cdot n=0~~~~\\text{on}~\\p\\Omega.\\end{cases}\\ee Then there exist a small time\n$00,$ with initial data $(\\n_0,u_0,\\te_0)$ satisfying \\eqref{2.1}.\nFor\n$\\si(t)\\triangleq\\min\\{1,t\\}, $ we define\n$A_i(T)(i= 1,2,3)$ as follows:\n \\be\\la{As1}\n A_1(T) \\triangleq \\sup_{t\\in[0,T] }\\|\\nabla u \\|_{L^2}^2\n + \\int_0^{T}\\int \\rho|\\dot{u}|^2dxdt,\n \\ee\n \\be\\label{AS1}\n A_2(T) \\triangleq \\frac{1}{2(\\ga-1)}\\sup_{t\\in[0,T] }\\int\\n (R\\te-\\bp)^2dx\n +\\int_0^T\\left( \\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right)dt,\n \\ee\n \\be\\ba \\label{AS2}\n A_3(T) \\triangleq &\\sup_{t\\in(0,T]}\\left(\\si \\|\\na u\\|_{L^2}^2+\\si^2\\int\\n |\\dot u|^2dx + \\si^2\\|\\na\\te\\|_{L^2}^2 \\right)\\\\\n & + \\int_0^T\\int\\left(\\si\\n |\\dot u|^2 +\\sigma^2|\\nabla\\dot{u}|^2 +\\sigma^2\\n|\\dot \\te|^2 \\right)dxdt.\n \\ea\\ee\n\n\n\nWe have the following key a priori estimates on $(\\n,u,\\te)$.\n\\begin{pro}\\la{pr1}\nFor given numbers $M>0$, $\\on> 2,$ and $\\bt> 1,$ assume further that $(\\rho_0,u_0,\\te_0) $ satisfies\n\\be \\la{3.1}\n0<\\inf \\rho_0 \\le\\sup \\rho_0 <\\on,\\quad 0<\\inf \\te_0 \\le\\sup \\te_0 \\le \\bt, \\quad \\|\\na u_0\\|_{L^2} \\le M.\n\\ee\nThen there exist positive constants $K$, $C^\\ast$, $\\al$, $\\te_\\infty$, and $\\ep_0$ all depending on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,$ $\\on,$ $\\bt$, $\\O$, and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\O\\times (0,T]$ satisfying\n\\be \\la{z1}\n0< \\rho\\le 2\\on, \\,\\,\\, A_1(T)\\le 3 K, \\,\\,\\, A_2(T) \\le 2C_0^{1\/4}, \\,\\,\\, A_3(T) \\le 2C_0^{1\/6},\n\\ee\nthe following estimates hold:\n\\be \\la{zs2}\n0< \\rho\\le 3\\on\/2,\\,\\,\\, A_1(T)\\le 2 K, \\,\\,\\, A_2(T) \\le C_0^{1\/4}, \\,\\,\\, A_3(T) \\le C_0^{1\/6},\n\\ee\nand for any $t\\geq1$,\n\\be\\la{h22}\n\\|\\n-1\\|_{L^2}+\\|u\\|_{W^{1,6}}^2+\\|\\te-\\te_\\infty\\|_{H^2}^2\\le C^\\ast e^{-\\al t},\n\\ee\nprovided \\be\\la{z01}C_0\\le \\ve_0.\\ee \\end{pro}\n\n\\begin{proof}\nProposition \\ref{pr1} is a straight consequence of\nthe following Lemmas \\ref{le2}, \\ref{le6}, \\ref{le3}, \\ref{le7}, and \\ref{pr2} with $\\ve_0$ as in (\\ref{t7}).\n\\end{proof}\n\nIn this section, we always assume that $C_0\\le 1$ and let $C$ denote some generic positive constant depending only on $\\mu$, $\\lambda$, $\\ka$, $R$, $\\ga$, $\\on$, $\\bt$, $\\O,$ and $M,$ and we write $C(\\al)$ to emphasize that $C$ may depend on $\\al.$\n\nTo begin with, we have the following uniform estimate on $\\bp$, which plays an important role in the whole analysis.\n\n\n\\begin{lemma}\\la{a13.1} Under the conditions of Proposition \\ref{pr1}, there exists a positive constant $C$ depending only on $\\mu$, $R$, and $\\on$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying\n\\be\\la{3.q2}\n0<\\n\\le 2\\on ,\\quad A_2(T)\\le 2C_0^{1\/4},\n\\ee\nthe following estimates hold:\n\\be \\la{a2.112}\n\\sup_{0\\le t\\le T}\\int\\left( \\n |u|^2+(\\n-\\tn)^2\\right)dx \\le C C_0^{1\/4},\n\\ee\nand\n\\be \\la{key}\n0<\\pi_1 \\le \\overline P (t)\\le \\pi_2,~~~\\mbox{for~any}~t\\in [0,T],\n\\ee\nwhere $\\pi_1 $ and $\\pi_2$ are positive constants depending only on $\\mu$, $\\gamma$, $R$, and $\\O$.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nFirst, it follows from (\\ref{3.1}) and (\\ref{mn2}) that, for all $ (x,t)\\in \\Omega\\times(0,T),$\n\\be \\la{3.2}\n\\te(x,t)>0 .\n\\ee\n\nNote that\n\\be\\notag\n\\Delta u = \\na \\div u - \\na \\times \\curl u,\n\\ee\none can rewrite $(\\ref{a1})_2$ as\n\\be\\la{a11}\\ba\n\\n (u_t+ u\\cdot \\na u )&=(2\\mu+\\lambda)\\na{\\rm div}u- \\mu \\na \\times \\curl u-\\na P.\\ea\\ee\nAdding $(\\ref{a11})$ multiplied by $u$ to $(\\ref{a1})_3$ multiplied by $1-\\te^{-1}$ and integrating the resulting equality over $\\Omega$ by parts, we obtain\nafter using $(\\ref{a1})_1$, \\eqref{h3}, \\eqref{3.2}, and the boundary conditions \\eqref{h1} that\n\\be\\la{la2.7}\\ba\nE'(t)\n&=-\\int \\left( \\frac{\\lambda(\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2}{{\\te}}+\\ka \\frac{|\\na \\te|^2}{\\te^2} \\right)dx \\\\\n&\\quad - \\mu \\int \\left(|\\curl u|^2+2(\\div u)^2 - 2 |\\mathfrak{D}(u)|^2\\right)dx\\\\\n&\\le 2\\mu \\int |\\na u|^2 dx,\n\\ea\\ee\nwhere $E(t)$ is the basic energy defined by \\eqref{enet}.\n\nThen, integrating \\eqref{la2.7} with respect to $t$ over $(0,T)$ and using \\eqref{3.q2}, one has\n \\be\\la{a2.8}\\ba\n&\\sup_{0\\le t\\le T} E(t\n\\le C_0+ 2\\mu \\int_{0}^{T} \\int |\\na u|^2 dxdt\\le C C_0^{1\/4},\\ea\\ee\nwhich together with\n\\be\\la{a2.9}\\ba\n (\\n-1)^2\\ge 1+\\n\\log\\n-\\n&=(\\n-1)^2\\int_0^1\\frac{1-\\al}{\\al (\\n-1)+1}d\\al \\ge \\frac{(\\n-1)^2}{ 2(2\\on+1) }\n \\ea\\ee\ngives (\\ref{a2.112}).\n\n\n\n\n\n\nNext, it is easy to deduce from $\\eqref{a1}_1$ and \\eqref{m} that for any $t\\in[0,T]$,\n\\be \\la{mmm}\n\\overline\\rho(t)=\\overline{\\n_0}=1.\n\\ee\nDenote $d\\mu\\triangleq|\\Omega|^{-1}\\n dx$. Then $d\\mu$ is a positive measure satisfying $\\mu(\\Omega)=1$ due to \\eqref{3.q2} and \\eqref{mmm}. Moreover, observe that $y-\\log y-1$ is a convex function in $(0,\\infty)$, it thus follows directly from Jensen's inequality \\eqref{jen} that for any $t\\in [0,T]$,\n\\be \\notag\n\\overline {\\rho \\te }(t) - \\log \\overline {\\rho \\te }(t) -1 \\le \\int(\\te-\\log\\te-1)\\frac{\\n dx}{|\\Omega|}\\le C\n\\ee due to \\eqref{a2.8}.\nThis in particular gives \\eqref{key} and finishes the proof of Lemma \\ref{a13.1}.\n\n\\end{proof}\n\n\\begin{remark}\n It should be pointed out that the following term in (\\ref{la2.7})\n \\begin{equation}\\label{xm}\n - \\mu \\int \\left( |\\curl u|^2+2(\\div u)^2 - 2 |\\mathfrak{D}(u)|^2 \\right)dx\n \\end{equation}\n is a sign-undetermined term due to the slip boundary condition (\\ref{bb}), which is in sharp contrast to the Cauchy problem \\cite{H-L} where the term (\\ref{xm}) vanishes after integration by parts.\n \n \n \n Thus, in this case, we can not bound the basic energy only by the initial energy. However, this term obviously can be bounded by $C \\int |\\nabla u|^2dx$, which implies a ``weaker\" basic energy estimate (\\ref{a2.8}).\n\\end{remark}\n\n\n\n\nThe next lemma provides an estimate on the term $A_1(T)$.\n\\begin{lemma}\\la{le2}\n\tUnder the conditions of Proposition \\ref{pr1}, there exist positive constants $K $ and $\\ep_1 $ both depending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\, \\O,$ and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying\n\t\\be\\la{3.q1} 0<\\n\\le 2\\on ,\\quad A_2(T)\\le 2C_0^{1\/4},\\quad A_1(T)\\le 3K,\\ee\n\tthe following estimate holds:\n\t\\be\\la{h23} A_1(T)\\le 2K , \\ee\n\tprovided $C_0\\le \\ep_1.$\n\\end{lemma}\n\n\n\n\\begin{proof}\n First, integrating $(\\ref{a11})$ multiplied by $2u_t $ over $\\Omega $ by parts gives\n \\be\\ba \\la{hh17}\n &\\frac{d}{dt}\\int \\left( {\\mu} |\\curl u|^2+ (2\\mu+\\lambda)(\\div u)^2\\right)dx+ \\int\\rho |\\dot u|^2dx \\\\\n&\\le 2\\int P\\div u_t dx+ \\int \\n|u\\cdot \\na u|^2dx\\\\\n &= 2\\frac{d}{dt}\\int (P-\\overline P) \\div u dx-2\\int (P-\\overline{P})_t \\div u dx+\\int \\n|u\\cdot \\na u|^2dx\\\\\n &= 2\\frac{d}{dt}\\int (P-\\overline P) \\div u dx-\\frac{1}{2\\mu+\\lambda}\\frac{d}{dt}\\int (P-\\overline P)^2 dx\\\\\n &\\quad-\\frac{2}{2\\mu+\\lambda}\\int (P-\\overline{P})_t G dx+ \\int \\n|u\\cdot \\na u|^2dx ,\n \\ea\\ee\nwhere in the last equality we have used\n(\\ref{hj1}).\n\nNext, straight calculations show\nthat for any $p\\in [2,6],$\n\\be\\la{pq}\\ba \\|R\\te- \\overline P \\|_{L^p}\n \\le R\\|\\te- \\overline\\te \\|_{L^p}+C|R\\overline\\te-\\overline P|\\le C(\\hat{\\n}) \\|\\na \\te\\|_{L^2},\\ea\\ee\nwhere one has used (\\ref{g1}) and the following fact:\n\\be\\notag\\ba\n|R\\overline\\te-\\overline P|=&\\frac{R}{|\\O|}\\xl|\\int(1-\\n)\\te dx\\xr|\n= \\frac{R}{|\\O|}\\xl|\\int(1-\\n)(\\te-\\overline\\te)dx\\xr|\\\\\n\\le&C\\|\\n-1\\|_{L^2}\\| \\te-\\overline\\te\\|_{L^2}\\\\\n\\le&C(\\on)C_0^{1\/8}\\|\\na \\te\\|_{L^2}\\ea\\ee\ndue to (\\ref{mmm}) and \\eqref{a2.112}. Thus, it follows from \\eqref{key}, \\eqref{3.q1}, \\eqref{a2.112}, and \\eqref{pq} that for any $p\\in[2,6]$,\n\\be\\la{p}\\ba \\|P-\\overline P \\|_{L^p}&= \\| \\n(R\\te-\n\\overline P )+ (\n \\n -1)\\overline P\\|_{L^p}\\\\&\\le \\|\\n(R\\te-\\overline P )\\|_{L^2}^{(6-p)\/(2p)}\n \\|\\n(R\\te-\\overline P )\\|_{L^6}^{ 3(p-2)\/(2p)}+ \\pi_2 \\|\\n-1\\|_{L^p}\n \\\\&\\le C(\\on)C_0^{(6-p)\/(16p)}\n \\|\\na\\te \\|_{L^2}^{ 3(p-2)\/(2p)}+ C(\\hat\\n)C_0^{1\/(4p)},\n \\ea\\ee\nwhich together with (\\ref{h17}) and \\eqref{3.q1} yields\n\\be \\la{3.30} \\|\\na\nu\\|_{L^6} \\le C(\\hat\\n) \\left( \\|\\n^{1\/2}\\dot\nu\\|_{L^2}+\\|\\na u\\|_{L^2}+ \\|\\na \\te\\|_{L^2}+C_0^{1\/24}\\right).\n\\ee\n\nNote that (\\ref{a1})$_3$ implies\n\\be \\la{op3} \\ba\nP_t=&-\\div (Pu) -(\\gamma-1) P\\div u+(\\ga-1)\\ka \\Delta\\te\\\\&+(\\ga-1)\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right),\n\\ea\\ee\nwhich along with \\eqref{h1} gives\n\\be \\ba \\la{pt}\n\\bp_t =&\n-(\\gamma-1) \\overline{P\\div u} +(\\ga-1)\\left(\\lambda\n\\overline{(\\div u)^2}+2\\mu \\overline{|\\mathfrak{D}(u)|^2}\\right).\n\\ea \\ee\nWe thus obtain after using integration by parts, (\\ref{key}), (\\ref{g1}), (\\ref{h19}), and (\\ref{p})--(\\ref{op3}) that\n\\be\\la{a16}\\ba\n&\\left|\\int P_t Gdx\\right| \\\\\n&\\le C\\int P(|G||\\na u|+ |u||\\na G|)dx+ C\\int\\left( |\\na\\te||\\na G|+|\\na u|^2|G|\\right)dx \\\\\n&\\le C\\int |P- \\bp| (|G||\\na u|+|u||\\na G|)dx +C\\bp\\int (|G||\\na u|+|u||\\na G|)dx \\\\\n&\\quad + C \\|\\na G\\|_{L^2} \\|\\na \\te\\|_{L^2} + C \\| G\\|_{L^6} \\|\\na u\\|_{L^2}^{3\/2} \\|\\na u\\|_{L^6}^{1\/2} \\\\\n&\\le C(\\hat \\n)(\\|\\na \\te\\|_{L^2}^{1\/2}+1)\\|\\na G\\|_{L^2}\\|\\na u\\|_{L^2}+ C \\|\\na G\\|_{L^2} \\|\\na \\te\\|_{L^2} \\\\\n& \\quad+ C(\\hat\\n) \\|\\na G\\|_{L^2} \\|\\na u\\|_{L^2}^{3\/2} \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+\\|\\na\\te\\|_{L^2}+1 \\right)^{1\/2} \\\\\n&\\le \\de \\|\\na G\\|_{L^2}^2 +\\de \\|\\rho^{1\/2}\\dot u\\|^2_{L^2}+C(\\de,\\on) \\left( \\|\\na u\\|_{L^2}^2+ \\|\\na \\te\\|_{L^2}^2+ \\|\\na u\\|^6_{L^2}\\right) \\\\\n&\\le C(\\on)\\de\\|\\rho^{1\/2} \\dot u\\|^2_{L^2} +C(\\de,\\on) \\left( \\|\\na u\\|_{L^2}^2+ \\|\\na \\te\\|_{L^2}^2 +\\|\\na u\\|^6_{L^2}\\right),\n\\ea\\ee\nand\n\\be\\la{a161}\\ba \\left|\\int \\bp_t Gdx\\right|\n\\le& C(\\on)(\\| \\na u\\|_{L^2}+\\| \\na u\\|_{L^2}^2)\\| G\\|_{L^2}\n\\\\ \\le& \\de \\|\\na G\\|_{L^2}^2 +C(\\de,\\on)(\\|\\na u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^4)\n\\\\\\le& C(\\on)\\de\\|\\rho^{1\/2}\n\\dot u\\|^2_{L^2} +C(\\de,\\on) (\\|\\na u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^4),\n\\ea\\ee\nwhere one has used\n\\be\\la{511} \\ba\n|\\overline P_t|\n&\\le C\\| P-\\bp\\|_{L^2} \\| \\na u\\|_{L^2} + C \\|\\na u\\|_{L^2}^2 \\le C(\\on)(C^{1\/8}_0\\| \\na u\\|_{L^2}+\\| \\na u\\|_{L^2}^2)\n\\ea \\ee\nowing to \\eqref{pt} and \\eqref{p}.\n\nThen, it follows from (\\ref{g1}), \\eqref{3.q1}, and (\\ref{3.30}) that\n\\be\\la{op1}\\ba\n\\int \\n|u\\cdot \\na u|^2dx&\\le C(\\on)\\|u\\|_{L^6}^2 \\|\\na u\\|_{L^2} \\|\\na u\\|_{L^6} \\\\\n&\\le \\de\\|\\rho^{1\/2} \\dot u\\|_{L^2}^2+ C(\\de,\\on)\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+\\|\\na\nu\\|_{L^2}^6\\right).\n\\ea\\ee\n\nFinally, substituting (\\ref{a16}), \\eqref{a161}, and (\\ref{op1}) into (\\ref{hh17}), one obtains after choosing $\\de$ suitably small that\n\\be\\ba \\la{1hh17}\n&\\frac{d}{dt}\\int \\left( {\\mu} |\\curl u|^2+ (2\\mu+\\lambda)(\\div u)^2\\right)dx+ \\frac{1}{2\\mu+\\lambda}\\frac{d}{dt} \\|P-\\overline P\\|_{L^2}^2 + \\frac{1}{2}\\int\\rho |\\dot u|^2dx \\\\\n& \\le 2\\frac{d}{dt}\\int (P-\\overline P) \\div u dx + C(\\on)\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+\\|\\na\nu\\|_{L^2}^6\\right).\n\\ea\\ee\n\nNote that it holds\n\\be\\la{cz}\\ba \\|P-\\overline P\\|_{L^2}^2(0)&=R^2\\int(\\n_0\\te_0-\\overline{\\n_0\\te_0})^2dx\\\\\n&\\le C(\\on)\\int\\n_0(\\te_0-\\overline{\\n_0\\te_0})^2dx+C\\int(\\n_0-1)^2dx\\\\\n&\\le C(\\on, \\bt)C_0,\\ea\\ee\nwhere we have used \\eqref{key}, \\eqref{a2.9}, and the following fact:\n\\be \\ba \\label{jia10} \\int \\n_0(\\te_0- \\overline{\\n_0\\te_0})^2dx &\\le C\\int \\n_0(\\te_0-1)^2dx+C|1-\\overline{\\n_0\\te_0}|^2\\\\\n&\\le C\\int \\n_0(\\te_0-1)^2dx+C\\left|\\int\\n_0(1-\\te_0)dx\\right|^2\\\\\n&\\le C(\\hat\\n,\\bt)\\int \\n_0(\\te_0-\\log\\te_0-1) dx\\\\\n&\\le C(\\hat\\n, \\bt)C_0\n\\ea\\ee\ndue to \\eqref{m} and\n \\be\\la{cz1}\\te-\\log\\te-1 =(\\te-1)^2\\int_0^1\\frac{\\al}{\\al (\\te-1)+1}d\\al\\geq\\frac{1}{2(\\|\\te(\\cdot,t)\\|_{L^{\\infty}}+1)}(\\te-1)^2.\\ee\nThen, integrating (\\ref{1hh17}) over $(0,T)$, one deduces from (\\ref{h18}), \\eqref{cz}, (\\ref{3.q1}), and \\eqref{p} that\n\\bnn\\la{h81} \\ba\n&\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}^2+ \\int_0^{T}\\int\\rho|\\dot{u}|^2dxdt\\\\\n&\\le CM^2+C(\\on,\\hat\\te)C_0^{1\/4} + C(\\on ) C_0^{1\/4}\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}^4 + C(\\on ) C_0^{1\/8}\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}\\\\\n&\\le CM^2+C(\\on,\\hat\\te)C_0^{1\/12}+ C(\\on ) C_0^{1\/4}\\sup_{0\\le t\\le T}\\|\\na u\\|_{L^2}^4 \\\\\n&\\le K+9C(\\on)C_0^{1\/4}K^2 \\\\\n&\\le 2K,\n\\ea \\enn\nwith $K\\triangleq CM^2+C(\\on,\\hat\\te) +1$, provided\n\\be\\notag C_0\\le \\ep_1 \\triangleq \\min\\left\\{1,\\xl(9C(\\on)K\\xr)^{-4}\\right\\}.\\ee\nThe proof of Lemma \\ref{le2} is completed.\n\\end{proof}\n\n\nNext, to estimate $A_3(T)$, we adopt the approach due to Hoff \\cite{Hof1} (see also Huang-Li \\cite{H-L}) to establish the following elementary estimates on $\\dot u$ and $\\dot \\te$, where the boundary terms are handled by the ideas due to \\cite{C-L}.\nThe estimate of $A_3(T)$ will be postponed to Lemma \\ref{le6}.\n\n\\begin{lemma}\\la{a113.4}\n\tUnder the conditions of Proposition \\ref{pr1}, let $(\\rho,u,\\te)$ be a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}. Then there exist positive constants $C$, $ C_1$, and $C_2$ depending only on $\\mu,\\,\\lambda, \\,k,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O,$ and $ M$ such that, for any $\\eta\\in (0,1]$ and $m\\geq0,$\nthe following estimates hold:\n\\be\\ba \\la{an1}\n(\\sigma B_1)'(t) + \\frac{1}{2}\\sigma \\int \\rho |\\dot u|^2dx\n\\le C C_0^{1\/4} \\sigma' + C\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right),\n\\ea\\ee\n\\be\\la{ae0}\\ba\n&\\left(\\sigma^{m}\\|\\rho^{1\/2}\\dot{u}\\|_{L^2}^2\\right)_t+C_1 \\sigma^{m}\\|\\na\\dot{u}\\|_{L^2}^2\\\\\n&\\le - 2\\left(\\int_{\\p \\O} \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t + C(\\si^{m-1}\\si'+\\si^m) \\|\\rho^{1\/2} \\dot u\\|_{L^2}^2 \\\\&\\quad+\nC_2 \\si^m \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2+ C\\|\\na u\\|^2_{L^2}+C \\si^m \\|\\na u\\|^4_{L^4} + C \\si^m \\|\\te \\na u\\|_{L^2}^2,\\\\\\ea\\ee\n and\n \\be\\la{nle7}\\ba &(\\si^mB_2 )'(t)+\\si^m \\int\\n|\\dot \\te|^2dx\\\\\n&\\le C \\eta \\si^m\\|\\na\\dot u\\|_{L^2}^2+C \\|\\na\n\\te \\|_{L^2}^2+C\\si^m \\|\\na u\\|_{L^4}^4+C(\\eta)\\si^m \\|\\te\\na u\\|_{L^2}^2,\\ea\\ee\nwhere\n\\be \\la{an2} \\ba B_1(t)\\triangleq& \\mu\\|\\curl u\\|_{L^2}^2+ (2\\mu+\\lambda) \\|\\div u\\|_{L^2}^2 \\\\&+ \\frac{1}{2\\mu+\\lambda} \\|P-\\overline P\\|_{L^2}^2 -2 \\int \\div u(P-\\bp) dx, \\ea\\ee\nand\n \\be\\la{e6}\nB_2(t)\\triangleq\\frac{\\ga-1}{R}\\left(\\ka \\|\\na\n\\te\\|_{L^2}^2-2 \\int (\\lambda (\\div u)^2+2\\mu|\\mathfrak{D}(u)|^2)\\te dx\\right).\\ee\n\\end{lemma}\n\n\n\\begin{proof} First, multiplying \\eqref{1hh17} by $\\sigma$ and using \\eqref{p} and \\eqref{z1} give \\eqref{an1} directly.\n\n\\iffalse\n\nMultiplying\n$(\\ref{a111}) $ by $\\sigma \\dot{u}$ and integrating the resulting\nequality over $\\Omega $ lead to \\be\\la{m0} \\ba \\int \\sigma\n\\rho|\\dot{u}|^2dx & = \\int (\\sigma \\dot{u}\\cdot\\nabla G - \\si \\mu \\na \\times \\curl u\\cdot\\dot{u})dx \\\\\n&=\\int_{\\partial \\O} \\sigma (u\\cdot\\na u \\cdot n) G dS - \\int \\sigma \\div \\dot{u} G dx - \\mu \\int \\si \\curl u \\cdot \\curl \\dot{u} dx \\\\\n& \\triangleq \\sum_{i=1}^{3}M_i. \\ea \\ee\n\nFor the term $M_1$, it can be deduced from \\eqref{pzw1}, \\eqref{g1}, \\eqref{h19}, and \\eqref{z1} that\n\\be \\ba \\la{b2}\nM_1&=-\\int_{\\partial \\O} \\sigma (u\\cdot \\na n\\cdot u) G dS\\\\\n &\\le C \\sigma\\|u\\|_{H^1}^2 \\|G\\|_{H^1}\\\\\n &\\le C \\sigma\\|\\na u\\|_{L^2}^2 \\|\\rho \\dot u\\|_{L^2}\\\\\n &\\le \\delta \\sigma\\|\\rho^{1\/2} \\dot u\\|_{L^2}^2+C(\\de,\\on,M) \\sigma\\|\\na u\\|_{L^2}^2.\n\\ea \\ee\n\n\nNotice that\n\\be \\ba \\label{jia1}\nP_t=(R\\n \\te)_t=R\\n \\dot{\\te}-\\div (Pu),~~~\\bp_t=R\\overline{\\n\\dot{\\te}},\n\\ea \\ee\nwhich along with some straight calculations gives\n\\be \\ba\\la{pt1}\n\\div \\dot u G=& (\\div u_t + \\div (u \\cdot \\na u))((2\\mu+\\lambda)\\div u - (P-\\bp))\\\\\n=& \\frac{2\\mu+\\lambda}{2} (\\div u)^2_t - ((P-\\bp) \\div u)_t +(P-\\overline{P})_t \\div u \\\\\n&+ (2\\mu+\\lambda) \\div (u \\cdot \\na u)\\div u - (P-\\bp) \\div (u \\cdot \\na u)\\\\\n=& \\frac{2\\mu+\\lambda}{2} (\\div u)^2_t - ((P-\\bp) \\div u)_t +R\\rho \\dot\\te \\div u - \\div(Pu) \\div u\\\\\n&-R\\overline{\\rho \\dot\\te} \\div u + (2\\mu+\\lambda) \\div (u \\cdot \\na u)\\div u - (P-\\bp) \\div (u \\cdot \\na u)\\\\\n=& \\frac{2\\mu+\\lambda}{2} (\\div u)^2_t - ((P-\\bp) \\div u)_t +R\\rho \\dot\\te \\div u -R\\overline{\\rho \\dot\\te} \\div u\\\\\n& + (2\\mu+\\lambda) \\na u : (\\na u)^{\\rm tr} \\div u + \\frac{2\\mu+\\lambda}{2} u \\cdot \\na(\\div u)^2 \\\\\n&- \\div((P-\\bp)u \\div u)- (P-\\bp) \\na u : (\\na u)^{\\rm tr}- \\bp (\\div u)^2.\n\\ea \\ee\nThis together with integration by parts, \\eqref{key}, and \\eqref{z1} implies that for any $\\beta\\in\n(0,1],$\n\\be\\la{m1} \\ba\nM_2 =& -\\frac{2\\mu+\\lambda}{2} \\left(\\int \\sigma (\\div u)^2 dx \\right)_t\n + \\frac{2\\mu+\\lambda}{2} \\si' \\int (\\div u)^2 dx \\\\\n&+\\left(\\int \\si (P-\\bp) \\div u dx\\right)_t - \\si' \\int (P-\\bp) \\div u dx - R\\si \\int \\rho \\dot\\te \\div u dx\\\\\n&-(2\\mu+\\lambda) \\si \\int \\na u : (\\na u)^{\\rm tr} \\div u dx + \\frac{2\\mu +\\lambda}{2} \\si \\int (\\div u)^3 dx\\\\\n&+ \\si \\int (P-\\bp) \\na u : (\\na u)^{\\rm tr}dx + \\si \\bp \\int (\\div u)^2 dx\\\\\n\\le & - \\frac{2\\mu+\\lambda}{2} \\left(\\int \\sigma (\\div u)^2 dx \\right)_t +\\left(\\int \\si (P-\\bp) \\div u dx\\right)_t \\\\\n& + C \\si' \\|P-\\bp\\|_{L^2}^2+ \\beta \\si^2 \\|\\rho^{1\/2} \\dot\\te\\|_{L^2}^2 \\\\\n&+ C \\si^2 \\|\\na u\\|_{L^4}^4 + C(\\on) \\beta^{-1} \\|\\na u\\|_{L^2}^2 + C(\\on) \\si \\int\n\\te |\\na u|^2dx\\\\\n\\le & - \\frac{2\\mu+\\lambda}{2} \\left(\\int \\sigma (\\div u)^2 dx \\right)_t +\\left(\\int \\si (P-\\bp) \\div u dx\\right)_t \\\\\n&+ C(\\on) C_0^{1\/4}\\si'+ \\beta \\si^2 \\|\\rho^{1\/2} \\dot\\te\\|_{L^2}^2 + C \\si^2 \\|\\na u\\|_{L^4}^4 \\\\\n&+ C(\\de,\\on,M) \\beta^{-1} ( \\|\\na u\\|_{L^2}^2 + \\|\\na \\te \\|_{L^2}^2)+ C(\\on) \\de \\si \\|\\rho^{1\/2} \\dot u \\|_{L^2}^2,\\\\\n\\ea \\ee where in the last inequality we have used (\\ref{p}) and\nthe following simple fact:\n\\be \\la{2.48}\\ba \\int\n\\te |\\na u|^2dx & \\le\nC\\int|R\\te-\\bp||\\na u|^2dx+C\\bp \\int |\\na u|^2dx\\\\ &\\le C\n\\|R\\te-\\bp \\|_{L^6}\\|\\na u\\|_{L^2}^{3\/2}\n\\|\\na u\\|_{L^6}^{1\/2}+ C \\|\\na u\\|_{L^2}^2\n\\\\ &\\le C(\\on)\\|\\na\\te\\|_{L^2}\\|\\na u\\|_{L^2}^{3\/2}\n\\left( \\|\\n \\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+ \\|\\na\\te\\|_{L^2}+1\\right)^{1\/2}\\\\\n&\\quad+ C \\|\\na u\\|_{L^2}^2\\\\ &\\le \\de\n\\left( \\|\\na\\te\\|^2_{L^2} + \\|\\n^{1\/2} \\dot\nu\\|_{L^2}^2 \\right) + C(\\de,\\on,M) \\|\\na\nu\\|_{L^2}^2 , \\ea\\ee\ndue to \\eqref{key}, (\\ref{pq}), (\\ref{3.30}), and (\\ref{z1}).\n\n\nFor the term $M_3$, it holds\u00a3\u00ba\n \\be\\la{m2} \\ba\nM_3\n& = -\\frac{\\mu }{2}\\int\\sigma |\\curl u|^2_t dx\n -\\mu \\sigma \\int \\curl u \\cdot \\curl (u\\cdot \\na u)dx \\\\\n& = -\\frac{\\mu }{2}\\left(\\sigma \\|\\curl u\\|_{L^2}^2\\right)_t +\n\\frac{\\mu }{2}\\si' \\|\\curl u\\|_{L^2}^2\n-\\mu \\sigma \\int \\curl u \\cdot (\\na u^i \\times \\na_i u) dx \\\\&\\quad+ \\frac{\\mu}{2}\\si\n\\int |\\curl u|^2\\div udx \\\\\n& \\le -\\frac{\\mu }{2}\\left(\\sigma \\|\\curl u\\|_{L^2}^2\\right)_t + C\n\\|\\na u\\|_{L^2}^2 + C\\sigma^2 \\|\\na u\\|_{L^4}^4 . \\ea \\ee\n\nNow, substituting \\eqref{b2}, (\\ref{m1}), and (\\ref{m2}) into (\\ref{m0}), we obtain (\\ref{an1}) after choosing $\\de$ suitably small.\n\\fi\n\nNow, we will prove (\\ref{ae0}).\n\n\nFirst, one can rewrite $(\\ref{a1})_2$ as\n\\be\\la{a111}\\ba\\n \\dot u&=\\na G- \\mu \\na \\times \\curl u,\n\\ea\\ee\nwith $G$ defined in \\eqref{hj1}.\nFor $m\\ge 0,$ operating $ \\si^m\\dot u^j[\\pa\/\\pa t+\\div (u\\cdot)]$ to $ (\\ref{a111})^j$ and integrating the resulting equality over $\\Omega$ by parts lead to\n\\be\\la{m4} \\ba\n& \\left(\\frac{\\sigma^m}{2}\\int\\rho|\\dot{u}|^2dx \\right)_t -\\frac{m}{2}\\sigma^{m-1}\\si'\\int\\rho|\\dot{u}|^2dx\\\\\n &= \\int_{\\p \\O} \\sigma^m \\dot{u} \\cdot n G_t dS - \\int \\sigma^m [\\div \\dot{u} G_t + u \\cdot \\na \\dot u \\cdot \\na G]dx \\\\\n&\\quad- \\mu \\int\\sigma^m\\dot{u}^j\\xl[(\\na \\times \\curl u)_t^j + \\div (u (\\na \\times \\curl u)^j)\\xr] dx \\triangleq\\sum_{i=1}^{3}N_i.\n\\ea \\ee\n\nNoticing that\n\\be\\la{pzw1} u\\cdot\\nabla u\\cdot n=-u\\cdot\\nabla n\\cdot u ~~\\quad \\mbox{on}~\\p \\O\\ee\ndue to $u \\cdot n|_{\\p \\O}=0$,\none can deduce from \\eqref{h1} and \\eqref{pzw1} that\n\\be \\la{bz8}\\ba\nN_1&=- \\int_{\\p \\O} \\sigma^m (u \\cdot \\na n \\cdot u) G_t dS\\\\\n&=- \\left(\\int_{\\p \\O} \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t + m \\si^{m-1} \\si' \\int_{\\p \\O}( u \\cdot \\na n \\cdot u) G dS \\\\\n&\\quad+ \\int_{\\p \\O} \\sigma^m (\\dot u \\cdot \\na n \\cdot u) G dS+ \\int_{\\p \\O} \\sigma^m ( u \\cdot \\na n \\cdot \\dot u) G dS\\\\\n&\\quad- \\int_{\\p \\O} \\sigma^m G( u \\cdot \\na) u \\cdot \\na n \\cdot u dS -\\int_{\\p \\O} \\sigma^m G u \\cdot \\na n \\cdot (u\\cdot \\na )u dS\\\\\n&\\le - \\left(\\int_{\\p \\O} \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t + C\\si^{m-1} \\si' \\| \\na u\\|_{L^2}^2\\|\\na G\\|_{L^2} \\\\\n&\\quad+\\de \\si^m \\|\\dot u\\|_{H^1}^2+ C(\\de) \\si^m \\|\\na u\\|_{L^2}^2 \\|\\na G\\|_{L^2}^2\\\\\n& \\quad- \\int_{\\p \\O} \\sigma^m G( u \\cdot \\na) u \\cdot \\na n \\cdot u dS -\\int_{\\p \\O} \\sigma^m G u \\cdot \\na n \\cdot (u\\cdot \\na )u dS,\n\\ea \\ee\nwhere one has used\n\\be \\ba\\notag\n\\left|\\int_{\\partial \\O} (\\dot u\\cdot \\na n\\cdot u+ u\\cdot \\na n\\cdot\\dot u) G dS\\right| &\\le C \\|\\dot u\\|_{H^1} \\| u\\|_{H^1} \\|G\\|_{H^1} \\\\&\\le C \\|\\dot u\\|_{H^1} \\|\\na u\\|_{L^2} \\|\\na G\\|_{L^2},\n\\ea \\ee\nand\n\\be\\la{b2}\n\\left|\\int_{\\partial \\O} ( u\\cdot \\na n\\cdot u) G dS \\right| \\le C \\|\\na u\\|_{L^2} ^2\\|\\na G\\|_{L^2}.\n\\ee\nNow, we will adopt the idea in \\cite{C-L} to deal with the last two boundary terms in \\eqref{bz8}. In fact, denote $u^{\\perp}\\triangleq-u\\times n$, it follows from\n$u \\cdot n|_{\\p \\O}=0$ that\n\\begin{align}\\notag\nu=u^{\\perp} \\times n~~~~~\\text{on}\\, \\p \\O,\n\\end{align}\nwhich along with \\eqref{g1}, \\eqref{cd}, and integration by parts yields\n\\be \\la{bz3}\\ba &- \\int_{\\partial\\Omega} G (u\\cdot \\na) u\\cdot\\na n\\cdot u dS \\\\&= -\\int_{\\partial\\Omega} G u^\\bot\\times n \\cdot\\na u^i \\nabla_i n\\cdot u dS \\\\&= - \\int_{\\partial\\Omega} G n\\cdot ( \\na u^i \\times u^\\bot) \\nabla_i n\\cdot u dS\\\\\n&= - \\int\\div( G( \\na u^i \\times u^\\bot) \\nabla_i n\\cdot u) dx \\\\\n&= - \\int \\na (\\nabla_i n\\cdot u G) \\cdot ( \\na u^i \\times u^\\bot) dx - \\int \\div( \\na u^i \\times u^\\bot) \\nabla_i n\\cdot u G dx \\\\\n&= - \\int \\na (\\nabla_i n\\cdot u G) \\cdot ( \\na u^i \\times u^\\bot) dx + \\int G \\na u^i \\cdot \\na\\times u^\\bot \\nabla_i n\\cdot u dx \\\\\n& \\le C \\int |\\na G||\\na u||u|^2dx+C \\int |G| (|\\na u|^2|u|+|\\na u||u|^2)dx\n\\\\& \\le C \\|\\na G\\|_{L^6}\\|\\na u\\|_{L^2}\\|u\\|^2_{L^6}\n+C \\| G\\|_{L^3}\\|\\na u\\|^2_{L^4}\\|u\\|_{L^6}+C\\| G\\|_{L^{6}} \\|\\na u\\|_{L^2}\\|u\\|_{L^6}^2\n\\\\& \\le \\de \\|\\na G\\|_{L^6}^2+C(\\de) \\|\\na u\\|^6_{L^2}+C\\|\\na u\\|^4_{L^4}+ C \\| \\na G\\|_{L^2}^2 (\\|\\na u\\|^2_{L^2} +1 ).\\ea\\ee\nSimilarly, it holds that\n\\be \\la{bz4}\\ba &- \\int_{\\partial\\Omega}G u\\cdot\\na n\\cdot ({u}\\cdot\\na) u dS\\\\& \\le \\de \\|\\na G\\|_{L^6}^2+C(\\de) \\|\\na u\\|^6_{L^2}+C\\|\\na u\\|^4_{L^4}+ C \\| \\na G\\|_{L^2}^2(\\|\\na u\\|^2_{L^2} +1 ).\\ea\\ee\n\n\n\n\n\n\nNext, it follows from \\eqref{hj1} that\n\\be \\ba \\la{pt11}\nG_t =& (2\\mu+\\lambda)\\div u_t - (P_t-\\bp_t )\\\\\n=& (2\\mu+\\lambda) \\div \\dot u - (2\\mu+\\lambda) \\div (u\\cdot \\na u) - R\\rho \\dot\\te + \\div(Pu) + R\\overline{\\rho \\dot\\te}\\\\\n=& (2\\mu+\\lambda) \\div \\dot u - (2\\mu+\\lambda) \\na u : (\\na u)^{\\rm tr} - u \\cdot \\na G + P \\div u - R\\rho \\dot\\te + R\\overline{\\rho \\dot\\te},\n\\ea \\ee\nwhere one has used\n\\be \\ba \\label{jia1}\nP_t=(R\\n \\te)_t=R\\n \\dot{\\te}-\\div (Pu),~~~\\bp_t=R\\overline{\\n\\dot{\\te}}.\n\\ea \\ee\nThen, integration by parts combined with \\eqref{pt11} gives\n\\be\\la{m5} \\ba\nN_2 =& - \\int \\sigma^m [\\div \\dot{u} G_t + u \\cdot \\na \\dot u \\cdot \\na G]dx \\\\\n=& - (2\\mu+\\lambda) \\int \\sigma^m (\\div \\dot{u} )^2 dx + (2\\mu+\\lambda) \\int \\sigma^m \\div \\dot{u} \\na u : (\\na u)^{\\rm tr} dx \\\\\n&+\\int \\sigma^m \\div \\dot{u} u \\cdot \\na G dx - \\int \\sigma^m \\div \\dot{u} P \\div u dx \\\\\n&+ R \\int \\sigma^m \\div \\dot{u} \\rho \\dot\\te dx - R \\overline{\\rho \\dot\\te} \\int \\sigma^m \\div \\dot{u} dx - \\int \\sigma^m u \\cdot \\na \\dot u \\cdot \\na Gdx \\\\\n\\le & - (2\\mu+\\lambda) \\int \\sigma^m (\\div \\dot{u} )^2 dx\\\\\n& + C \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\na u\\|_{L^4}^2 + C \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\na G\\|_{L^2}^{1\/2} \\|\\na G\\|_{L^6}^{1\/2} \\|u\\|_{L^6} \\\\\n&+ C(\\on) \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\te \\na u\\|_{L^2}+ C(\\on) \\si^m \\|\\na \\dot{u}\\|_{L^2} \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}.\\ea \\ee\n\n\n\nNote that\n$$\\curl u_t=\\curl \\dot u-u\\cdot \\na \\curl u-\\na u^i\\times \\nabla_iu,$$\nwhich together with some straight calculations yields\n\\be\\la{ax3999}\\ba\nN_3 &=- \\mu \\int\\sigma^{m}|\\curl\\dot{u}|^{2}dx+\\mu \\int\\sigma^{m}\\curl\\dot{u}\\cdot(\\nabla u^i\\times\\nabla_i u) dx \\\\&\\quad+\\mu \\int\\sigma^{m} u\\cdot\\na \\curl u \\cdot\\curl\\dot{u} dx +\\mu \\int\\sigma^{m} u \\cdot \\na \\dot{u}\\cdot (\\nabla\\times\\curl u) dx \\\\\n&\\le - \\mu \\int\\sigma^{m}|\\curl\\dot{u}|^{2}dx+\\de \\sigma^{m}(\\|\\na \\dot u\\|_{L^2}^2 +\n\\|\\na \\curl u\\|_{L^6}^2)\\\\&\\quad +C(\\de) \\sigma^{m} \\|\\na u\\|_{L^4}^4 +C(\\de) \\sigma^{m}\\|\\na u\\|_{L^2}^4\\|\\na \\curl u\\|_{L^2}^2.\\\\\n\\ea\\ee\n\n\n\nFinally, it is easy to deduce from Lemmas \\ref{le4} and \\ref{uup1} that\n\\be \\la{bz5} \\|G\\|_{H^1}+\\|\\curl u\\|_{H^1}\\le C(\\|\\n \\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}), \\ee\nand that\n\\be \\la{bz6}\\ba \\|\\na G\\|_{L^6}+\\|\\na\\curl u\\|_{L^6} +\\| \\dot u\\|_{H^1}\n &\\le C(\\|\\n \\dot u\\|_{L^6}+\\|\\na u\\|_{L^2})+\\| \\dot u\\|_{H^1}\n\\\\&\\le C(\\on) (\\|\\na \\dot u\\|_{L^2}+ \\|\\na u\\|_{L^2}+ \\|\\na u\\|_{L^2}^2).\\ea \\ee\n\n\\iffalse\nCombining \\eqref{m5} with \\eqref{bz8}, \\eqref{bz3}, and \\eqref{bz4}, one obtains after using \\eqref{bz5}, \\eqref{bz6}, and \\eqref{z1} that\n\\be\\la{ax399}\\ba\nN_1\n\\le &- \\left(\\int_{\\p \\O} \\sigma^m u \\cdot \\na n \\cdot u G dS\\right)_t - (2\\mu+\\lambda) \\int \\sigma^m (\\div \\dot{u} )^2 dx \\\\\n&+ C(\\on) \\de \\si^m \\|\\na \\dot{u}\\|_{L^2}^2+C(\\de,\\on,M) (\\si^{m-1} \\si'+\\si^m) \\| \\rho^{1\/2} \\dot u\\|_{L^2}^2\\\\\n&+C(\\de,\\on) \\si^m \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2 + C(\\de,\\on,M) \\| \\na u\\|_{L^2}^2\\\\\n&+C(\\de) \\si^m \\| \\na u\\|_{L^4}^4 + C(\\de, \\on) \\si^m \\|\\te \\na u\\|_{L^2}^2.\n\\ea\\ee\n\\fi\n\nHence, submitting \\eqref{bz8}, \\eqref{m5}, and \\eqref{ax3999} into \\eqref{m4}, one obtains after using \\eqref{bz3}, \\eqref{bz4}, \\eqref{z1}, \\eqref{bz5}, and \\eqref{bz6} that\n\\be\\la{ax40}\\ba\n&\\left(\\frac{\\sigma^{m}}{2}\\|\\rho^{1\/2}\\dot{u}\\|_{L^2}^2\\right)_t+(2\\mu+\\lambda)\\sigma^{m}\\|\\div\\dot{u}\\|_{L^2}^2+\\mu\\sigma^{m}\\|\\curl\\dot{u}\\|_{L^2}^2\\\\\n&\\le -\\left(\\int_{\\p \\O} \\sigma^m (u \\cdot \\na n \\cdot u) G dS\\right)_t+ C(\\on) \\de \\si^m \\|\\na \\dot{u}\\|_{L^2}^2 \\\\\n&\\quad+ C(\\de,\\on) \\si^m \\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2 + C(\\de, \\on, M)(\\si^{m-1}\\si'+\\si^m) \\|\\rho^{1\/2} \\dot u\\|_{L^2}^2\\\\\n&\\quad+C(\\de, \\on, M) \\|\\na u\\|^2_{L^2} +C(\\de)\\si^m \\|\\na u\\|^4_{L^4}+ C(\\de,\\on) \\si^m \\|\\te \\na u\\|_{L^2}^2.\n\\ea\\ee\nApplying \\eqref{tb11} to \\eqref{ax40} and choosing $\\de$ small enough infer \\eqref{ae0} directly.\n\n\n\n\n\n\n\n\n\nFinally, we will prove (\\ref{nle7}).\n\nFor $m\\ge 0,$\nmultiplying $(\\ref{a1})_3 $ by $\\sigma^m \\dot\\te$ and integrating\nthe resulting equality over $\\Omega $ yield that\n \\be\\la{e1} \\ba &\\frac{\\ka\n{\\sigma^m}}{2}\\left( \\|\\na\\te\\|_{L^2}^2\\right)_t+\\frac{R\\sigma^m}{\\ga-1} \\int\\rho|\\dot{\\te}|^2dx\n\\\\&=-\\ka\\sigma^m\\int\\na\\te\\cdot\\na(u\\cdot\\na\\te)dx\n+\\lambda\\sigma^m\\int (\\div u)^2\\dot\\te dx\\\\&\\quad\n+2\\mu\\sigma^m\\int |\\mathfrak{D}(u)|^2\\dot\\te dx-R\\si^m\\int\\n\\te \\div\nu\\dot\\te dx \\triangleq \\sum_{i=1}^4I_i . \\ea\\ee\n\n First, combining (\\ref{g1}) and (\\ref{z1}) gives\n\\be\\la{e2} \\ba\nI_\n&\\le C\\sigma^{m} \\|\\na u\\|_{L^2}\\|\\na\\te\\|^{1\/2}_{L^2}\n\\|\\na^2\\te\\|^{3\/2}_{L^2} \\\\\n&\\le \\de\\sigma^{m} \\|\\n^{1\/2}\\dot\\te\\|^2_{L^2}+\\si^m \\left(\\|\\na u\\|_{L^4}^4+\\|\\te\\na u\\|_{L^2}^2\\right) +C(\\de,\\on,M)\\sigma^{m}\n\\|\\na\\te\\|^2_{L^2} ,\\ea\\ee\nwhere in the last inequality we have used the following estimate:\n\\be \\la{lop4}\\ba\n\t\\|\\na^2\\te\\|_{L^2} &\\le C (\\on)\\left(\\|\\n^{1\/2}\\dot \\te\\|_{L^2}+ \\|\\na u\\|_{L^4}^2+\\|\\te\\na u\\|_{L^2}\\right),\n\t\\ea\\ee\nwhich is derived from the standard $L^2$-estimate to the following elliptic problem:\n\t\\be\\la{3.29}\\begin{cases}\n\t\t\\ka\\Delta \\te=\\frac{R}{\\ga-1}\\n\\dot\\te +R\\n\\te\\div\n\t\tu-\\lambda (\\div u)^2-2\\mu |\\mathfrak{D}(u)|^2 ,\\\\\n\t\t\\na \\theta\\cdot n|_{\\p\\O \\times (0,T)}=0 .\n\t\\end{cases}\\ee\n\n\n\n\n\nNext, it holds that for any $\\eta\\in (0,1],$\n\\be\\la{e3}\\ba I_2 =&\\lambda\\si^m\\int (\\div u)^2 \\te_t\ndx+\\lambda\\si^m\\int (\\div u)^2u\\cdot\\na\\te\ndx\\\\=&\\lambda\\si^m\\left(\\int (\\div u)^2 \\te\ndx\\right)_t-2\\lambda\\si^m \\int \\te \\div u \\div (\\dot u-u\\cdot\\na u)\ndx\\\\&+\\lambda\\si^m\\int (\\div u)^2u\\cdot\\na\\te\ndx \\\\=&\\lambda\\si^m\n\\left(\\int (\\div u)^2 \\te dx\\right)_t-2\\lambda\\si^m\\int \\te \\div u\n\\div \\dot udx\\\\&+2\\lambda\\si^m\\int \\te \\div u \\pa_i u^j\\pa_j u^i dx\n+ \\lambda\\si^m\\int u \\cdot\\na\\left(\\te (\\div u)^2 \\right)dx\n \\\\\n\\le &\\lambda\\left(\\si^m\\int (\\div u)^2 \\te dx\\right)_t-\\lambda\nm\\si^{m-1}\\si'\\int (\\div u)^2 \\te dx\\\\& +\\eta\\si^m\\|\\na \\dot\nu\\|_{L^2}^2+C(\\eta)\\si^m\\|\\te\\na u\\|_{L^2}^2+C\\si^m\\|\\na u\\|_{L^4}^4,\\ea\\ee\nand\n \\be \\la{e5}\\ba I_3&\\le 2\\mu\\left(\\si^m\\int\n|\\mathfrak{D}(u)|^2 \\te dx\\right)_t-2\\mu m\\si^{m-1}\\si'\\int\n|\\mathfrak{D}(u)|^2 \\te dx\n \\\\&\\quad+ \\eta\\si^m\\|\\na \\dot\nu\\|_{L^2}^2+C(\\eta)\\si^m\\|\\te\\na u\\|_{L^2}^2+C\\si^m\\|\\na u\\|_{L^4}^4 . \\ea\\ee\n\nFinally, Cauchy's inequality gives\n \\be\\la{e39}\\ba\n |I_4| \\le \\delta \\si^m \\int \\n |\\dot\\te|^2dx+C( \\delta,\\on)\\si^m \\|\\te\\na u\\|_{L^2}^2. \\ea\\ee\n\n\nSubstituting (\\ref{e2}) and \\eqref{e3}--(\\ref{e39}) into (\\ref{e1}), we obtain \\eqref{nle7}\nafter using \\eqref{h3} and choosing $\\de$ suitably\nsmall.\nThe proof of Lemma \\ref{a113.4} is completed.\n\\end{proof}\n\n\n\nWith the estimates \\eqref{an1}--\\eqref{nle7} (see Lemma \\ref{a113.4}) at hand, we are now in a position to prove the following estimate on $A_3(T).$\n\n\n\n\n\n\\begin{lemma}\\la{le6} Under the conditions of Proposition \\ref{pr1}, there exists a positive constant $\\ve_2$\ndepending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O,$ and $M$\nsuch that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as\nin Lemma \\ref{le2}, the following estimate holds: \\be\\la{b2.34} A_3(T) \\le C_0^{1\/6},\\ee provided $C_0\\le\n\\ve_2.$\n\\end{lemma}\n\n\n\\begin{proof}\n\tFirst, it follows from \\eqref{an2}, (\\ref{h18}), and (\\ref{p}) that\n\t\\be\\ba \\notag B_1(t)\n\t&\\ge C \\|\\na u\\|_{L^2}^2- C \\|P-\\bp\\|^2_{L^2} \\ge C \\|\\nabla u\\|_{L^2}^2-C(\\on) C_0^{1\/4},\\ea\\ee\n\twhich together with \\eqref{an1} and \\eqref{z1} implies that\n\t\\begin{align}\\label{dd}\n\t\t\\sup_{t\\in(0,T]}\\left(\\si \\|\\na u\\|_{L^2}^2\\right) + \\int_0^T\\int \\si\\n |\\dot u|^2 dx dt \\le C(\\hat{\\rho}, M)C_0^{1\/4}.\n\t\\end{align}\n\n\t\n\tFor $C_2$ as in (\\ref{ae0}), adding (\\ref{nle7}) multiplied by $C_2+1$ to (\\ref{ae0}) and choosing $ \\eta$ suitably small give\n\t\\be\\la{e8}\\ba\n\t&\\left(\\sigma^m\\varphi\\right)'(t) + \\sigma^m \\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dx\\\\\n\t&\\le - 2\\left(\\int_{\\p \\O} \\sigma^m( u \\cdot \\na n \\cdot u )G dS\\right)_t + C( \\on, M) (\\si^{m-1}\\si'+\\si^m ) \\|\\rho^{1\/2} \\dot u\\|_{L^2}^2\\\\\n\t&\\quad+ C(\\on, M) (\\|\\na u\\|_{L^2}^2 + \\|\\na \\te\\|_{L^2}^2)+C\\si^m \\|\\na u\\|^4_{L^4} + C (\\on)\\si^m \\|\\te\\na u\\|_{L^2}^2,\\\\\n\n\t\\ea\\ee\n\twhere $\\varphi(t)$ is defined by\n\t\\be\\la{wq3} \\varphi(t) \\triangleq \\|\\n^{1\/2}\\dot u\\|_{L^2}^2+(C_2+1) B_2(t).\\ee\n\tThen it follows from (\\ref{e6}) that\n\t\\be\\la{wq2} \\varphi(t) \\ge \\frac{1}{2}\\|\\n^{1\/2}\\dot u\\|_{L^2}^2 +\\frac{\\ka(\\ga-1)}{2R}\\|\\na\\te\\|_{L^2}^2-C(\\on, M)\\|\\na u\\|_{L^2}^2, \\ee\n\twhere one has used that for any $\\de\\in(0,1]$,\n\t\\be \\la{2.48}\\ba \\int\n\t\\te |\\na u|^2dx & \\le\n\tC\\int|R\\te-\\bp||\\na u|^2dx+C\\bp \\int |\\na u|^2dx\\\\ &\\le C\n\t\\|R\\te-\\bp \\|_{L^6}\\|\\na u\\|_{L^2}^{3\/2}\n\t\\|\\na u\\|_{L^6}^{1\/2}+ C \\|\\na u\\|_{L^2}^2\n\t\\\\ &\\le C(\\on)\\|\\na\\te\\|_{L^2}\\|\\na u\\|_{L^2}^{3\/2}\n\t\\left(\\|\\n^{1\/2} \\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+ \\|\\na\\te\\|_{L^2}+1\\right)^{1\/2}\\\\\n\t&\\quad+ C \\|\\na u\\|_{L^2}^2\\\\ &\\le \\de\n\t\\left( \\|\\na\\te\\|^2_{L^2} + \\|\\n^{1\/2} \\dot\n\tu\\|_{L^2}^2 \\right) + C(\\de,\\on,M) \\|\\na\n\tu\\|_{L^2}^2 \\ea\\ee\n\tdue to \\eqref{key}, (\\ref{pq}), (\\ref{3.30}), and (\\ref{z1}).\n\t\n\t\n\tNext, it follows from \\eqref{key}, (\\ref{pq}), \\eqref{z1}, and (\\ref{3.30}) that\n\t\\be \\la{m20}\\ba\n\t\\|\\te\\na u\\|_{L^2}^2\n\t&\\le C\\|R\\te-\\bp \\|_{L^6}^2 \\|\\na u\\|_{L^2} \\|\\na u\\|_{L^6} + C\\bp^2 \\|\\na u\\|_{L^2}^2 \\\\\n\t&\\le C(\\on,M) \\left( \\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right)\\left( \\|\\n^{1\/2}\\dot u\\|^2_{L^2} +\n\t\\|\\na\\te\\|^2_{L^2} +1\\right).\\ea\\ee\n\tAnd, by virtue of (\\ref{h17}), (\\ref{pq}), and (\\ref{z1}), one gets\n\t\\be\\la{ae9}\\ba\n\t&\\|\\na u\\|_{L^4}^4\\\\\n\t&\\le C\\|\\n\\dot u\\|_{L^2}^3(\\|\\na u\\|_{L^2}+\\|P-\\bp\\|_{L^2})+C(\\|\\na u\\|_{L^2}^4+\\|P-\\bp\\|_{L^4}^4)\\\\\n\t&\\le C(\\on)\\|\\n^{1\/2}\\dot u\\|_{L^2}^3\\left(\\|\\na u\\|_{L^2}+1\\right) +C(\\on)\\|\\na\\te\\|_{L^2}^3+C\\|\\n-1\\|_{L^4}^4 +C\\|\\na u\\|_{L^2}^4\\\\\n\t&\\le C(\\on,M)\\left( \\|\\n^{1\/2}\\dot u\\|_{L^2}^3 + \\|\\na \\te\\|_{L^2}^3\\right) +C(\\on)\\|\\n-1\\|_{L^2}^2+C(\\on,M)\\|\\na u\\|_{L^2}^2,\n\t\\ea\\ee\n\twhich together with (\\ref{z1}) yields\n\t\\be \\la{m22}\\ba\n\t\\si \\|\\na u\\|_{L^4}^4 &\\le C(\\on,M) \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2 + \\|\\na \\te\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2\\right)\n\t+C(\\on)\\si\\|\\n-1\\|_{L^2}^2.\n\t\\ea\\ee\n\n\nThus, taking $m=2$ in \\eqref{e8}, one obtains after using \\eqref{z1}, \\eqref{m20}, and \\eqref{m22} that\n\\be\\la{e25}\\ba\n& \\left(\\sigma^2\\varphi\\right)'(t) + \\sigma^2 \\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dx\\\\\n&\\le - 2\\left(\\int_{\\p \\O} \\sigma^2 (u \\cdot \\na n \\cdot u) G dS\\right)_t+ C(\\on,M)\\si\\|\\n^{1\/2}\\dot u\\|_{L^2}^2\\\\\n&\\quad + C(\\on, M)\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right)+C(\\on)\\si\\|\\n-1\\|_{L^2}^2.\n\t\\ea\\ee\n\t\n\n\t\n\tNow, we deduce from \\eqref{h19}, \\eqref{z1}, and \\eqref{b2} that\n\t\\be \\ba\\label{jia4}\n\t\\sup_{ 0\\le t\\le T} \\left|\\int_{\\p \\O} \\sigma^2 u \\cdot \\na n \\cdot u G dS\\right|\n\n\t\\le & C(\\on) \\sup_{ 0\\le t\\le T} (\\si \\|\\na u\\|_{L^2}^2) \\sup_{ 0\\le t\\le T} (\\si \\|\\rho^{1\/2} \\dot u\\|_{L^2})\\\\\n\t\\le & C(\\on) C_0^{1\/4}.\n\t\\ea \\ee\n\tFurthermore, note that (\\ref{hj1}) is equivalent to\n \\be\\nota\n\\bp(\\n-1)=-G+(2\\mu+\\lambda)\\div u-\\n(R\\te-\\bp),\n\\ee\n this together with \\eqref{key}, \\eqref{z1}, \\eqref{1h19}, \\eqref{pq}, and \\eqref{dd} implies\n\\be\\ba\\la{67}\n&\\int_0^T\\si\\|\\n-1\\|_{L^2}^2 dt\\\\\n&\\le C\\int_0^T\\si(\\|G\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2) dt+C(\\on)\\int_0^T\\|R\\te-\\bp\\|_{L^2}^2 dt\\\\\n&\\le C(\\on)\\int_0^T\\left(\\si\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) dt\\\\\n&\\le C(\\on,M)C_0^{1\/4}.\n\\ea\\ee\n\t\n\tThus, integrating \\eqref{e25} over $(0,T)$, one obtains after using \\eqref{dd}, \\eqref{wq2}, (\\ref{jia4}), (\\ref{67}), and (\\ref{z1}) that\n\t\\be\\ba\\notag\n\tA_3(T)\\le C(\\on,M)C_0^{1\/4}\\le C_0^{1\/6},\n\t\\ea\\ee\n provided\n\t\\be \\ba\\notag C_0\\le\\ve_2\\triangleq \\min\\{1,(C(\\on,M))^{-12}\\}.\\ea\\ee\n\n \n\n\n The proof of Lemma \\ref{le6} is completed.\n \\end{proof}\n\n\n\n\n\n\nNext, in order to control $A_2(T)$, we first re-establish the basic energy estimate for short time $[0, \\si(T)]$, and then show that the spatial $L^2$-norm of $R\\te-\\overline P$ could be bounded by the combination of the initial energy and the spatial $L^2$-norm of $\\na \\te$, which is indeed the key ingredient to estimate $A_2(T)$.\n\n\\begin{lemma}\\la{a13} Under the conditions of Proposition \\ref{pr1},\n\tthere exist positive constants $C$ and $\\varepsilon_{3,1}$\n\tdepending only on\n\t$\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O, $ and $M$ such\n\tthat if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as\nin Lemma \\ref{le2},\n\tthe following estimates hold:\n\t\\be \\la{a2.121} \\ba\n\t&\\sup_{0\\le t\\le \\si(T)}\\int\\left( \\n |u|^2+(\\n-1)^2 + \\n(\\te-\\log \\te-1) \\right)dx\\le C C_0,\\ea\\ee\n\nand\n\\be \\la{a2.17} \\ba\n\\|(R\\te-\\bp)(\\cdot,t)\\|_{L^2} \\le C \\left(C_0^{1\/2} +C_0^{1\/3}\\|\\na\\te(\\cdot,t)\\|_{L^2}\\right),\n\t\\ea\\ee\nfor all $t\\in(0,\\si(T)]$, provided $C_0\\le\\varepsilon_{3,1}.$\n\t\\end{lemma}\n\\begin{proof}\n\tThe proof is divided into the following two steps.\n\t\n{\\it Step 1: The proof of (\\ref{a2.121}).}\n\t\n\tFirst, multiplying (\\ref{a11}) by $u$, one deduces from integration by parts, \\eqref{a1}$_1$, and \\eqref{cz1} that\n\t\\be \\la{a2.12} \\ba\n\t&\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+R(1+\\n\\log \\n-\\n)\n\t\\right)dx+ \\int(\\mu|\\curl u|^2+(2\\mu+\\lambda)(\\div u)^2)dx \\\\\n\t&= R \\int \\rho (\\te -1) \\div u dx\\\\\n\t&\\le \\de \\|\\na u\\|_{L^2}^2 + C(\\de, \\on) \\int \\n(\\te-1)^2dx\\\\\n&\\le\\de \\|\\na u\\|_{L^2}^2 + C(\\de, \\on)(\\|\\te(\\cdot,t)\\|_{L^{\\infty}} +1) \\int \\rho (\\te -\\log \\te -1)dx.\n\t\\ea\\ee\n\nUsing \\eqref{h18} and choosing $\\de$ small enough in \\eqref{a2.12}, it holds tha\n\t\\be \\la{a2.222} \\ba\n\t&\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+R(1+\\n\\log \\n-\\n)\n\t\\right)dx + C_3 \\int|\\na u|^2dx \\\\\n&\\le C(\\on) (\\|\\te(\\cdot,t)\\|_{L^{\\infty}} +1) \\int \\rho (\\te -\\log \\te -1)dx.\n\t\\ea\\ee\nThen, adding \\eqref{a2.222} multiplied by $(2\\mu+1) {C_3}^{-1}$ to \\eqref{la2.7}, one has\n\t\\be \\la{a2.22} \\ba\n\t\n\t\\xl((2\\mu+1) {C_3}^{-1}+1\\xr)\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+R(1+\\n\\log \\n-\\n)\\right)dx\n\t\\\\&+ \\frac{R}{\\ga-1} \\frac{d}{dt} \\int \\n(\\te-\\log \\te-1)dx+\\int|\\na u|^2dx\\\\\n\n\t &\\le C(\\on) (\\|\\te(\\cdot,t)\\|_{L^{\\infty}} +1) \\int \\rho (\\te -\\log \\te -1)dx.\n\t\\ea\\ee\n\t\nNext, we claim that\n\t\\be \\ba\\la{k}\n\t\\int_0^{\\si(T)}\\|\\te\\|_{L^\\infty}dt \\le C(\\on, M).\n\t\\ea \\ee\n\tCombining this with \\eqref{a2.22}, \\eqref{a2.9}, and Gr\\\"onwall inequality implies \\eqref{a2.121} directly.\n\t\n\tFinally, it remains to prove \\eqref{k}.\n\tTaking $m=1$ in \\eqref{e8} and integrating the resulting inequality, one deduces from \\eqref{wq2}, \\eqref{m20}, \\eqref{m22}, (\\ref{z1}), \\eqref{67}, \\eqref{h19}, and \\eqref{b2} that\n\\bnn \\ba\n& \\sigma \\varphi + \\int_0^t \\sigma\\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dxd\\tau\\\\\n&\\le 2\\sigma\\left|\\int_{\\p \\O} (u \\cdot \\na n \\cdot u )G dS\\right|(t)\n + C(\\on,M)\\int_0^t (\\|\\n^{1\/2} \\dot u\\|_{L^2}^2 +\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)d\\tau \\\\\n&\\quad+ C(\\on)\\int_0^t \\si \\|\\n-1\\|_{L^2}^2d\\tau+C(\\on)\\int_0^t \\left(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) \\si\\varphi d\\tau\\\\\n& \\le C(\\on) (\\sigma \\|\\na u\\|_{L^2}^2 \\|\\rho^{1\/2} \\dot u\\|_{L^2})(t)+ C(\\on,M)\\\\\n&\\quad+C(\\on,M)\\int_0^t \\left(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) \\si\\varphi d\\tau\\\\\n& \\le C(\\on,M)+C(\\on,M)\\int_0^t \\left(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right) \\si\\varphi d\\tau.\n\\ea\\enn\nThen Gr\\\"onwall inequality together with (\\ref{z1}) and \\eqref{wq2} yields\n\\be\\ba\\label{ae26}\n\t\\sup_{0\\le t\\le T}\\si \\left(\\int\\rho|\\dot{u}|^2dx+\\|\\na\\te\\|_{L^2}^2\\right)+\\int_0^T\\si \\int\\left(|\\na\\dot u|^2+\\n|\\dot\\te|^2\\right)dxdt\n\\le C(\\on,M).\\ea\\ee\n\n\n\n\n\n\n\n\n\n\n\t\n\t Next, it follows from (\\ref{lop4}), \\eqref{m22}, \\eqref{m20}, \\eqref{ae26}, (\\ref{z1}), and (\\ref{67}) that\n\n\n\n\n\n\n\t\\be\\ba \\la{k1}\n\t\\int_0^{T }\\si \\|\\na^2\\te\\|_{L^2}^2dt\n\t&\\le C(\\on,M)\\int_0^{T } \\left(\\si \\|\\n^{1\/2}\\dot\\te\\|_{L^2}^2+\n\t\\|\\n^{1\/2}\\dot u \\|_{L^2}^2 \\right) dt\\\\\n\t&\\quad+ C(\\on,M)\\int_0^{T } \\left( \\| \\na u\\|_{L^2}^2+\\| \\na\\te\\|_{L^2}^2+\\si\\|\\n-1\\|_{L^2}^2\\right) dt \\\\\n\t&\\le C(\\on,M).\n\t\\ea\\ee\n\tFurthermore, one deduces from \\eqref{g2}, \\eqref{g1}, and \\eqref{pq} that\n\t\\begin{equation}\n\t \\label{6yue}\\ba\n\t \\|R\\te-\\bp\\|_{L^\\infty} \\le& C \\|R\\te-\\bp\\|_{L^6}^{1\/2} \\|\\na\\te\\|_{L^6}^{1\/2} + \\|R\\te-\\bp\\|_{L^2}\\\\\n\t \\le& C(\\hat{\\rho}) \\|\\na \\te\\|_{L^2}^{1\/2} \\|\\na^2 \\te\\|_{L^2}^{1\/2} + C(\\hat{\\n}) \\|\\na \\te\\|_{L^2},\n\t \\ea\n\t\\end{equation}\n\twhich together with (\\ref{z1}) and \\eqref{k1} gives that\n\t\\be\\la{3.88}\\ba\n\t& \\int_0^{\\si(T)}\\|R\\te-\\bp\\|_{L^\\infty}dt \\\\\n\n\t&\\le C(\\hat{\\rho}) \\int_0^{\\si(T)}\\|\\na\\te\\|_{L^2}^{1\/2} \\left(\\si\\|\\na^2\\te\\|^2_{L^2}\\right)^{1\/4}\\si^{-1\/4}dt + C(\\hat{\\rho}) \\left(\\int_0^{\\si(T)} \\|\\na\\te\\|_{L^2}^2 dt\\right)^{1\/2}\\\\\n\t&\\le C(\\hat{\\rho}) \\left(\\int_0^{\\si(T)} \\|\\na \\te\\|_{L^2}^2dt \\int_0^{\\si(T)}\\si\\|\\na^2\\te\\|_{L^2}^2dt\\right)^{1\/4}\n+ C(\\on)C_0^{1\/8}\\\\\n\t&\\le C(\\on,M)C_0^{1\/16}.\n\t\\ea\\ee\n Combining this with \\eqref{key} yields \\eqref{k} directly.\n\n\n\n{\\it Step 2: The proof of (\\ref{a2.17}).}\n\nDirect calculations together with \\eqref{cz1} lead to\n \\be\\notag\\ba\n \\te-\\log\\te-1\n\n\\ge \\frac{1}{8} (\\te-1)1_{(\\te(\\cdot,t)>2)\n}+\\frac{1}{12}(\\te-1)^21_{(\\te(\\cdot,t)<3)}, \\ea\\ee with\n$(\\te(\\cdot,t)> 2)\\triangleq \\left.\\left\\{x\\in\n\\Omega\\right|\\te(x,t)> 2\\right\\}$ and $(\\te(\\cdot,t)< 3)\\triangleq\n\\left.\\left\\{x\\in \\Omega\\right|\\te(x,t)<3\\right\\}.$\n Combining this with \\eqref{a2.121} gives\n\t\\be \\la{a2.11}\\ba\n\t\\sup_{0\\le t\\le \\si(T)}\\int \\left(\\n(\\te-1)1_{(\\te(\\cdot,t)>2)}+\\n(\\te-1)^21_{(\\te(\\cdot,t)<3)}\\right)dx \\le C(\\hat\\n,M) C_0.\n\t\\ea\\ee\n\t\n\tNext, it follows from \\eqref{a2.11}, \\eqref{a2.121}, and the Sobolev inequality that for $t \\in (0,\\sigma(T)]$,\n\t\\begin{equation}\\la{la2.19}\n\t\\begin{aligned}\n\t &\\|\\te -1\\|_{L^2(\\te(\\cdot,t)<3)}^2 \\\\\n\t &\\le\\int \\n (\\te-1)^2 1_{(\\te(\\cdot,t)<3)}dx + \\left|\\int (\\n-1) (\\te -1)^2 dx \\right|\\\\\n\t&\\le C(\\on,M)C_0 + C\\|\\n-1\\|_{L^2} \\| \\te-1 \\|_{L^2}^{1\/2} \\|\\te -1\\|_{L^6}^{3\/2}\\\\\n\t&\\le C(\\on,M)C_0 + C(\\on,M)C_0^{1\/2} \\| \\te-1 \\|_{L^2}^{1\/2} \\left(\\| \\te-1 \\|_{L^2}+ \\|\\na \\te\\|_{L^2}\\right)^{3\/2}\\\\\n\t&\\le C(\\on,M)\\left(C_0+ C(\\delta) C_0^{2\/3} \\|\\na \\te\\|_{L^2}^2 + (\\delta+C_0^{1\/2}) \\| \\te-1 \\|_{L^2}^2 \\right),\n\t\\end{aligned}\n\t\\end{equation}\n\tand\n\t\\be\\la{a2.18}\\ba\n\t &\\|\\te-1\\|_{L^2(\\te(\\cdot,t)> 2)}^2\\\\\n\t&\\le \\|\\te-1\\|^{4\/5}_{L^1(\\te(\\cdot,t)> 2)} \\| \\te-1\\|_{L^6}^{6\/5}\\\\\n\n\t&\\le C(\\on,M) \\left(C_0 +C_0^{1\/2} \\|\\te-1\\|_{L^2}\\right)^{4\/5} (\\| \\te-1\\|_{L^2}+ \\|\\na \\te\\|_{L^2})^{6\/5}\\\\\n\t&\\le C(\\on,M)\\left(C_0+C(\\delta)C_0^{2\/3}\\|\\na \\te\\|_{L^2}^2 + (\\delta+C_0^{2\/5} ) \\|\\te -1\\|_{L^2}^2\\right),\n\t\\ea\\ee\n\twhere in the second inequality one has used\n\t\\be \\notag\\ba\n\t\\|\\te-1\\|_{L^1(\\te(\\cdot,t)>2)} \\le &\\int \\n (\\te-1) 1_{(\\te(\\cdot,t)>2)}dx + \\int \\left|(\\n-1) (\\te -1) \\right|dx \\\\\n\t\\le & C(\\on,M)(C_0 +C_0^{1\/2} \\|\\te-1\\|_{L^2}).\n\t\\ea\\ee\n\tHence, adding \\eqref{la2.19} with \\eqref{a2.18} together and choosing $\\delta$ small enough in the resulting inequality, one has for any $t\\in (0,\\sigma(T)],$\n\t\\be\\ba\\notag\n\t\\|\\te-1\\|_{L^2}^2 \\le C(\\on,M)\\left(C_0+C_0^{2\/3}\\|\\na \\te\\|_{L^2}^2 + C_0^{2\/5} \\|\\te -1\\|_{L^2}^2\\right),\n\t\\ea\\ee\n\twhich implies that\n\t\\be\\ba\\la{nnn1}\n\t\\|\\te-1\\|_{L^2}^2 \\le C(\\on,M)\\left(C_0+C_0^{2\/3}\\|\\na \\te\\|_{L^2}^2 \\right),\n\t\\ea\\ee\n\tprovided\n\t\\begin{equation}\\label{31}\n\tC_0\\le \\ve_{3,1} \\triangleq\\min\\left\\{1,(2C(\\on,M))^{-5\/2}\\right\\}.\n\t\\end{equation}\n\t\n\tFinally, note that\n\t\\be \\ba\\notag\n\t\\| R\\te-\\bp \\|_{L^2} &\\le R \\|\\te -1\\|_{L^2}\t+C|1-\\overline{\\n\\te}|\\\\\n\t&\\le R \\|\\te -1\\|_{L^2}+C\\left|\\int\\n(1-\\te)dx\\right|\\\\\n\t&\\le C(\\on) \\|\\te -1\\|_{L^2},\n\t\\ea\\ee\n\tthis together with \\eqref{nnn1} yields (\\ref{a2.17}).\n\t\n\tThe proof of Lemma \\ref{a13} is completed.\n\\end{proof}\n\n\nNext, with the help of \\eqref{a2.17}, the estimate on $A_2(T)$ will be handled smoothly.\n\n\n\n\\begin{lemma}\\la{le3}\nUnder the conditions of Proposition \\ref{pr1}, there exists a positive constant $\\ve_3$ depending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O$, and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}, the following estimate holds:\n\\be\\la{a2.34} A_2(T) \\le C_0^{1\/4},\\ee\nprovided $C_0\\le \\ve_3.$\n\\end{lemma}\n\n\n\n\\begin{proof}\nTo begin with, multiplying (\\ref{a11}) by $u$ and integrating by parts give that\n\\be \\la{a2.225}\\ba\n&\\frac{d}{dt}\\int\\left(\\frac{1}{2}\\n |u|^2+\\bp(1+\\n\\log \\n-\\n)\\right)dx\\\\\n&\\quad+ \\int\\left(\\mu|\\curl u|^2+(2\\mu+\\lambda)(\\div u)^2\\right)dx \\\\\n&= \\bp_t \\int (1+\\n\\log \\n-\\n) dx + \\int \\rho (R \\te -\\bp) \\div u dx.\n\\ea\\ee\nNext, multiplying $(\\ref{a1})_3$ by $\\bp^{-1}(R\\te-\\bp)$, one obtains after integrating the resulting\nequality over $\\Omega $ by parts that\n\\be\\la{a2.231} \\ba\n& \\frac{1}{2(\\ga-1)} \\frac{d}{dt}\\int\n{\\bp}^{-1}\\n {(R\\te-\\bp)^2}dx+ {\\ka}R \\bp^{-1} \\|\\na\\te\\|_{L^2}^2\\\\\n&= - \\frac{1}{\\ga-1} {\\bp}^{-1} \\bp_t \\int\n\\n {(R\\te-\\bp)} dx - \\frac{1}{2(\\ga-1)} \\bp^{-2} \\bp_t \\int \\n {(R\\te-\\bp)^2} dx\\\\\n& \\quad- {\\bp}^{-1} \\int \\n {(R\\te-\\bp)^2} \\div u dx - \\int \\rho (R\\te -\\bp) \\div u dx \\\\\n&\\quad+ {\\bp}^{-1} \\int {(R\\te-\\bp)} (\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2) dx.\n\\ea\\ee\nAdding \\eqref{a2.225} and \\eqref{a2.231} together yields that\n\\be\\la{a2.23} \\ba\n& \\frac{d}{dt}\\int\n\\left(\\frac{1}{2}\\n |u|^2+\\bp (1+\\n\\log \\n-\\n)+\\frac{1}{2(\\ga-1)} \\n \\bp^{-1}{(R\\te-\\bp)^2} \\right)dx\\\\\n&\\quad+ \\mu \\|\\curl u\\|_{L^2}^2 +(2\\mu+\\lambda)\\|\\div u\\|_{L^2}^2+ \\ka R \\bp^{-1} \\|\\na\\te\\|_{L^2}^2\\\\\n&= - \\frac{1}{\\ga-1} {\\bp}^{-1} \\bp_t \\int\n\\n {(R\\te-\\bp)} dx - \\frac{1}{2(\\ga-1)} {\\bp}^{-2} \\bp_t \\int \\n {(R\\te-\\bp)^2} dx\\\\\n& \\quad+\\bp_t \\int (1+\\n\\log \\n-\\n) dx - \\bp^{-1}\\int \\n {(R\\te-\\bp)^2} \\div u dx \\\\\n&\\quad+ \\bp^{-1} \\int {(R\\te-\\bp)} (\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2) dx\\triangleq \\sum_{i=1}^{5} J_i.\n\\ea\\ee\n\nThe terms $J_i \\,(i=1,\\cdots,5)$ can be estimated as follows.\n\nIt follows from \\eqref{key}, \\eqref{511}, \\eqref{z1}, and \\eqref{pq} that\n\\be \\ba\\la{aa1}\nJ_1+J_2 \\le & C|\\overline P_t|\\left(\\| \\n (R\\te-\\bp)\\|_{L^2}+\\| \\n^{1\/2} (R\\te-\\bp)\\|_{L^2}^2\\right)\\\\\n\\le &C(\\on)\\left(C_0^{1\/8} \\|\\na u \\|_{L^2}+\\| \\na u\\|_{L^2}^2 \\right)\\|\\n^{1\/2}(R\\te-\\bp)\\|_{L^2}\\\\\n\\le & C(\\on) C_0^{1\/8} \\left( \\|\\na u\\|_{L^2}^2+\\| \\na \\te \\|_{L^2}^2\\right),\n\\ea \\ee\nand\n\\be \\ba\\la{aa4}\nJ_4 \\le & C \\|\\n^{1\/2} {(R\\te-\\bp)}\\|_{L^2}^{1\/2} \\|\\n^{1\/2} {(R\\te-\\bp)}\\|_{L^6}^{3\/2} \\|\\na u\\|_{L^2} \\\\\n\\le & C(\\on) A_2^{1\/4}(T) \\|\\na \\te\\|_{L^2}^{3\/2} \\|\\na u\\|_{L^2}\\\\\n\\le & C(\\on,M) C_0^{1\/16} (\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^{2}).\n\\ea \\ee\nFurthermore, by virtue of \\eqref{a2.9}, \\eqref{511}, and \\eqref{a2.112}, we have\n\\be \\ba\\la{aa3}\nJ_3 \\le & |\\overline P_t|\\left|\\int (1+\\n \\log\\n-\\n )dx\\right|\\\\\n\\le & C(\\on)\\left( C_0^{1\/8}\\|\\na u \\|_{L^2}+\\| \\na u\\|_{L^2}^2 \\right)\\| \\n -1\\|_{L^2}^2\\\\\n\\le & C(\\on) C_0^{1\/4} \\|\\na u\\|_{L^2}^2 + C(\\on) C_0^{1\/4} \\| \\n -1\\|_{L^2}^2.\n\\ea \\ee\n\n\nNow, we will estimate the term $J_5$ for the short time $t\\in [0, \\si(T))$ and the large time $t\\in [\\si(T),T]$, respectively.\n\nFor $t\\in[0, \\si(T))$, it follows from \\eqref{key}, \\eqref{3.30}, \\eqref{pq}, \\eqref{a2.17}, and \\eqref{z1} that\n\n\\be \\ba\\la{aa5}\nJ_5 \\le& C \\int |R\\te-\\bp| |\\na u|^2dx\\\\\n\\le & C \\| {R\\te-\\bp}\\|_{L^2}^{1\/2} \\|R\\te-\\bp\\|_{L^6}^{1\/2} \\|\\na u\\|_{L^2} \\|\\na u\\|_{L^6}\\\\\n\\le & C(\\on) \\|{R\\te-\\bp}\\|_{L^2}^{1\/2} \\|\\na \\te\\|_{L^2}^{1\/2} \\|\\na u\\|_{L^2}\\\\\n & \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}+\\|\\na u\\|_{L^2}+\\|\\na\\te\\|_{L^2}+C_0^{1\/24}\\right)\\\\\n\\le & C (\\on) \\| {R\\te-\\bp}\\|_{L^2}^{1\/2} \\|\\na \\te\\|_{L^2}^{1\/2} \\|\\na u\\|_{L^2}\\|\\n^{1\/2}\\dot u\\|_{L^2}\\\\\n&+C(\\on,M) C_0^{1\/24} (\\|\\na \\te\\|_{L^2}^{2}+ \\|\\na u\\|_{L^2}^2)\\\\\n\\le & C(\\on,M) C_0^{7\/24} \\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 + C(\\on,M)C_0^{1\/24} (\\|\\na u\\|_{L^2}^2 + \\|\\na \\te\\|_{L^2}^2),\n\\ea \\ee\nwhere we have used following calculations:\n\\be \\ba\\notag\n&\\| {R\\te-\\bp}\\|_{L^2}^{1\/2} \\|\\na \\te\\|_{L^2}^{1\/2} \\|\\na u\\|_{L^2} \\|\\n^{1\/2}\\dot u\\|_{L^2} \\\\\n&\\le C(\\on,M)(C_0^{1\/4}\\|\\na\\te\\|_{L^2}^{1\/2}+C_0^{1\/6}\\|\\na\\te\\|_{L^2})\\|\\na u\\|_{L^2}\\|\\n^{1\/2}\\dot u\\|_{L^2}\\\\\n&\\le C(\\on,M) C_0^{7\/24} \\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 + C(\\on,M)C_0^{1\/12} \\|\\na u\\|_{L^2}^2 + C(\\on,M)C_0^{1\/24} \\|\\na \\te\\|_{L^2}^2\n\\ea \\ee\nowing to \\eqref{a2.17}.\n\nFor $t\\in[\\si(T), T]$, it holds that\n\\be \\ba\\la{aa6}\nJ_5\n\\le &C\\|R\\te-\\bp\\|_{L^3}\\|\\na u\\|_{L^2}\\|\\na u\\|_{L^6}\n\\le C(\\on)C_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2),\n\\ea \\ee\nwhere one has used \\eqref{pq} and the following fact:\n\\be\\notag\\ba \\sup_{0\\le t\\le T}\\xl(\\si\\|\\na\nu\\|_{L^6}\\xr)\n&\\le C(\\on)C_0^{1\/24} \\ea\\ee\ndue to \\eqref{z1} and \\eqref{3.30}.\n\n\nFinally, substituting \\eqref{aa1}--\\eqref{aa6} into \\eqref{a2.23}, one obtains after using \\eqref{z1}, \\eqref{key}, and \\eqref{jia10} that\n\\be \\ba \\label{jia9}\n& \\sup_{ 0\\le t\\le T} \\int \\left(\\frac{1}{2}\\n |u|^2+\\bp (1+\\n\\log \\n-\\n)+\\frac{1}{2(\\ga-1)} \\n \\bp^{-1}{(R\\te-\\bp)^2}\n\\right)dx\\\\\n&+ \\int_{0}^{T} (\\mu \\|\\curl u\\|_{L^2}^2 +(2\\mu+\\lambda)\\|\\div u\\|_{L^2}^2+ {R\\ka}{\\bp}^{-1} \\|\\na\\te\\|_{L^2}^2)dt\\\\\n&\\le C(\\on,M) C_0^{1\/24} \\int_{ 0}^{T} ( \\|\\na u\\|_{L^2}^2+\\| \\na \\te \\|_{L^2}^2 ) dt+ C(\\on) C_0^{1\/4}\\int_{ 0}^{T} \\| \\n -1\\|_{L^2}^2 dt\n\\\\&\\quad + C(\\on,M) C_0^{7\/24} \\int_{0}^{\\si(T)} \\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 dt+C(\\on,\\bt)C_0\\\\\n&\\le C(\\on,\\bt, M) C_0^{7\/24},\n\\ea \\ee\nwhere one has used\n\\be\\la{xx}\n\\int_{ 0}^{T} \\| \\n -1\\|_{L^2}^2 dt \\le \\sup_{0\\le t\\le\\si(T)} \\|\\n -1\\|_{L^2}^2 + \\int_{\\si(T)}^{T} \\| \\n -1\\|_{L^2}^2 dt \\le C(\\on,M) C_0^{1\/4}\n\\ee\ndue to \\eqref{a2.121} and \\eqref{67}. Thus, one deduces from \\eqref{jia9}, \\eqref{h18}, and \\eqref{key} that\n\\be \\la{kyu1} A_2(T)\\le C(\\on,\\hat{\\theta}, M) C_0^{7\/24} \\ee which implies \\eqref{a2.34}\nprovided \\be \\ba \\notag C_0\\le \\ve_3\\triangleq\\min\n\\left\\{\\ve_{3,1}, (C(\\on,\\hat{\\theta},M))^{-24}\\right\\},\\ea\\ee\nwith $\\ve_{3,1}$ as in \\eqref{31}. The proof\nof Lemma \\ref{le3} is completed.\n\\end{proof}\n\n\\begin{remark}\\la{r2}\n\tIt's worth noticing that the energy-like estimate $A_2(T)$ is a little subtle, since $A_2(T)$ is not a conserved quantity for the full Navier-Stokes system owing to the nonlinear coupling of $\\te$ and $u$. Thus, further consideration is needed to handle this issue.\n\tMore precisely,\n\t\\begin{itemize}\n\t\t\\item on the one hand, while deriving the kinetic energy (see (\\ref{a2.225})), we need to deal with the following term\n\t\t$$\\int \\rho (R\\te -\\bp) \\div u dx.$$\n\t\tUnfortunately, this term is troublesome for large time $t\\in [\\sigma(T), T]$. In fact, this term could be bounded by $\\|\\na \\te \\|_{L^2}\\|\\na u \\|_{L^2}$, which will only be of the same order as $C_0^{1\/4}$ with the help of all a priori estimates (\\ref{z1}). Therefore, we can not handle this term directly. Here, based on careful analysis on system (\\ref{a1}), we find that this term can be cancelled by a suitable combination of kinetic energy and thermal energy, see (\\ref{a2.225})--(\\ref{a2.23});\n\t\t\\item on the other hand, while deriving the thermal energy (see (\\ref{a2.231})), we need to handle the following term\n\t\t$$\\int (R\\te - \\bp) (\\div u)^2 dx.$$\n\t\tNote that for short time $t\\in [0, \\sigma(T))$, the ``weaker\" basic energy estimate (\\ref{a2.112}) is not enough, hence it's necessary to re-establish the basic energy estimate (\\ref{a2.121}), which is obtained by the a priori $L^1(0,\\sigma(T);L^\\infty)$-norm of $\\te$ (see (\\ref{k})). Consequently, we can obtain (\\ref{a2.17}) as a consequence of (\\ref{a2.121}) and then handle this term for short time $t\\in [0, \\sigma(T))$ (see (\\ref{aa5})).\t\n\t\n\t\n\t\\end{itemize}\n\t\n\tMoreover, it should be mentioned that the uniform positive lower and upper bounds of $\\bp$ also play a critical role in estimating $A_2(T)$.\n\\end{remark}\n\n\n\n\n\nWe now proceed to derive a uniform (in time) upper bound for the density, which turns out to be the key to obtaining all the higher order estimates and thus extending the classical solution globally.\n\n\\begin{lemma}\\la{le7}\nUnder the conditions of Proposition \\ref{pr1}, there exists a positive constant $\\ve_4$ depending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O$, and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}, the following estimate holds:\n\\be \\la{a3.7}\n\\sup_{0\\le t\\le T}\\|\\n(\\cdot,t)\\|_{L^\\infty} \\le\n\\frac{3\\on }{2},\n\\ee\nprovided $C_0\\le \\ve_4$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, it follows from \\eqref{k1}, \\eqref{6yue}, and \\eqref{z1} that\n\\be\\la{3.89}\\ba\n\\int_{\\si(T)}^T\\|R\\te-\\bp\\|^2_{L^\\infty}dt\n\\le& C(\\on) \\left(\\int_{\\si(T)}^T\\|\\na\\te\\|^2_{L^2}dt\\right)^{1\/2}\\left(\\int_{\\si(T)}^T \\|\\na^2\\te\\|^2_{L^2}dt\\right)^{1\/2} \\\\\n&+ C(\\on) \\int_{\\si(T)}^T\\|\\na\\te\\|^2_{L^2}dt \\\\\n\\le& C(\\on,M) C_0^{1\/8}.\n\\ea\\ee\n\n\nNext, it follows from (\\ref{h19}), \\eqref{tb90}, (\\ref{ae26}), and (\\ref{z1}) that\n\\be\\la{3.90}\\ba &\\int_0^{\\si(T)}\\|G\\|_{L^\\infty}dt\\\\\n&\\le C\\int_0^{\\si(T)}\\|\\na G\\|_{L^2}^{1\/2} \\|\\na G\\|_{L^6}^{1\/2}dt\\\\\n&\\le C(\\on)\\int_0^{\\si(T)}\\|\\n \\dot u\\|_{L^2}^{1\/2}(\\|\\na\\dot u\\|_{L^2}+ \\|\\na u\\|_{L^2}^2)^{1\/2}dt\\\\\n&\\le C(\\on)\\int_0^{\\si(T)}\\left(\\si\\|\\n \\dot u\\|_{L^2}\\right)^{1\/4} \\left(\\si\\|\\n \\dot u\\|^{2}_{L^2}\\right)^{1\/8} \\left(\\si\\|\\na \\dot u\\|^2_{L^2}\\right)^{1\/4}\\si^{-5\/8}dt\\\\\n& \\quad+ C(\\on) \\int_0^{\\si(T)}\\left(\\si \\|\\n \\dot u\\|_{L^2}\\right)^{1\/2} \\|\\na u\\|_{L^2} \\si^{-1\/2} dt\\\\\n&\\le C(\\on,M)C_0^{1\/48}\\left(\\int_0^{\\si(T)} \\si\\|\\na \\dot u\\|^2_{L^2} dt\\right)^{1\/4}\\left(\\int_0^{\\si(T)} \\si^{-5\/6}dt\\right)^{3\/4} \\\\\n&\\quad+ C(\\on,M) C_0^{1\/24} \\int_0^{\\si(T)}\n\\si^{-1\/2} dt\\\\\n&\\le C(\\on,M)C_0^{1\/48},\n\\ea\\ee\nand\n \\be\\la{3.91}\\ba \\int_{\\si(T)}^T\\|G\\|^2_{L^\\infty}dt\n &\\le C\\int_{\\si(T)}^T\\|\\na G\\|_{L^2} \\|\\na G\\|_{L^6} dt\n \\\\ &\\le C(\\on,M)\\int_{\\si(T)}^T\\left(\\|\\n^{1\/2} \\dot u\\|^2_{L^2}+\n \\|\\na\\dot u\\|_{L^2}^2 + \\|\\na u\\|_{L^2}^2\\right)dt\\\\ &\\le C(\\on,M)C_0^{1\/6}.\n \\ea\\ee\n\n\n\nDenoting $ D_t\\n=\\n_t+u \\cdot\\nabla \\n $ and using (\\ref{hj1}), one can rewrite $(\\ref{a1})_1$ as follows\n\\bnn\\ba\n(2\\mu+\\lambda) D_t \\n&=-\\bp \\n(\\n-1)- \\n^2(R\\te-\\bp)-\\n G\\\\\n&\\le -\\bp (\\n-1)+C(\\on)\\|R\\te-\\bp\\|_{L^\\infty}+C(\\on)\\| G\\|_{L^\\infty},\n\\ea\\enn\nwhich gives\n\\be\\la{3.92}\\ba\nD_t (\\n-1)+\\frac{\\bp }{2\\mu+\\lambda} (\\n-1)\\le C(\\on)\\|R\\te-\\bp\\|_{L^\\infty}+C(\\on)\\| G\\|_{L^\\infty}.\n\\ea\\ee\n\n\n\nFinally, applying Lemma \\ref{le1} with\n$$y=\\n-1, \\quad\\al=\\frac{\\bp}{2\\mu+\\lambda} ,\\quad g=C(\\on)\\|R\\te-\\bp\\|_{L^\\infty}+C(\\on)\\| G\\|_{L^\\infty},\\quad T_1=\\si(T),$$\n we thus deduce from (\\ref{3.92}), (\\ref{3.88}), \\eqref{3.89}--(\\ref{3.91}), (\\ref{2.34}), and \\eqref{key} that\n \\bnn\\ba\\n\n & \\le \\on+1 +C\\left(\\|g\\|_{L^1(0,\\si(T))}+\\|g\\|_{L^2(\\si(T),T)}\\right) \\le \\on+1 +C(\\on,M)C_0^{1\/48} ,\n \\ea\\enn\n which gives \\eqref{a3.7}\n provided\n \\be \\notag C_0\\le \\ve_4\\triangleq\\min\\left\\{1,\\left(\\frac{\\hat \\n-2 }{2C(\\on,M) }\\right)^{48}\\right\\}.\\ee\n The proof of Lemma \\ref{le7} is completed.\n\\end{proof}\n\n\n\n\n\n\n\nNext, we summarize some uniform estimates on $(\\n,u,\\te)$ which will be useful for higher-order ones in the next section.\n\\begin{lemma}\\la{le8}\nUnder the conditions of Proposition \\ref{pr1}, there exists a positive constant $C $ depending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R$, $\\ga,\\, \\on,\\,\\bt,\\, \\O$, and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}, the following estimate holds:\n\t\\be \\la{ae3.7}\\sup_{0< t\\le T}\\si^2\\int \\n|\\dot\\te|^2dx + \\int_0^T\\si^2 \\|\\na\\dot\\te\\|_{L^2}^2dt\\le C.\\ee\nMoreover, it holds that\n\\be\\la{vu15}\\ba\n&\\sup_{0< t\\le T}\\left( \\si\\|\\na u \\|^2_{L^6}+\\si^2\\|\\te\\|^2_{H^2}\\right)\\\\\n&+\\int_0^T(\\si \\|\\na u \\|_{L^4}^4+\\si\\|\\na\\te \\|_{H^1}^2+\\si\\|u_t\\|_{L^2}^2+\\si^2\\|\\te_t\\|^2_{H^1}+\\|\\n -1\\|_{L^2}^2)dt\\le C.\n\\ea\\ee\n\\end{lemma}\n\n\\begin{proof}\nFirst, applying the operator $\\pa_t+\\div(u\\cdot) $ to (\\ref{a1})$_3 $ and using (\\ref{a1})$_1$, one gets\n\\be\\la{3.96}\\ba\n&\\frac{R}{\\ga-1} \\n \\left(\\pa_t\\dot \\te+u\\cdot\\na\\dot \\te\\right)\\\\\n&=\\ka \\Delta \\te_t +\\ka \\div (\\Delta \\te u)+\\left( \\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)\\div u +R\\n \\te \\pa_ku^l\\pa_lu^k\\\\\n&\\quad -R\\n \\dot\\te \\div u-R\\n \\te\\div \\dot u +2\\lambda \\left( \\div\\dot u-\\pa_ku^l\\pa_lu^k\\right)\\div u\\\\\n&\\quad + \\mu (\\pa_iu^j+\\pa_ju^i)\\left( \\pa_i\\dot u^j+\\pa_j\\dot u^i-\\pa_iu^k\\pa_ku^j-\\pa_ju^k\\pa_ku^i\\right).\n\\ea\\ee\nDirect calculations show that\n\\be \\ba\\la{bea}\n \\int (\\Delta \\te_t + \\div (\\Delta \\te u)) \\dot \\te dx &= - \\int (\\na \\te_t \\cdot \\na \\dot\\te + \\Delta \\te u \\cdot \\na \\dot \\te) dx\\\\\n&= - \\int |\\na \\dot\\te|^2 dx + \\int ( \\na(u\\cdot \\na \\te) \\cdot \\na \\dot \\te - \\Delta \\te u \\cdot \\na \\dot \\te) dx.\n\\ea \\ee\nMultiplying (\\ref{3.96}) by $\\dot \\te$ and integrating the resulting equality over $\\O$, it holds that\n\\be\\la{3.99}\\ba\n& \\frac{R}{2(\\ga-1)}\\left(\\int \\n |\\dot\\te|^2dx\\right)_t + \\ka \\|\\na\\dot\\te\\|_{L^2}^2 \\\\\n&\\le C \\int|\\na \\dot \\te|\\left( |\\na^2\\te||u|+ |\\na \\te| |\\na u|\\right)dx+C\\int \\n|R\\te-\\bp| |\\na\\dot u| |\\dot \\te|dx\\\\\n&\\quad +C(\\on) \\int|\\na u|^2|\\dot\\te|\\left(|\\na u|+|R\\te-\\bp| \\right)dx+C \\int |\\na\\dot u|\\n|\\dot \\te| dx \\\\\n&\\quad +C (\\on) \\int\\left( |\\na u|^2|\\dot \\te|+\\n |\\dot\n\\te|^2|\\na u|+|\\na u| |\\na\\dot u| |\\dot \\te|\\right)dx \\\\\n&\\le C\\|\\na u\\|^{1\/2}_{L^2}\\|\\na u\\|^{1\/2}_{L^6}\\|\\na^2\\te\\|_{L^2}\\|\\na \\dot \\te\\|_{L^2}+C(\\on)\\|\\na\\te\\|_{L^2} \\|\\na\\dot u\\|_{L^2} \\|\\dot\\te\\|_{L^6}\\\\\n&\\quad+C(\\on) \\|\\na u\\|_{L^2}\\|\\na u\\|_{L^6}\\left(\\|\\na u\\|_{L^6}+\\|\\na \\te\\|_{L^2}\\right)\n\\|\\dot\\te\\|_{L^6} +C \\|\\na\\dot u\\|_{L^2} \\|\\n\\dot\\te\\|_{L^2} \\\\\n&\\quad+C(\\on) \\|\\na u\\|^{1\/2}_{L^6}\\|\\na u\\|^{1\/2}_{L^2} \\|\\dot\\te\\|_{L^6}\\left(\\|\\na u\\|_{L^2}\n+\\|\\n\\dot\\te\\|_{L^2}+\\|\\na\\dot u\\|_{L^2}\\right) \\\\\n&\\le\\frac{\\ka}{2}\\|\\na\\dot\\te\\|_{L^2}^2+C(\\on)\\|\\na u\\|_{L^2}^2\\left(\\|\\na u\\|_{L^6}^4+\\|\\na\\te \\|_{L^2}^4\\right)+C(\\on,M) \\|\\na u\\|_{L^6}\\|\\na u\\|_{L^2}^2\\\\\n&\\quad+C(\\on,M) \\left(1+\\|\\na u\\|_{L^6}+\\|\\na\\te\\|_{L^2}^2\\right) \\left(\\|\\na^2\\te\\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+\\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2\\right),\n\\ea\\ee\nwhere we have used \\eqref{bea}, \\eqref{g1}, \\eqref{g2}, \\eqref{z1}, \\eqref{pq}, and the following Poincar\\'e-type inequality (\\cite[Lemma 3.2]{feireisl1}):\n\\be \\la{kk}\n\\|f\\|_{L^p}\\le C(\\on)(\\|\\n^{1\/2}f\\|_{L^2}+\\|\\na f \\|_{L^2}),~~~p\\in[2,6],\n\\ee\nfor any $f\\in\\{h\\in H^1 \\left|\\n^{1\/2}h\\in L^2\\}\\right.$.\n\n\nMultiplying (\\ref{3.99}) by $\\si^2$ and integrating the resulting inequality over $(0,T),$\nwe obtain after integrating by parts that\n\\bnn\\ba\n& \\sup_{0\\le t\\le T}\\si^2\\int \\n|\\dot\\te|^2dx + \\int_0^T\\si^2 \\|\\na\\dot\\te\\|_{L^2}^2dt \\\\\n&\\le C(\\on) \\sup_{0\\le t\\le T} \\left(\\si^2(\\|\\na u\\|_{L^6}^4+\\|\\na\\te\\|_{L^2}^4)\\right)\\int_0^T\\|\\na u\\|_{L^2}^2dt \\\\\n&\\quad + C(\\on,M) \\sup_{0\\le t\\le T} \\left(\\si\\left(1+\\|\\na u\\|_{L^6}+\\|\\na\\te\\|_{L^2}^2\\right)\\right)\\\\\n&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\cdot\\int_0^T\\si\\left(\\|\\na^2\\te\\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+\\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2\\right)dt\\\\\n&\\quad +C(\\on,M) \\sup_{0\\le t\\le T} \\left(\\si\\|\\na u\\|_{L^6} \\right) \\int_0^T\\|\\na u\\|_{L^2}^2dt+C\\int_0^T\\si\\|\\rho^{1\/2} \\dot \\te\\|_{L^2}^2dt\\\\\n&\\le C(\\on,M), \\ea\\enn\nwhere we have used (\\ref{z1}), (\\ref{ae26}), (\\ref{k1}), and the following fact:\n\\be\\ba\\la{ong}\\sup_{0\\le t\\le T}(\\si\\|\\na u\\|_{L^6}^2)\\le C(\\on,M)\\ea\\ee\ndue to \\eqref{3.30}, \\eqref{ae26}, and \\eqref{z1}.\n\nNext, it follows from (\\ref{z1}), \\eqref{ae26}, (\\ref{lop4}), (\\ref{m20}), (\\ref{m22}), (\\ref{ae3.7}), (\\ref{xx}), \\eqref{nnn1}, and (\\ref{k1}) that\n\\be \\la{vu02}\\ba\n\\sup_{0\\le t\\le T}\\left(\\si^2\\|\\te \\|^2_{H^2}\\right)+\\int_0^T \\left(\\si\\|\\na u \\|_{L^4}^4+\\si\\|\\na\\te \\|_{H^1}^2+\\|\\n -1\\|_{L^2}^2\\right)d\n \\le C(\\on,M), \\ea\\ee\nwhich along with (\\ref{z1}), \\eqref{ae26}, (\\ref{k1}), \\eqref{kk}, \\eqref{ong}, and \\eqref{ae3.7} gives\n \\be\\la{vu12}\\ba \\int_0^T \\si \\|u _t\\|_{L^2}^2dt\n &\\le C\\int_0^T \\si(\\| \\dot u \\|_{L^2}^2+\\|u\\cdot\\na u \\|_{L^2}^2)dt\\\\\n &\\le C(\\hat\\n)\\int_0^T \\si(\\|\\n^{1\/2} \\dot u \\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+\\|u \\|_{L^\\infty}^2\\|\\na u \\|_{L^2}^2)dt\\\\\n &\\le C(\\hat \\n,M) ,\\ea\\ee\n\\be\\la{vu11}\\ba\n\\int_0^T \\si^2 \\| \\te _t\\|_{L^2}^2dt\n&\\le C\\int_0^T \\si^2(\\| \\dot \\te \\|_{L^2}^2+\\|u \\cdot\\na \\te \\|_{L^2}^2)dt\\\\\n&\\le C(\\hat\\n)\\int_0^T \\si^2(\\|\\n^{1\/2} \\dot \\te\\|_{L^2}^2+\\|\\na\\dot\\te\\|_{L^2}^2+\\|u\\|_{L^6}^2\\|\\na\\te\\|_{L^3}^2)dt\\\\\n&\\le C(\\hat \\n,M) ,\\ea\\ee\n and\n\\be\\la{vu01}\\ba\n\\int_0^T \\si^2 \\| \\na\\te _t\\|_{L^2}^2dt\n&\\le C\\int_0^T \\si^2\\|\\na \\dot \\te \\|_{L^2}^2 dt+ C\\int_0^T \\si^2\\|\\na(u \\cdot\\na \\te )\\|_{L^2}^2dt\\\\\n&\\le C(\\on,M) +C\\int_0^T\\si^2\\left(\\|\\na u \\|_{L^3}^2+\\|u \\|_{L^\\infty}^2\\right)\\|\\na^2 \\te \\|_{L^2}^2dt \\\\ &\\le C(\\on,M).\\ea\\ee\nHence, (\\ref{vu15}) is derived from (\\ref{ong})--\\eqref{vu01} immediately.\nThe proof of Lemma \\ref{le8} is finished.\n\\end{proof}\n\n\nFinally, we end this section by establishing the exponential decay-in-time for the classical solutions.\n\\begin{lemma}\\la{pr2}\nUnder the conditions of Proposition \\ref{pr1}, there exist positive constants $\\ve_0$, $C^\\ast$, $\\al$, and $\\te_\\infty$ depending only on $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\, \\bt,\\,\\O,$ and $M$ such that if $(\\rho,u,\\te)$ is a smooth solution to the problem (\\ref{a1})--(\\ref{h1}) on $\\Omega\\times (0,T] $ satisfying (\\ref{z1}) with $K$ as in Lemma \\ref{le2}, (\\ref{h22}) holds for any $t\\geq 1$, provided $C_0\\le\\ve_0.$\n\\end{lemma}\n\\begin{proof}\nFirst, it follows from \\eqref{a2.23}, \\eqref{z1}, \\eqref{key}, \\eqref{a2.9}, \\eqref{511}, \\eqref{pq}, \\eqref{6yue}, and \\eqref{vu15} that for any $t\\geq 1$,\n\\be\\la{ao}\\ba\n&\\frac{1}{2}W'(t)+ \\mu \\|\\curl u\\|_{L^2}^2 +(2\\mu+\\lambda)\\|\\div u\\|_{L^2}^2+ \\ka R \\bp^{-1} \\|\\na\\te\\|_{L^2}^2\\\\\n&\\le C|\\bp_t|(\\|R\\te-\\bp\\|_{L^2}+\\|R\\te-\\bp\\|_{L^2}^2)+|\\bp_t| \\|\\n-1\\|_{L^2}^2 \\\\\n&\\quad+C\\|R\\te-\\bp\\|_{L^\\infty}(\\|R\\te-\\bp\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2)\\\\\n&\\le C(C_0^{1\/8}\\|\\na u\\|_{L^2}+\\|\\na u\\|_{L^2}^2)(\\|\\na\\te\\|_{L^2}+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2)\\\\\n&\\quad+C(\\|\\na\\te\\|_{L^2}^{1\/2}\\|\\na^2\\te\\|_{L^2}^{1\/2}+\\|\\na\\te\\|_{L^2})(\\|\\na\\te\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2)\\\\\n&\\le CC_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2),\n\\ea\\ee\nwhere\n\\be\\la{t3}\\ba\nW(t)&\\triangleq\\int\\left(\\n |u|^2+2\\bp (1+\\n\\log \\n-\\n)+\\frac{1}{\\ga-1} \\n \\bp^{-1}{(R\\te-\\bp)^2} \\right)dx\\\\\n&\\le \\hat C_3(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2)\n\\ea\\ee\nowing to \\eqref{z1}, \\eqref{key}, \\eqref{a2.9}, and \\eqref{pq}.\nCombining \\eqref{ao} with \\eqref{h18} and \\eqref{key} yields that\n\\be\\la{t5}\\ba &W'(t)+\\hat C_1(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)\n\\\\&\\le \\hat C_2C_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2).\\ea\\ee\n\n\nNext, rewriting $\\eqref{a1}_2$ as\n\\bnn\\la{t2}\\ba &(\\n u)_t+\\div(\\n u\\otimes u)\\\\&=\\mu\\Delta u+(\\mu+\\lambda)\\na(\\div u)-\\na(\\n(R\\te-\\bp))-\\bp\\na(\\n-1),\\ea\\enn\n multiplying this by $\\mathcal{B}[\\n-1]$ and using Lemma \\ref{th00}, \\eqref{z1}, and \\eqref{pq}, one gets that for any $t\\geq 1$,\n\\be\\notag \\ba\n&\\bp\\int(\\n-1)^2 dx \\\\\n&= \\left(\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right)_t -\\int\\rho u\\cdot\\mathcal{B}[\\n_t]dx\n -\\int\\rho u\\cdot\\nabla\\mathcal{B}[\\n-1]\\cdot u dx \\\\\n& \\quad +\\mu\\int\\p_j u\\cdot\\p_j\\mathcal{B}[\\n-1] dx +(\\mu+\\lambda)\\int(\\rho-1)\\div udx -\\int\\n(R\\te-\\bp)(\\n-1)dx\\\\\n& \\le\\left(\\int\\n u\\cdot\\mathcal{B}[\\n-1]dx\\right)_t+C\\|\\n u\\|_{L^2}^2+C\\|u\\|_{L^{4}}^{2}\\|\\n-1\\|_{L^2}\\\\\n& \\quad +C\\|\\rho-1\\|_{L^2}\\|\\na u\\|_{L^2}+C\\|\\rho-1\\|_{L^2}\\|R\\te-\\bp\\|_{L^2} \\\\\n& \\leq \\left(\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right)_t+\\frac{\\pi_1}{2}\\|\\n-1\\|_{L^2}^2+C(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2),\n\\ea\\ee\nwhich as well as \\eqref{key} leads to\n\\be\\la{t6}\\|\\rho-1\\|_{L^2}^2\\le\\frac{2}{\\pi_1}\\left(\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right)_t+\\hat C_4(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2).\\ee\nBy virtue of \\eqref{key}, \\eqref{a2.9}, and Lemma \\ref{th00}, it holds\n\\be \\la{t4} \\ba\n\\left|\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx\\right|\n&\\leq C \\left(\\|\\n u\\|^2_{L^2}+\\|\\n-1\\|_{L^2}^2\\right)\\\\\n&\\le \\hat C_5\\left(\\|\\n^{1\/2} u\\|^2_{L^2}+2\\bp (1+\\n\\log \\n-\\n)\\right).\n\\ea\\ee\nAdding \\eqref{t5} to \\eqref{t6} multiplied by $\\hat C_6$ with $\\hat C_6=\\min\\{\\frac{\\pi_1}{4\\hat C_5},\\frac{\\hat C_1}{4\\hat C_4}\\}$ yields\n\\be\\ba\\la{t1}\n&W_1'(t)+\\frac{3\\hat C_1}{4}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)+\\hat C_6\\|\\n-1\\|_{L^2}^2\\\\\n&\\le \\hat C_2C_0^{1\/24}(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+ \\|\\n-1\\|_{L^2}^2),\\ea\\ee\nwhere\n$$W_1(t)\\triangleq W(t)-\\frac{2\\hat C_6}{\\pi_1}\\int\\rho u\\cdot\\mathcal{B}[\\n-1] dx,$$\nsatisfies\n\\be\\la{t8}\\frac{1}{2}W(t)\\le W_1(t)\\le 2W(t)\\ee\ndue to \\eqref{t4}.\nThus we infer from \\eqref{t1} that\n\\be\\la{t9} W_1'(t)+\\frac{\\hat C_1}{2}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)+\\frac{\\hat C_6}{2}\\|\\n-1\\|_{L^2}^2\\le 0,\\ee\nprovided\n\\be\\la{t7}C_0\\le\\ve_0\\triangleq \\min\\left\\{\\ve_1,\\cdots,\\ve_4,\\left(\\frac{\\hat C_6}{2\\hat C_2}\\right)^{24},\\left(\\frac{\\hat C_1}{4\\hat C_2}\\right)^{24}\\right\\}.\\ee\nThen by \\eqref{t3}, one derives that for $\\alpha=\\frac{1}{3}\\min\\{\\frac{\\hat C_1}{2\\hat C_3},\\frac{\\hat C_6}{2\\hat C_3}\\}$,\n\\be\\notag W_1'(t)+3\\alpha W_1(t)\\le 0,\\ee\nwhich along with \\eqref{t8}, \\eqref{a2.9}, \\eqref{key}, \\eqref{t3}, and \\eqref{z1} shows that for any $t\\geq 1$,\n\\be\\la{t10}\\|\\n^{1\/2} u\\|_{L^2}^2+\\|\\n-1\\|_{L^2}^2+\\|\\n^{1\/2}(R\\te-\\bp)\\|_{L^2}^2\\le C W_1(t)\\le Ce^{-3\\al t}.\\ee\nMoreover, we deduce from \\eqref{t9} and \\eqref{t10} that for any $1\\le t\\le T<\\infty$,\n\\be\\la{t11}\\int_1^Te^{\\al t}(\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2)dt\\le C.\\ee\n\nNext, multiplying \\eqref{1hh17} by $e^{\\al t}$ and using \\eqref{z1} imply that for $B_1$ defined in \\eqref{an2},\n\\be\\ba \\la{t12}\n&(e^{\\al t}B_1(t))'+ \\frac{1}{2}e^{\\al t}\\int\\rho |\\dot u|^2dx\\le Ce^{\\al t}\\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2+\\|P-\\overline P\\|_{L^2}^2\\right).\n\\ea\\ee\nNote that by \\eqref{z1}, \\eqref{key}, and \\eqref{t10},\n\\be\\la{t16}\\|P-\\overline P\\|_{L^2}\\le \\|\\n(R\\te-\\bp)\\|_{L^2}+\\bp\\|\\n-1\\|_{L^2}\\le Ce^{-\\al t},\\ee which together with \\eqref{t11}, \\eqref{t12}, and \\eqref{h18} gives for any $1\\le t\\le T<\\infty$,\n\\be\\la{t13}\\ba\n\\sup_{1\\leq t\\leq T}\\left(e^{\\al t}\\|\\na u\\|_{L^2}^2\\right)+\\int_1^T e^{\\al t}\\|\\n^{1\/2}\\dot u\\|^2_{L^2}dt\\le C.\n\\ea \\ee\n\nFurthermore, choosing $m=0$ in \\eqref{e8}, it follows from \\eqref{m20}, \\eqref{ae9}, and \\eqref{z1} that for any $t\\geq1$,\n\\be\\notag\\ba\n&\\varphi'(t)+ 2\\left(\\int_{\\p \\O}( u \\cdot \\na n \\cdot u )G dS\\right)_t +\\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dx\\\\\n&\\le C(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2 + \\|\\na \\te\\|_{L^2}^2+\\|\\n-1\\|_{L^2}^2),\n\\ea\\ee\nwhere $\\varphi(t)$ is defined in \\eqref{wq3}. Multiplying this by $e^{\\al t}$ along with \\eqref{e6}, \\eqref{wq3}, \\eqref{wq2}, \\eqref{b2}, \\eqref{t10}, \\eqref{t11}, \\eqref{t13}, \\eqref{h19}, and \\eqref{z1} yields that for any $1\\le t\\le T<\\infty$,\n\\be\\la{t15}\\ba\n\\sup_{1\\leq t\\leq T}\\left(e^{\\al t}(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2)\\right)+\\int_1^T e^{\\al t}(\\|\\na\\dot u\\|_{L^2}^2+\\|\\n^{1\/2}\\dot \\te\\|^2_{L^2})dt\\le C.\n\\ea \\ee\nAdopting the analogous method and applying \\eqref{3.99}, \\eqref{vu15}, \\eqref{t10}, \\eqref{t11}, \\eqref{t13}, \\eqref{t15}, \\eqref{lop4}, \\eqref{m20}, \\eqref{z1}, and \\eqref{ae9}, we obtain that\n\\be\\la{t17}\\ba\n\\sup_{1\\leq t\\leq T}\\left(e^{\\al t}(\\|\\n^{1\/2}\\dot\\te\\|_{L^2}^2+\\|\\na^2\\te\\|_{L^2}^2)\\right)+\\int_1^T e^{\\al t}\\|\\na\\dot \\te\\|_{L^2}^2dt\\le C.\n\\ea \\ee\n\nFinally, it remains to determine the limit of $\\te$ as $t$ tends to infinity. Combining \\eqref{pt}, \\eqref{t16}, and \\eqref{t13} shows that for any $t\\geq 1$,\n\\be\\ba\\notag\n|\\bp_t|\\le C(\\|\\na u\\|_{L^2}^2+\\|P-\\bp\\|_{L^2}^2)\\le C e^{-\\al t},\n\\ea\\ee\nwhich implies there exists a constant $P_\\infty$ such that $\\lim_{t\\rightarrow \\infty}\\overline P=P_\\infty$ and\n\\be\\la{t18}|\\overline P-P_\\infty|\\le Ce^{-\\al t}.\\ee\nDenoting $\\te_\\infty\\triangleq P_\\infty\/R$, we have\n\\be\\la{t19}\\|\\te-\\te_\\infty\\|_{L^2}^2\\le C\\|R\\te-\\bp\\|_{L^2}^2+C|\\bp-R\\te_\\infty|^2\\le Ce^{-\\al t},\\ee\nwhere we have used \\eqref{pq}, \\eqref{t15}, and \\eqref{t18}.\nTherefore, the combination of \\eqref{t10}, \\eqref{t13}--\\eqref{t17}, \\eqref{t19}, \\eqref{h17}, and \\eqref{p} concludes \\eqref{t7} and finishs the proof of Lemma \\ref{pr2}.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{\\la{se4} A priori estimates (II): higher-order estimates}\n\nIn this section, we will derive the higher-order estimates of smooth solution $(\\rho, u, \\te)$ to problem (\\ref{a1})--(\\ref{h1}) on $ \\Omega\\times (0,T]$ with initial data $(\\n_0 ,u_0,\\te_0)$ satisfying (\\ref{co3}) and (\\ref{3.1}).\n\n\nWe shall assume that (\\ref{z1}) and (\\ref{z01}) both hold as well. To proceed,\nwe define $\\tilde g $ as\n\\be \\la{co12}\\tilde g\\triangleq\\n_0^{-1\/2}\\left(\n-\\mu \\Delta u_0-(\\mu+\\lambda)\\na\\div u_0+R\\na (\\n_0\\te_0)\\right).\\ee\nThen it follows from (\\ref{co3}) and (\\ref{3.1}) that\n\\be\\la{wq01}\\tilde g\\in L^2.\\ee\nFrom now on, the generic constant $C $ will depend only on \\bnn\nT, \\,\\, \\| \\tilde g\\|_{L^2}, \\,\\|\\n_0\\|_{W^{2,q}} , \\, \\,\\|\\na u_0\\|_{H^1}, \\ \\,\n\\| \\na\\te_0\\|_{L^2} , \\enn\nbesides $\\mu,\\,\\lambda,\\, \\ka,\\, R,\\, \\ga,\\, \\on,\\,\\bt,\\,\\O,$ and $M.$\n\n\nWe begin with the following estimates on the spatial gradient of\nthe smooth solution $(\\rho,u,\\te).$\n\n\n\\begin{lemma}\\la{le11}\n\tThe following estimates hold:\n\t\\be\\label{lee2}\\ba\n\t&\\sup_{0\\le t\\le T} \\left(\\|\\rho^{1\/2}\\dot u\\|_{L^2}^2 + \\sigma\\|\\rho^{1\/2}\\dot \\te\\|_{L^2}^2 +\\|\\te\\|_{H^1}^2 + \\sigma \\|\\na^2 \\theta\\|_{L^2}^2\n \\right) \\\\\n\t &\\quad+\\ia\\left( \\|\\nabla\\dot u\\|_{L^2}^2 + \\|\\rho^{1\/2}\\dot \\te\\|_{L^2}^2+ \\|\\nabla^2 \\theta\\|_{L^2}^2 +\\sigma \\|\\nabla\\dot \\te\\|_{L^2}^2 \\right) dt\\le C,\n\t\\ea\\ee\n\tand\n\t\\be\\la{qq1}\n\t\\sup_{0\\le t\\le T}\\left(\\|u\\|_{H^2} +\\|\\n\\|_{H^2}\\right)\n\t+ \\int_0^{T}\\left( \\|\\nabla u\\|_{L^{\\infty}}^{3\/2} + \\si \\| \\na^3 \\te\\|_{L^2}^2+\\|u\\|_{H^3}^2 \\right)dt\\le C.\n\t\\ee\n\\end{lemma}\n\n\n\\begin{proof}\nThe proof is divided into the following two steps.\n\t\n{\\it Step 1: The proof of (\\ref{lee2}).}\nFirst, for $\\varphi(t)$ as in \\eqref{wq3}, taking $m=0$ in \\eqref{e8}, one gets\n\\be\\la{ae8}\\ba\n& \\varphi'(t) + \\int\\left( \\frac{C_1}{2}|\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dx \\\\\n&\\le -2 \\left(\\int_{\\p\\O} G\\left( u\\cdot \\na n \\cdot u \\right)dS\\right)_t+C\\left(\\|\\n^{1\/2} \\dot u\\|_{L^2}^2+\\|\\na u\\|_{L^2}^2+\\|\\na \\te\\|_{L^2}^2\\right)\\\\\n&\\quad +C\\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^3+\\|\\na\\te\\|_{L^2}^3+\\|\\na u\\|_{L^2}^2+\\|\\n-1\\|_{L^2}^2\\right) \\\\\n& \\quad +C \\left(\\|\\na u\\|_{L^2}^2+\\|\\na\\te\\|_{L^2}^2\\right)\\left( \\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2+1 \\right) \\\\\n&\\le -2 \\left(\\int_{\\p\\O} G\\left( u\\cdot \\na n \\cdot u \\right)dS\\right)_t+C \\left( \\|\\n^{1\/2}\\dot u\\|_{L^2}^2 + \\|\\na\\te\\|_{L^2}^2\\right) (\\varphi+1)+C\n\\ea \\ee\ndue to (\\ref{z1}), \\eqref{wq2}, (\\ref{ae9}), and \\eqref{m20}.\nTaking into account the compatibility condition \\eqref{co2}, we can define\n\\bnn \\ba\n\\sqrt{\\n} \\dot u(x,t=0) \\triangleq\n-\\tilde g,\n\\ea\\enn\nwhich along with (\\ref{e6}), (\\ref{2.48}), and \\eqref{wq01} yields that\n\\be \\la{wq02}\n|\\varphi(0)|\\le C\\| \\tilde g\\|_{L^2}^2+C\\le C.\n\\ee\nThen, integrating \\eqref{ae8} over $(0,t)$, one obtains after using \\eqref{z1}, \\eqref{b2}, \\eqref{h19}, \\eqref{wq2}, and \\eqref{wq02} that\n\\be\\ba \\la{q1}\n& \\varphi(t)+\\int_0^t\\int\\left(\\frac{C_1}{2} |\\nabla\\dot{u}|^2 +\\n|\\dot \\te|^2\\right)dxds\\\\\n&\\le 2\\left| \\int_{\\p\\O} G\\left( u\\cdot \\na n \\cdot u \\right)dS\\right|(t)+C\\int_0^t \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2\\right)\\varphi ds+C\\\\\n&\\le C (\\|\\na u\\|_{L^2}^2\\|\\n^{1\/2}\\dot u\\|_{L^2})(t)+C\\int_0^t \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2\\right)\\varphi ds+C\\\\\n&\\le \\frac{1}{2} \\varphi(t) + C \\int_{0}^{t} \\left(\\|\\n^{1\/2}\\dot u\\|_{L^2}^2+ \\|\\na \\te \\|_{L^2}^2\\right)\\varphi ds+C.\n\\ea\\ee\nApplying Gr\\\"{o}nwall's inequality to \\eqref{q1} and\n using \\eqref{z1} and (\\ref{wq2}), it holds\n\\be\\label{lee3}\n\\sup_{0\\le t\\le T} \\left(\\|\\rho^{1\/2}\\dot u\\|_{L^2}^2\n+\\|\\na \\te\\|_{L^2}^2 \\right) + \\ia\\int\\left(|\\nabla\\dot\nu|^2+\\n|\\dot\\te|^2\\right)dxdt\\le C,\n\\ee\nwhich together with \\eqref{key} and \\eqref{pq} implies\n\\be\\la{ff1} \\ba\n\\|\\te\\|_{L^2} &\\le C( \\|R \\te-\\overline P \\|_{L^2} +\\overline P ) \\le C(\\|\\na \\te\\|_{L^2}+1) \\le C.\n\\ea \\ee\n\n\n\nNext,\nmultiplying (\\ref{3.99}) by $\\sigma$ and integrating over $(0,T)$ lead to\n\\be \\ba \\la{a5}\n&\\sup\\limits_{0\\le t\\le T} \\si \\int \\n|\\dot\\te|^2dx+\\int_0^T \\si \\|\\na\\dot\\te\\|_{L^2}^2dt\\\\\n&\\le\nC\\int_0^T\\left( \\|\\na^2\\te\\|_{L^2}^2+\\|\\na\\dot u\\|_{L^2}^2+ \\|\\n^{1\/2}\\dot\\te\\|_{L^2}^2 \\right)dt + C\\\\\n&\\le C,\n\\ea\\ee\nwhere we have used (\\ref{lee3}), (\\ref{z1}), (\\ref{m20}), \\eqref{ae9}, (\\ref{lop4}), and the following fact:\n\\be\\la{w1}\\|\\na u\\|_{L^6}\\le C\\ee\ndue to \\eqref{3.30}, \\eqref{z1}, and \\eqref{lee3}.\nThen, it follows from (\\ref{lop4}), \\eqref{lee3}, \\eqref{a5}, \\eqref{w1}, (\\ref{m20}), and (\\ref{z1}) that\n\\be \\notag \\sup\\limits_{0\\le t\\le T} \\si\\|\\na^2\\te\\|_{L^2}^2 + \\int_{0}^{T}\\|\\na^2\\te\\|_{L^2}^2 dt \\le C,\\ee\nwhich along with \\eqref{lee3}--\\eqref{a5} gives \\eqref{lee2}.\n\n\n\n{\\it Step 2: The proof of (\\ref{qq1}).}\nFirst, standard calculations show that for $ 2\\le p\\le 6$,\n\\be\\la{L11}\\ba\n\\partial_t\\norm[L^p]{\\nabla\\rho}\n&\\le C\\norm[L^{\\infty}]{\\nabla u} \\norm[L^p]{\\nabla\\rho}+C\\|\\na^2u\\|_{L^p}\\\\\n&\\le C\\left(1+\\|\\na u\\|_{L^{\\infty}}+\\|\\na^2\\te \\|_{L^2}\\right)\n\\norm[L^p]{\\nabla\\rho}\\\\\n&\\quad +C\\left(1+\\|\\na\\dot u\\|_{L^2}+\\|\\na^2\\te \\|_{L^2}\\right), \\ea\\ee\nwhere we have used\n\\be\\ba\\la{ua1}\n\\|\\na^2 u\\|_{L^p} & \\le C(\\|\\rho \\dot u\\|_{L^p} + \\|\\na P\\|_{L^p} + \\|\\na u\\|_{L^2} ) \\\\\n& \\le C\\left(1+\\|\\na\\dot u\\|_{L^2}+\\|\\na \\te\\|_{L^p}+\\|\\te\\|_{L^\\infty}\\|\\nabla\\n\\|_{L^p}\\right)\\\\\n&\\le C\\left(1+\\|\\na\\dot u\\|_{L^2}+\\|\\na^2\\te\\|_{L^2}+(\\|\\na^2\\te \\|_{L^2} + 1)\\|\\nabla\\n\\|_{L^p}\\right)\n\\ea\\ee\ndue to \\eqref{g1}, \\eqref{tb90}, \\eqref{rmk1}, \\eqref{z1}, and \\eqref{lee2}.\nIt follows from Lemma \\ref{le9}, \\eqref{z1}, and (\\ref{ua1}) that\n\\be\\la{u13}\\ba\n\\|\\na u\\|_{L^\\infty }\n&\\le C\\left(\\|{\\rm div}u\\|_{L^\\infty}+\\|\\curl u\\|_{L^\\infty}\\right)\\log(e+\\|\\na^2 u\\|_{L^6}) +C\\|\\na u\\|_{L^2}+C \\\\\n&\\le C\\left( \\|{\\rm div}u\\|_{L^\\infty } + \\|\\curl u\\|_{L^\\infty }\n\\right)\\log(e+ \\|\\na\\dot u\\|_{L^2 } + \\|\\na^2\\te \\|_{L^2})\\\\\n&\\quad +C\\left(\\|{\\rm div}u\\|_{L^\\infty }+ \\|\\curl u\\|_{L^\\infty } \\right)\n\\log\\left(e + \\|\\na \\n\\|_{L^6}\\right)+C.\n\\ea\\ee\nDenote\n\\bnn\\la{gt}\\begin{cases}\n\tf(t)\\triangleq e+\\|\\na\t\\n\\|_{L^6},\\\\\n\th(t)\\triangleq 1+ \\|{\\rm div}u\\|_{L^\\infty }^2+ \\|\\curl u\\|_{L^\\infty }^2\n\t+ \\|\\na\\dot u\\|_{L^2 }^2 +\\|\\na^2\\te \\|_{L^2}^2.\n\\end{cases}\\enn\nOne obtains after submitting \\eqref{u13} into (\\ref{L11}) with $p=6$ that\n\\bnn f'(t)\\le C h(t) f(t)\\ln f(t) ,\\enn\nwhich implies \\be\\la{w2} (\\ln(\\ln f(t)))'\\le C h(t).\\ee\nNote that by virtue of\n(\\ref{hj1}), (\\ref{key}), (\\ref{lee2}), (\\ref{z1}), \\eqref{bz5}, and (\\ref{bz6}), one gets\n\\be \\la{p2}\\ba\n& \\int_0^T\\left(\\|\\div u\\|^2_{L^\\infty}+\\|\\curl u\\|^2_{L^\\infty} \\right)dt \\\\\n& \\le C\\int_0^T\\left(\\|G\\|^2_{L^\\infty}+ \\|\\curl u\\|^2_{L^\\infty}+\\|P-\\bp\\|^2_{L^\\infty}\\right)dt \\\\\n&\\le C\\ia\\left(\\| G\\|^2_{ W^{1,6} } + \\| \\curl u\\|^2_{W^{1,6}} + \\|\\te\\|_{L^\\infty}^2\\right)dt + C \\\\\n& \\le C\\ia\\left(\\|\\na G\\|^2_{L^6}+\\|\\na\\curl u\\|^2_{L^6}+\\|\\na^2\\te \\|_{L^2}^2\\right)dt+C \\\\\n&\\le C\\ia(\\|\\na \\dot u\\|^2_{L^2}+\\|\\na^2\\te \\|_{L^2}^2)dt+C \\\\\n&\\le C,\n\\ea\\ee\nwhich as well as \\eqref{w2} and \\eqref{lee2} yields that\n\\be \\la{u113} \\sup\\limits_{0\\le t\\le T}\\|\\nabla \\rho\\|_{L^6}\\le C.\\ee\nCombining this with (\\ref{u13}), \\eqref{p2}, and (\\ref{lee2}) leads to\n\\be \\la{v6}\\ia\\|\\nabla u\\|_{L^\\infty}^{3\/2}dt \\le C.\\ee\nMoreover, it follows from (\\ref{rmk1}), \\eqref{z1}, \\eqref{u113}, and (\\ref{lee2}) that\n\\be\\ba\\la{aa95}\n\\sup\\limits_{0\\le t\\le T} \\| u\\|_{H^2}\n\\le &C \\sup\\limits_{0\\le t\\le T}\\left(\\|\\n\\dot u\\|_{L^2}+\\|\\nabla P\\|_{L^2}+\\| \\na u\\|_{L^2}\\right)\\le C.\n\\ea\\ee\n\n\n\n\nNext, applying operator $\\p_{ij}~(1\\le i,j\\le 3)$ to $(\\ref{a1})_1$ gives\n\\be\\la{4.52} (\\p_{ij} \\n)_t+\\div (\\p_{ij} \\n u)+\\div (\\n\\p_{ij} u)+\\div(\\p_i\\n\\pa_j u+\\p_j\\n\\p_i u)=0. \\ee\nMultiplying (\\ref{4.52}) by $2\\p_{ij} \\n$ and integrating the resulting equality over $\\O,$ it holds\n\\be\\la{ua2}\\ba\n\\frac{d}{dt}\\|\\na^2\\n\\|^2_{L^2}\n& \\le C(1+\\|\\na u\\|_{L^{\\infty}})\\|\\na^2\\n\\|_{L^2}^2+C\\|\\na u\\|^2_{H^2}\\\\\n& \\le C(1+\\|\\na u\\|_{L^{\\infty}} +\\|\\na^2\\te \\|_{L^2}^2)(1+\\|\\na^2\\n\\|_{L^2}^2) +C\\|\\na\\dot u \\|^2_{L^2}, \\ea\\ee\nwhere one has used \\eqref{z1}, \\eqref{u113}, and the following estimate:\n\\be\\ba\\la{va2}\n\\| u\\|_{H^3}&\\le C\\left(\\|\\na(\\n\\dot u)\\|_{L^2}+\\|\\n\\dot u\\|_{L^2}+\\| \\na P\\|_{H^1}+\\|\\na u\\|_{L^2}\\right)\n\\\\&\\le C\\|\\na\\n\\|_{L^3}\\|\\dot u\\|_{L^6}+C\\|\\na \\dot u\\|_{L^2}+C\\|\\na \\te\\|_{H^1}\n\\\\&\\quad+C\\||\\na\\n||\\na\\te|\\|_{L^2}+C\\|\\te\\|_{L^\\infty}\\|\\na \\n\\|_{H^1}+C\n\\\\ &\\le C\\|\\na\\dot u \\|_{L^2}+ C (1+\\|\\na^2\\te \\|_{L^2})(1+\\|\\na^2\\n\\|_{L^2}) +C\n\\ea\\ee\ndue to (\\ref{rmk1}), (\\ref{tb90}), (\\ref{lee2}), (\\ref{u113}), and (\\ref{z1}).\nThen applying Gr\\\"{o}nwall's inequality to \\eqref{ua2} and using (\\ref{lee2}), (\\ref{v6}) yield\n\\be\\la{ja3} \\sup_{0\\le t\\le T} \\|\\na^2\\n \\|_{L^2} \\le C,\\ee\nwhich together with \\eqref{va2} and \\eqref{lee2} gives\n\\be\\la{ja4} \\int_0^T\\|u\\|_{H^3}^2 dt\\le C . \\ee\n\n\nFinally,\napplying the standard $H^1$-estimate to elliptic problem (\\ref{3.29}), one derives from \\eqref{z1}, \\eqref{lee2}, \\eqref{u113}, \\eqref{kk}, and \\eqref{aa95} that\n\\be\\la{ex4}\\ba\n\\|\\na^2\\te\\|_{H^1}\n& \\le C\\left(\\|\\n \\dot \\te\\|_{H^1}+\\|\\n\\te\\div u\\|_{H^1}+\\||\\na u|^2\\|_{H^1}\\right)\\\\\n& \\le C\\left(1+ \\|\\na \\dot \\te\\|_{L^2} + \\|\\rho^{1\/2} \\dot \\theta\\|_{L^2} + \\|\\na(\\n\\te\\div u)\\|_{L^2}+ \\||\\na u||\\na^2u|\\|_{L^2} \\right) \\\\\n& \\le C\\left(1+ \\|\\na \\dot \\te\\|_{L^2} + \\|\\rho^{1\/2} \\dot \\theta\\|_{L^2} +\\|\\na^2 \\theta\\|_{L^2}+\\|\\na^3 u\\|_{L^2} \\right),\n\\ea\\ee\nwhich along with \\eqref{z1}, \\eqref{ja3}, \\eqref{ja4}, \\eqref{u113}--\\eqref{aa95}, and \\eqref{lee2} yields (\\ref{qq1}).\n\n\n\n The proof of Lemma \\ref{le11} is finished.\n\\end{proof}\n\n\\begin{lemma}\\la{le9-1}\n\tThe following estimates hold:\n\t\\be\\la{va5}\\ba&\n\t\\sup\\limits_{0\\le t\\le T}\n\t\\|\\n_t\\|_{H^1}\n\t+\\int_0^T\\left(\\| u_t\\|_{H^1}^2+\\si \\| \\te_t\\|_{H^1}^2+\\| \\n u_t\\|_{H^1}^2+\\si \\|\\n \\te_t\\|_{H^1}^2\n\t\\right)dt\\le C,\n\t\\ea \\ee\n\tand\n\t\\be\\la{vva5}\\ba\n\t\\int_0^T \\sigma \\left( \\|(\\n u_t)_t\\|_{H^{-1}}^2+\\|(\\n \\te_t)_t\\|_{H^{-1}}^2\n\t\\right)dt\\le C.\\ea\\ee\n\\end{lemma}\n\n\n\n\\begin{proof}\n\n\n\nFirst, it follows from (\\ref{lee2}) and (\\ref{qq1}) that\n\\be\\label{va1}\\ba\n&\\sup_{0\\le t\\le T}\\int\\left( \\n|u_t|^2 + \\si\n\\n\\te_t^2\\right)dx +\\int_0^T \\left(\\|\\na u_t\\|_{L^2}^2+ \\si \\|\\na\\te_t\\|_{L^2}^2\\right)dt\\\\\n&\\le \\sup\\limits_{0\\le t\\le T}\\int \\left(\\n|\\dot u|^2+ \\si\\n|\\dot\\te|^2 \\right)dx\n+\\int_0^T\\left(\\|\\na\\dot u\\|_{L^2}^2+\\si \\|\\na\\dot\\te\\|_{L^2}^2 \\right)dt\\\\\n&\\quad + \\sup\\limits_{0\\le t\\le T}\\int \\n\\left(|u\\cdot\\na u|^2+\\si |u\\cdot\\na\\te|^2\\right)dx\\\\\n&\\quad +\\int_0^T\\left(\\|\\na(u\\cdot\\na u)\\|_{L^2}^2+\\si \\|\\na(u\\cdot\\na \\te)\\|_{L^2}^2\\right)dt\\\\\n&\\le C,\\ea\\ee\nwhich together with (\\ref{lee2}) and (\\ref{qq1}) gives\n\\be\\la{vva1} \\ba\n& \\int_0^T\\left(\\|\\na(\\n u_t)\\|_{L^2}^2+ \\si \\|\\na(\\n\\te_t)\\|_{L^2}^2\\right)dt\\\\\n& \\le C\\int_0^T\\left(\\| \\na u_t \\|_{L^2}^2+\\| \\na \\n\\|_{L^3}^2\\| u_t \\|_{L^6}^2\n+ \\si \\| \\na \\te_t \\|_{L^2}^2+ \\si \\| \\na \\n\\|_{L^3}^2\\| \\te_t \\|_{L^6}^2 \\right)dt\\\\\n&\\le C,\\ea\\ee\nwhere we have used\n\\be\\la{w3}\\|\\te_t\\|_{L^6}\\le C\\|\\n^{1\/2}\\te_t\\|_{L^2}+C\\|\\na\\te_t\\|_{L^2\n\\ee\ndue to \\eqref{kk}.\n\nNext, one deduces from $(\\ref{a1})_1$, (\\ref{qq1}), and \\eqref{hs} that\n\\bnn\\ba\\la{sp1}\n\\|\\n_t\\|_{H^1}\\le& \\|\\div (\\rho u)\\|_{H^1}\n\\le C \\|u\\|_{H^2}\\|\n\\n\\|_{H^2}\n\\le C,\n\\ea \\enn\nwhich as well as (\\ref{va1})--\\eqref{w3} shows (\\ref{va5}).\n\nFinally, differentiating $(\\ref{a1})_2$ with respect to $t$ yields that\n\\be \\la{va7}\\ba (\\n u_t)_t\n=-(\\n u\\cdot\\na u)_t + \\left(\\mu\\Delta u+(\\mu+\\lambda)\\na\\div u \\right)_t -\\na P_t.\\ea\\ee\nIt follows from (\\ref{va5}), \\eqref{qq1}, \\eqref{lee2}, \\eqref{lop4}, and \\eqref{w3} that\n\\be \\la{va9}\\ba \\|(\\n u\\cdot\\na u)_t \\|_{L^{2}}\n&=\\| \\n_t u\\cdot\\na u+ \\n u_t\\cdot\\na u + \\n u\\cdot\\na u_t \\|_{L^{2}}\\\\\n&\\le C\\|\\n_t\\|_{L^6} \\|\\na u\\|_{L^3}+ C\\|u_t\\|_{L^6} \\|\\na u\\|_{L^3}+ C\\|u \\|_{L^\\infty} \\|\\na u_t\\|_{L^2}\\\\\n&\\le C+ C\\|\\na u_t\\|_{L^2},\\ea\\ee\nand\n\\be \\la{va10}\\ba \\|\\na P_t \\|_{L^2}\n&=R\\|\\n_t\\na\\te +\\n \\na\\te_t +\\na\\n_t\\te +\\na\\n\\te_t\\|_{L^2}\\\\\n&\\le C\\left(\\|\\n_t\\|_{L^6}\\|\\na\\te\\| _{L^3}+\\|\\na\\te_t\\|_{L^2}\n+\\|\\te\\|_{L^\\infty}\\|\\na \\n_t\\|_{L^2}+\\|\\na\\n\\|_{L^6}\\|\\te_t\\| _{L^3}\\right)\\\\\n&\\le C+C\\|\\na\\te_t\\|_{L^2}+C\\|\\n^{1\/2}\\te_t\\|_{L^2}.\\ea\\ee\nCombining (\\ref{va7})--(\\ref{va10}) with (\\ref{va5}) shows\n\\be\\la{vva04}\\ba\\int_0^T \\si \\|(\\n u_t)_t\\|_{H^{-1}}^2dt\\le C.\\ea\\ee\nSimilarly, we have \\bnn\\int_0^T \\si \\|(\\n \\te_t)_t\\|_{H^{-1}}^2dt\\le C, \\enn\nwhich combined with (\\ref{vva04}) implies (\\ref{vva5}). The proof of Lemma \\ref{le9-1} is completed.\n\\end{proof}\n\n\n\\begin{lemma}\\la{pe1}\n\tThe following estimate holds:\n\t\\be\\la{nq1}\n\t\\sup\\limits_{0\\le t\\le T} \\si\\left(\\|\\nabla u_t\\|^2_{L^2}+\\|\\n_{tt} \\|^2_{L^2} + \\|u\\|_{H^3}^2\\right)\n\t+ \\int_0^T\\si \\left(\\|\\rho^{1\/2} u_{tt}\\|_{L^2}^2+ \\|\\nabla u_t\\|_{H^1}^2\\right)dt\\le C.\n\t\\ee\n\\end{lemma}\n\\begin{proof}\nDifferentiating $(\\ref{a11})$ with respect to $t$ leads to \\be\\la{nt0}\\ba \\begin{cases}\n(2\\mu+\\lambda)\\na\\div u_t-\\mu \\na\\times \\curl u_t\\\\\n= \\n u_{tt} +\\n_tu_t+\\n_tu\\cdot\\na u+\\n u_t\\cdot\\na u +\\n u\\cdot\\na u_t+\\na P_t\\triangleq \\tilde f\n,&\\, \\text{in}\\,\\O\\times[0,T],\\\\\nu_{t}\\cdot n=0,\\ \\curl u_{t}\\times n=0, &\\,\\text{on}\\,\\p\\O\\times[0,T]. \\end{cases}\\ea\\ee\nMultiplying (\\ref{nt0})$_1$ by $u_{tt}$ and integrating the resulting equality by parts, one gets \\be\\la{sp9} \\ba&\n\\frac{1}{2}\\frac{d}{dt}\\int \\left(\\mu|\\curl u_t|^2 + (2\\mu +\\lambda)({\\rm div}u_t)^2\\right)dx\n+\\int \\rho| u_{tt}|^2dx\\\\\n&=\\frac{d}{dt}\\left(-\\frac{1}{2}\\int_{ }\\rho_t |u_t|^2 dx- \\int_{}\\rho_t u\\cdot\\nabla u\\cdot u_tdx\n+ \\int_{ }P_t {\\rm div}u_tdx\\right)\\\\\n& \\quad + \\frac{1}{2}\\int_{ }\\rho_{tt} |u_t|^2 dx+\\int_{ }(\\rho_{t} u\\cdot\\nabla u )_t\\cdot u_tdx\n-\\int_{ }\\rho u_t\\cdot\\nabla u\\cdot u_{tt}dx\\\\\n&\\quad - \\int_{ }\\rho u\\cdot\\nabla u_t\\cdot u_{tt}dx - \\int_{ }\\left(P_{tt}-\n\\ka(\\ga-1)\\Delta\\te_t\\right){\\rm div}u_tdx\\\\\n&\\quad +\\ka(\\ga-1)\\int_{ } \\na\\te_t\\cdot\\na {\\rm div}u_tdx\n\\triangleq\\frac{d}{dt}\\tilde{I}_0+ \\sum\\limits_{i=1}^6 \\tilde{I}_i. \\ea \\ee\n\n Each term $\\tilde{I}_i(i=0,\\cdots,6)$ can be estimated as follows:\n\nFirst, it follows from simple calculations, $(\\ref{a1})_1,$ (\\ref{va5}), \\eqref{qq1}, \\eqref{lee2}, and (\\ref{va1}) that\n\\be \\ba \\la{sp10}\n|\\tilde{I}_0|&=\\left|-\\frac{1}{2}\\int\\rho_t |u_t|^2 dx- \\int \\rho_t u\\cdot\\nabla\nu\\cdot u_tdx+ \\int P_t {\\rm div}u_tdx\\right|\\\\\n&\\le C\\int \\n |u||u_t||\\nabla u_t| dx+C\\norm[L^3]{\\rho_t}\\norm[L^2]\n{\\nabla u}\\norm[L^6]{u_t}+C\\|(\\n\\te)_t\\|_{L^2}\\|\\nabla u_t\\|_{L^2}\\\\\n&\\le C \\|\\n^{1\/2}u_t\\|_{L^2} \\|\\nabla u_t\\|_{L^2} +C(1+\\|\\n^{1\/2} \\te_t\\|_{L^2}+\\|\\n_t\\|_{L^3}\\|\\te\\|_{L^6})\\|\\nabla u_t\\|_{L^2}\\\\\n&\\le C (1+\\|\\n^{1\/2}\\te_t\\|_{L^2}) \\|\\nabla u_t\\|_{L^2} ,\\ea\\ee\n\\be \\la{sp11}\\ba\n2|\\tilde{I}_1|&=\\left|\\int \\rho_{tt} |u_t|^2 dx\\right| \\le C \\|\\n_{tt}\\|_{L^2}\\|u_t\\|_{L^4}^{2}\n \\le C\\|\\n_{tt}\\|_{L^2}^2+C \\|\\na u_t\\|_{L^2}^4,\n\\ea \\ee\n\\be \\la{sp12}\\ba\n|\\tilde{I}_2|&=\\left|\\int \\left(\\rho_t u\\cdot\\nabla u \\right)_t\\cdot u_{t}dx\\right|\\\\\n& = \\left| \\int\\left(\\rho_{tt} u\\cdot\\nabla u\\cdot u_t +\\rho_t\nu_t\\cdot\\nabla u\\cdot u_t+\\rho_t u\\cdot\\nabla u_t\\cdot u_t\\right)dx\\right|\\\\\n&\\le C\\norm[L^2]{\\rho_{tt}}\\norm[L^6]{\\na u}\\norm[L^6]{u}\\norm[L^6]{u_t}\n+C\\norm[L^2]{\\rho_t}\\norm[L^6]{u_t}^2\\norm[L^6]{\\nabla u} \\\\\n&\\quad+C\\norm[L^3]{\\rho_t}\\norm[L^{\\infty}]{u}\\norm[L^2]{\\nabla u_t}\\norm[L^6]{u_t}\\\\\n& \\le C\\norm[L^2]{\\rho_{tt}}^2 + C\\norm[L^2]{\\nabla u_t}^2, \\ea \\ee\nand\n\\be\\ba\\la{sp13}\n|\\tilde{I}_3|+|\\tilde{I}_4|&= \\left| \\int \\rho u_t\\cdot\\nabla u\\cdot u_{tt} dx\\right|\n+\\left| \\int \\rho u\\cdot\\nabla u_t\\cdot u_{tt}dx\\right|\\\\\n& \\le C\\|\\n^{1\/2}u_{tt}\\|_{L^2}\\left(\\|u_t\\|_{L^6}\\|\\na u\\|_{L^3}\n+\\|u\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}\\right) \\\\\n& \\le \\frac{1}{4}\\norm[L^2]{\\rho^{{1\/2}}u_{tt}}^2 + C \\norm[L^2]{\\nabla u_t}^2.\\ea\\ee\nThen, by virtue of\n(\\ref{op3}), (\\ref{va5}), \\eqref{va10}, and Lemma \\ref{le11}, it holds\n\\bnn\\ba & \\|P_{tt}-\\ka(\\ga-1)\\Delta \\te_t\\|_{L^2}\\\\\n&\\le C\\|(u\\cdot\\na P)_t\\|_{L^2}+C\\|(P\\div u)_t\\|_{L^2}+C\\||\\na u||\\na u_t|\\|_{L^2}\\\\\n&\\le C\\|u_t\\|_{L^6}\\|\\na P\\|_{L^3}+C\\|u\\|_{L^\\infty}\\|\\na P_t\\|_{L^2}\n+C\\|P_t\\|_{L^6}\\|\\na u\\|_{L^3}\\\\\n&\\quad +C\\|P\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}+C\\|\\na u\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}\\\\\n&\\le C\\left(1+\\|\\na u\\|_{L^\\infty}+\\|\\na^2\\te \\|_{L^2}\\right)\\|\\na u_t\\|_{L^2} +C\\left(1+\\|\\na\\te_t\\|_{L^2}+\\|\\n^{1\/2}\\te_t\\|_{L^2}\\right),\n\\ea\\enn\nwhich yields\n\\be\\ba\\la{sp15}\n|\\tilde{I}_5|&=\\left|\\int\\left(P_{tt}-\\ka(\\ga-1)\\Delta \\te_t\\right){\\rm div}u_tdx\\right|\\\\\n&\\le\\norm[L^2]{P_{tt}-\\ka(\\ga-1)\\Delta \\te_t}\\norm[L^2]{\\na u_t}\\\\\n&\\le C\\left(1+\\|\\na u\\|_{L^\\infty}+\\|\\na^2\\te \\|_{L^2}\\right)\\|\\na u_t\\|_{L^2}^2\\\\\n&\\quad+C\\left(1+\\|\\na\\te_t\\|^2_{L^2}+\\|\\n^{1\/2}\\te_t\\|_{L^2}^2\\right).\n\\ea\\ee\nNext, combining Lam\\'{e}'s system \\eqref{nt0} with Lemma \\ref{zhle}, \\eqref{qq1}, \\eqref{va5}, and (\\ref{va10}) gives\n\\be\\la{nt4}\\ba\n\\|\\na^2u_t\\|_{L^2}&\\le C\\|\\tilde f\\|_{L^2}+C\\|\\na u_t\\|_{L^2}\\\\\n&\\le C\\|\\n u_{tt}\\|_{L^2}\n+C\\|\\n_t\\|_{L^3}\\|u_t\\|_{L^6}+C\\|\\n_t\\|_{L^3}\\|\\na u\\|_{L^6}\\|u\\|_{L^\\infty}\\\\\n&\\quad +C\\|u_t\\|_{L^6}\\|\\na u\\|_{L^3}+C\\|\\na u_t\\|_{L^2}\\|u\\|_{L^\\infty}+C\\|\\na P_t\\|_{L^2}\\\\\n&\\le C\\left(\\|\\n u_{tt}\\|_{L^2}+\\|\\na u_t\\|_{L^2}+\\|\\n^{1\/2} \\te_{t}\\|_{L^2}+\\|\\na \\te_t\\|_{L^2}+1\\right),\\ea\\ee\nwhich immediately leads to\\be \\la{asp16}\\ba\n|\\tilde{I}_6| &=\\ka(\\ga-1) \\left| \\int_{ } \\na\\te_t\\cdot\\na{\\rm div}u_tdx \\right| \\\\\n& \\le C \\|\\na^2u_t\\|_{L^2}\\|\\na\\te_t\\|_{L^2}\\\\\n& \\le \\frac{1}{4} \\|\\n^{1\/2} u_{tt}\\|^2_{L^2}+ C\\left(1+\\|\\na u_t\\|^2_{L^2}+\\|\\n^{1\/2} \\te_t\\|^2_{L^2}+\\|\\na\\te_t\\|^2_{L^2}\\right).\n\\ea\\ee\n\nPutting (\\ref{sp11})--(\\ref{sp15}) and (\\ref{asp16}) into\n(\\ref{sp9}) yields\n\\be\\la{4.052} \\ba\n& \\frac{d}{dt}\\int \\left(\\mu|\\curl u_t|^2 + (2\\mu +\\lambda)({\\rm div}u_t)^2-2\\tilde{I}_0\\right)dx\n+\\int \\rho| u_{tt}|^2dx\\\\\n&\\le C\\left(1+\\|\\na u\\|_{L^\\infty}+\\|\\na u_t\\|_{L^2}^2+\\|\\na^2\\te \\|_{L^2}^2 \\right)\\|\\na u_t\\|_{L^2}^2 \\\\\n&\\quad +C\\left(1+\\|\\n_{tt}\\|_{L^2}^2+\\|\\n^{1\/2} \\te_t\\|^2_{L^2}+\\|\\na \\te_t\\|_{L^2}^2\\right).\\ea\\ee\nFurthermore, we infer from $(\\ref{a1})_1$, \\eqref{qq1}, and (\\ref{va5}) that\n\\be \\la{s4} \\ba\n\\|\\n_{tt}\\|_{L^2} &= \\|\\div(\\rho u)_t\\|_{L^2} \\\\\n& \\le C\\left(\\|\\n_t\\|_{L^6}\\|\\nabla u\\|_{L^3}+ \\|\\nabla u_t\\|_{L^2}\n+\\|u_t\\|_{L^6}\\|\\nabla \\n\\|_{L^3}+\\|\\nabla \\n_t\\|_{L^2}\\right) \\\\ &\\le C+C\\|\\na u_t\\|_{L^2}.\\ea\\ee\nMultiplying \\eqref{4.052} by $\\sigma$ and integrating the resulting inequality over $(0,T)$, one thus deduces from \\eqref{nn2}, \\eqref{lee2}, (\\ref{qq1}), (\\ref{va1}), (\\ref{sp10}), (\\ref{s4}),\nand Gr\\\"{o}nwall's inequality that\n\\be\\la{nq11}\n\\sup\\limits_{0\\le t\\le T} \\si \\|\\nabla u_t\\|^2_{L^2}\n+ \\int_0^T\\si\\int\\rho |u_{tt}|^2dxdt\n\\le C.\n\\ee\n\nFinally, by Lemma \\ref{le11}, \\eqref{s4}, (\\ref{va2}), \\eqref{nt4}, \\eqref{va1}, and (\\ref{nq11}), we have\n\\be\\notag\n\\sup\\limits_{0\\le t\\le T}\\si\\left(\\|\\n_{tt}\\|_{L^2}^2+\\|u\\|^2_{H^3}\\right) + \\int_0^T \\si\\|\\nabla u_t\\|_{H^1}^2 dt\\le C,\n\\ee\nwhich along with \\eqref{nq11} gives (\\ref{nq1}).\nWe complete the proof of Lemma \\ref{pe1}.\n\\end{proof}\n\n\\begin{lemma}\\la{pr3} For $q\\in (3,6)$ as in Theorem \\ref{th1}, it holds that\n\t\\be\\la{y2}\\ba\n\t&\\sup_{0\\le t\\le T} \\|\\n\\|_{W^{2,q}} +\\int_0^T\n\t \\|\\na^2u\\|_{W^{1,q}}^{p_0} dt\\le C,\n\t\\ea \\ee\n\twhere \\be 1< \\la{pppppp} p_0<\\frac{4q }{5q-6} \\in (1,4\/3).\\ee\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nFirst, it follows from \\eqref{rmk1}, \\eqref{tb90}, and Lemma \\ref{le11} that\n\\be\\la{a4.74}\\ba\n\\|\\na^2u\\|_{W^{1,q}}\n\\le & C\\left(\\|\\n \\dot u\\|_{W^{1,q}}+\\|\\na P\\|_{W^{1,q}}+\\|\\na u\\|_{L^2}\\right)\\\\\n\\le & C\n(\\|\\na\\dot u\\|_{L^2}+\\|\\na(\\n\\dot u)\\|_{L^q}+ \\|\\na^2\\te\\|_{L^2}+ \\|\\te\\na^2\\n\\|_{L^q}\\\\\n& +\\| |\\na\\n||\\na\\te|\\|_{L^q}+ \\|\\na^2\\te\\|_{L^q}+1)\\\\\n\\le & C\\left(\\|\\na\\dot u\\|_{L^2}+\\|\\na(\\n\\dot u)\\|_{L^q} + \\| \\na^2 \\theta\\|_{L^q}+1 \\right)\\\\\n&+ C(1+ \\|\\na^2 \\theta\\|_{L^2})\\| \\na^2 \\n\\|_{L^q}.\n\\ea\\ee\nNext, multiplying (\\ref{4.52}) by $q|\\p_{ij} \\n|^{q-2}\\p_{ij} \\n$ and integrating the resulting equality over $\\O,$ we obtain after using (\\ref{qq1}) and (\\ref{a4.74}) that\n\\be\\la{sp28}\\ba\n&\\frac{d}{dt}\\|\\na^2\\n\\|_{L^q}^q\\\\\n&\\le C\\|\\na u\\|_{L^\\infty}\\|\\na^2\\n\\|_{L^q}^q\n+C\\|\\na^{2} u\\|_{W^{1,q}}\\|\\na^2\\n\\|_{L^q}^{q-1}(\\|\\na\\n\\|_{L^q}+1)\\\\\n&\\le C\\| u\\|_{H^3}\\|\\na^2\\n\\|_{L^q}^q+C\\|\\na^2 u\\|_{W^{1,q}}\\|\\na^2\\n\\|_{L^q}^{q-1}\\\\\n&\\le C\\left(\\| u\\|_{H^3} + \\|\\na\\dot u\\|_{L^2} +\\|\\na(\\n\\dot u)\\|_{L^q}+\\|\\na^2 \\theta\\|_{L^q}+1\\right)\\left(\\|\\na^2 \\n\\|_{L^q}^q+1\\right).\n\\ea\\ee\n\nNote that Lemma \\ref{le11}, (\\ref{g1}), \\eqref{tb90}, and \\eqref{nq1} imply\n\\be\\la{4.49}\\ba \\|\\na(\\n\\dot u)\\|_{L^q}\n&\\le C\\|\\na \\n\\|_{L^6}\\|\\dot u \\|_{L^{6q\/(6-q)}}+C\\|\\na\\dot u \\|_{L^q}\\\\\n&\\le C\\|\\dot u \\|_{W^{1,6q\/(6+q)}}+C\\|\\na\\dot u \\|_{L^q}\\\\\n&\\le C\\|\\na u_t \\|_{L^q}+C\\|\\na(u\\cdot \\na u ) \\|_{L^q}+C\\\\\n&\\le C\\|\\na u_t \\|_{L^2}^{(6-q)\/2q}\\|\\na u_t \\|_{L^6}^{3(q-2)\/{2q}}\\\\\n& \\quad+C\\|\\na u \\|_{L^q}\\| \\na u \\|_{L^\\infty}+C\\| u \\|_{L^\\infty}\\|\\na^2 u \\|_{L^q}+C\\\\\n&\\le C\\si^{-1\/2} \\left(\\si\\|\\na u_t \\|^2_{H^1}\\right)^{3(q-2)\/{4q}}\n+C\\|u\\|_{H^3}+C,\\ea \\ee\nand\n\\begin{equation}\\notag\n \\begin{aligned}\n \\| \\na^2 \\theta\\|_{L^q} \\le& C \\|\\na^2 \\theta\\|_{L^2}^{(6-q)\/2q} \\|\\na^3 \\theta\\|_{L^2}^{3(q-2)\/2q} +C\\|\\na^2 \\theta\\|_{L^2}\\\\\n \\le & C \\si^{-1\/2} \\left(\\si\\|\\na^3 \\theta \\|^2_{L^2} \\right)^{3(q-2)\/{4q}} +C \\|\\na^2 \\theta\\|_{L^2},\n\\end{aligned}\n\\end{equation}\nwhich combined with Lemma \\ref{le11} and \\eqref{nq1} shows that, for $p_0$ as in (\\ref{pppppp}),\n\\be \\la{4.53}\\int_0^T \\left(\\|\\na(\\n \\dot u)\\|^{p_0}_{L^q} + \\|\\na^2 \\theta\\|_{L^q}^{p_0} \\right) dt\\le C. \\ee\n\nFinally, applying Gr\\\"{o}nwall's\ninequality to (\\ref{sp28}), we obtain after using Lemma \\ref{le11} and (\\ref{4.53}) that\n\\bnn \\sup\\limits_{0\\le t\\le T}\\|\\na^2 \\n\\|_{L^q}\\le C,\\enn\nwhich together with Lemma \\ref{le11}, (\\ref{4.53}),\nand (\\ref{a4.74}) gives (\\ref{y2}). We finish the proof of Lemma \\ref{pr3}.\n\\end{proof}\n\n\\begin{lemma}\\la{sq90} For $q\\in (3,6)$ as in Theorem \\ref{th1}, the following estimate holds:\n\t\\be \\ba\\la{eg17}\n\t&\\sup_{ 0\\le t\\le T}\\si \\left(\\|\\te_t\\|_{H^1}+\\|\\na^2\\te\\|_{H^1}+\\| u_t\\|_{H^2}\n\t+\\| u\\|_{W^{3,q}}\\right) +\\int_0^T \\si^2\\|\\na u_{tt}\\|_{L^2}^2 dt\\le C.\\ea \\ee\n\t\n\\end{lemma}\n\n\\begin{proof}\nFirst, differentiating $(\\ref{nt0})$ with\nrespect to $t$ gives\n\\be\\la{sp30}\\ba \\begin{cases}\n\\n u_{ttt}+\\n u\\cdot\\na u_{tt}-(2\\mu+\\lambda)\\nabla{\\rm div}u_{tt} +\\mu\\na \\times \\curl u_{tt}\\\\\n= 2{\\rm div}(\\n u)u_{tt} +{\\rm div}(\\n u)_{t}u_t-2(\\n u)_t\\cdot\\na u_t\\\\\\quad -(\\n_{tt} u+2\\n_t u_t) \\cdot\\na u\n- \\n u_{tt}\\cdot\\na u-\\na P_{tt}, & \\text{in}\\,\\O\\times[0,T],\\\\\nu_{tt} \\cdot n=0,\\quad \\curl u_{tt} \\times n=0, & \\text{on}\\,\\p\\O\\times[0,T].\n\\end{cases}\\ea \\ee\nMultiplying (\\ref{sp30})$_1$ by $u_{tt}$ and integrating the resulting equality over ${\\Omega}$ by parts imply that\n\\be \\la{sp31}\\ba\n&\\frac{1}{2}\\frac{d}{dt}\\int \\n |u_{tt}|^2dx\n+\\int \\left((2\\mu+\\lambda)({\\rm div}u_{tt})^2+\\mu|\\curl u_{tt}|^2\\right)dx\\\\\n&=-4\\int u^i_{tt}\\n u\\cdot\\na u^i_{tt} dx\n-\\int (\\n u)_t\\cdot \\left(\\na (u_t\\cdot u_{tt})+2\\na u_t\\cdot u_{tt}\\right)dx\\\\\n&\\quad -\\int (\\n_{tt}u+2\\n_tu_t)\\cdot\\na u\\cdot u_{tt}dx\n-\\int \\n u_{tt}\\cdot\\na u\\cdot u_{tt} dx\\\\\n& \\quad+\\int P_{tt}{\\rm div}u_{tt}dx\\triangleq\\sum_{i=1}^5\\tilde{J}_i.\\ea\\ee\nIt follows from Lemmas \\ref{le11}--\\ref{pe1}, (\\ref{va1}), \\eqref{w3}, and \\eqref{s4} that, for $\\eta\\in(0,1],$\n\\be \\la{sp32} \\ba\n|\\tilde{J}_1| &\\le C\\|\\n^{1\/2}u_{tt}\\|_{L^2}\\|\\na u_{tt}\\|_{L^2}\\| u \\|_{L^\\infty} \\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta) \\|\\n^{1\/2}u_{tt}\\|^2_{L^2},\\ea\\ee\n\\be \\la{sp33}\\ba\n|\\tilde{J}_2| &\\le C\\left(\\|\\n u_t\\|_{L^3}+\\|\\n_t u\\|_{L^3}\\right)\n\\left(\\| \\na u_{tt}\\|_{L^2}\\| u_t\\|_{L^6}+\\| u_{tt}\\|_{L^6}\\| \\na u_t\\|_{L^2}\\right)\\\\\n&\\le C\\left(\\|\\n^{1\/2} u_t\\|^{1\/2}_{L^2}\\|u_t\\|^{1\/2}_{L^6}+\\|\\n_t\\|_{L^6}\\| u\\|_{L^6}\\right)\\| \\na u_{tt}\\|_{L^2}\\| \\na u_t\\|_{L^2}\\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\| \\na u_t\\|_{L^2}^{3}+C(\\eta)\\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\si^{-3\/2} ,\\ea\\ee\n\\be \\la{sp34}\\ba\n|\\tilde{J}_3| &\\le C\\left(\\|\\n_{tt}\\|_{L^2}\\|u\\|_{L^6}+\n\\|\\n_{t}\\|_{L^2}\\|u_{t}\\|_{L^6} \\right)\\|\\na u\\|_{L^6}\\|u_{tt}\\|_{L^6} \\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\si^{-1} ,\\ea\\ee\nand\\be \\la{sp36}\\ba &\n|\\tilde{J}_4|+|\\tilde{J}_5|\\\\\n&\\le C\\|\\n u_{tt}\\|_{L^2} \\|\\na u\\|_{L^3}\\|u_{tt}\\|_{L^6}\n+C \\|(\\n_t\\te+\\n\\te_t)_t\\|_{L^2}\\|\\na u_{tt}\\|_{L^2}\\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta) \\left(\\|\\n^{1\/2}u_{tt}\\|^2_{L^2}\n+\\|\\n_{tt}\\te\\|_{L^2}^2+\\|\\n_{t}\\te_t\\|_{L^2}^2 +\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}^2\\right) \\\\\n&\\le \\eta \\|\\na u_{tt}\\|_{L^2}^2+C(\\eta)\\left( \\|\\n^{1\/2}u_{tt}\\|^2_{L^2}\n+\\|\\na\\te_t\\|_{L^2}^2 +\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}^2+\\sigma^{-2}\\right). \\ea\\ee\nSubstituting (\\ref{sp32})--(\\ref{sp36}) into (\\ref{sp31}), we obtain after using \\eqref{nn2} and choosing $\\eta$ suitably small that\n\\be \\la{ex12}\\ba\n& \\frac{d}{dt}\\int \\n |u_{tt}|^2dx+C_4 \\int|\\na u_{tt}|^2dx \\\\\n& \\le C\\si^{-2} +C\\|\\n^{1\/2}u_{tt}\\|^2_{L^2}\n+C\\|\\na\\te_t\\|_{L^2}^2+C_5\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}^2.\\ea\\ee\n\nNext, differentiating \\eqref{3.29} with respect to $t$ infers\n\\be\\la{eg1}\\ba \\begin{cases}\n-\\frac{\\ka(\\ga-1)}{R}\\Delta \\te_t+\\n\\te_{tt}\\\\\n=-\\n_t\\te_{t}- \\n_t\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)-\\n\\left( u\\cdot\\na\n\\te+(\\ga-1)\\te\\div u\\right)_t\\\\\n\\quad+\\frac{\\ga-1}{R}\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_t, \\\\\n\\na \\te_t\\cdot n|_{ \\p\\O\\times(0,T)}=0 .\n\\end{cases}\\ea\\ee\nMultiplying (\\ref{eg1})$_1$ by $\\te_{tt}$ and integrating the resulting\nequality over $\\Omega$ lead to\n\\be\\la{ex5}\\ba\n& \\left(\\frac{\\ka(\\ga-1)}{2R}\\|\\na \\te_t\\|_{L^2}^2+H_0\\right)_t+ \\int\\n\\te_{tt}^2dx \\\\\n&=\\frac{1}{2}\\int\\n_{tt}\\left( \\te_t^2\n+2\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)\\te_t\\right)dx\\\\\n&\\quad + \\int\\n_t\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u \\right)_t\\te_{t}dx\\\\\n& \\quad-\\int\\n\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u\\right)_t\\te_{tt}dx\\\\\n& \\quad -\\frac{\\ga-1}{R}\\int \\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_{tt}\\te_t dx\n\\triangleq\\sum_{i=1}^4H_i,\\ea\\ee\nwhere\n\\bnn\\ba H_0\\triangleq & \\frac{1}{2}\\int \\n_t\\te_{t}^2dx\n+\\int\\n_t\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u\\right) \\te_tdx\\\\\n&- \\frac{\\ga-1}{R}\\int\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2 \\right)_t\\te_t dx. \\ea\\enn\nIt follows from $(\\ref{a1})_1,$ (\\ref{va1}), \\eqref{w3}, \\eqref{s4}, and Lemmas \\ref{le11}--\\ref{pe1} that\n\\be\\la{ex6}\\ba\n|H_0|\\le & C\\int \\n|u||\\te_{t}||\\na\\te_{t}|dx+C\\|\\n_t\\|_{L^3}\\|\\te_t\\|_{L^6}\\left( \\|\\na\\te\\|_{L^2} \\|u\\|_{L^\\infty}+ \\|\\theta\\|_{L^6} \\|\\na u\\|_{L^3}\\right)\\\\\n&+ C\\|\\na u\\|_{L^3}\\|\\na u_t\\|_{L^2} \\|\\te_t\\|_{L^6} \\\\\n\\le &C (\\|\\rho^{1\/2}\\theta_t \\|_{L^2}+\\|\\na\\te_t\\|_{L^2})\\left(\\|\\n^{1\/2}\\te_t\\|_{L^2}+\\|\\na u_t\\|_{L^2}+1\\right)\\\\\n\\le &\\frac{\\ka(\\ga-1)}{4R} \\|\\na\\te_t\\|_{L^2}^2+C\\si^{-1},\\ea\\ee\nand\n\\be\\la{ex7}\\ba\n|H_1|& \\le C\\|\\n_{tt}\\|_{L^2}\\left(\\|\\te_t\\|_{L^4}^{2}\n+\\|\\te_t\\|_{L^6}\\left(\\|u\\cdot\\na \\te\\|_{L^3}+\\| \\te\\div u\\|_{L^3} \\right)\\right)\\\\\n&\\le C\\|\\n_{tt}\\|_{L^2}\\left(\\|\\rho^{1\/2} \\te_t\\|_{L^2}^{2} + \\|\\na \\te_t\\|_{L^2}^{2}\n+\\si^{-1} \\right) \\\\\n& \\le C(1+\\|\\na u_{t}\\|_{L^2} )\\|\\na \\te_t\\|^2_{L^2}+C\\si^{-3\/2}.\\ea\\ee\nCombining Lemma \\ref{le11} with (\\ref{w3}) gives\n\\be\\la{eg12}\\ba\n &\\|\\left(u\\cdot\\na\\te+(\\ga-1)\\te\\div u \\right)_t\\|_{L^2}\\\\\n& \\le C\\left(\\|u_t\\|_{L^6}\\|\\na\\te\\|_{L^3}+\\|\\na\\te_t\\|_{L^2}+\\|\\te_t\\|_{L^6}\\|\\na u\\|_{L^3}\n+\\|\\te\\|_{L^\\infty}\\|\\na u_t\\|_{L^2}\\right)\\\\\n& \\le C\\|\\na u_t\\|_{L^2}(\\|\\na^2 \\te\\|_{L^2}+1)+ C\\|\\na \\te_t\\|_{L^2}+C\\|\\rho^{1\/2} \\theta_t \\|_{L^2},\\ea\\ee\nwhich together with \\eqref{lee2}, \\eqref{va5}, (\\ref{va1}), and \\eqref{w3} shows\n\\be\\la{ex9}\\ba\n|H_2|+|H_3|&\\le C\\left(\\si^{-1\/2}(\\|\\na u_t\\|_{L^2}+1)+\\|\\na\\te_t\\|_{L^2}\\right)\n\\left(\\|\\n_t\\|_{L^3} \\|\\te_t\\|_{L^6}+\\|\\n \\te_{tt}\\|_{L^2}\\right)\\\\\n&\\le \\frac{1}{2}\\int\\n\\te_{tt}^2dx+C\\|\\na\\te_t\\|_{L^2}^2+C\\si^{-1 } \\|\\na u_t\\|^2_{L^2}+C\\si^{-1 } . \\ea\\ee\nOne deduces from \\eqref{qq1}, (\\ref{va1}), \\eqref{w3}, and (\\ref{nq1}) that\n\\be\\la{ex10}\\ba\n|H_4|&\\le C\\int \\left(|\\na u_t|^2+|\\na u||\\na u_{tt}|\\right)|\\te_t|dx\\\\\n&\\le C\\left(\\|\\na u_t\\|_{L^2}^{3\/2}\\|\\na u_t\\|_{L^6}^{1\/2}\n+ \\|\\na u\\|_{L^3} \\|\\na u_{tt}\\|_{L^2}\\right)\\|\\te_t\\|_{L^6}\\\\\n&\\le \\de\\|\\na u_{tt}\\|^2_{L^2}+C\\|\\na^2 u_t\\|^2_{L^2}\n+C(\\de)(\\|\\na\\te_t\\|_{L^2}^2+\\si^{-1 })+C\\si^{-2}\\|\\na u_t\\|_{L^2}^2.\\ea\\ee\nSubstituting (\\ref{ex7}), (\\ref{ex9}), and (\\ref{ex10}) into (\\ref{ex5}) gives\n\\be\\la{ex11}\\ba\n& \\left(\\frac{\\ka(\\ga-1)}{2R}\\|\\na \\te_t\\|_{L^2}^2+H_0\\right)_t\n+\\frac{1}{2}\\int\\n\\te_{tt}^2dx \\\\\n&\\le \\de\\|\\na u_{tt}\\|^2_{L^2}+C(\\de)((1+\\|\\na u_{t}\\|_{L^2}) \\|\\na \\te_t\\|^2_{L^2}+\\si^{-3\/2})\\\\\n&\\quad +C(\\|\\na^2 u_t\\|^2_{L^2} +\\si^{-2}\\|\\na u_t\\|_{L^2}^2).\\ea\\ee\n\n\n\nFinally, for $C_5$ as in (\\ref{ex12}), adding (\\ref{ex11}) multiplied by\n$2 (C_5+1) $ to (\\ref{ex12}) and choosing $\\de$ suitably small yield that\n\\bnn\\la{ex13}\\ba\n& \\left[ 2 (C_5+1)\\left(\\frac{\\ka(\\ga-1)}{2R}\\|\\na \\te_t\\|_{L^2}^2+H_0\\right)\n+\\int \\n |u_{tt}|^2dx\\right]_t\\\\\n&\\quad + \\int\\n\\te_{tt}^2dx+\\frac{C_4}{2}\\int |\\na u_{tt}|^2dx\\\\\n&\\le C (1+\\|\\na u_{t}\\|_{L^2}^2) (\\si^{-2} +\\|\\na \\te_t\\|^2_{L^2})\n+C\\|\\n^{1\/2}u_{tt}\\|^2_{L^2} + C\\|\\na^2 u_t\\|^2_{L^2}.\\ea\\enn\nMultiplying this by $\\si^2$ and integrating the resulting inequality over $(0,T),$\nwe obtain after using (\\ref{ex6}), (\\ref{nq1}), (\\ref{va5}), and Gr\\\"{o}nwall's inequality that\n\\be \\la{eg10}\n\\sup_{ 0\\le t\\le T}\\si^2\\int \\left(|\\na\\te_t|^2+\\n |u_{tt}|^2\\right)dx\n+\\int_{0}^T\\si^2\\int \\left(\\n\\te_{tt}^2+|\\nabla u_{tt}|^2\\right)dxdt\\le C,\\ee\nwhich together with Lemmas \\ref{le11}, \\ref{pe1}, \\ref{pr3}, (\\ref{nt4}), (\\ref{ex4}), \\eqref{va1}, (\\ref{a4.74}),\nand (\\ref{4.49}) gives\n\\be\\la{sp20} \\sup_{ 0\\le t\\le T}\\si \\left(\\|\\na u_t\\|_{H^1}\n+ \\|\\na^2\\te\\|_{H^1}+\\|\\na^2u\\|_{W^{1,q}} \\right)\\le C.\\ee\n\nWe thus derive (\\ref{eg17}) from (\\ref{eg10}), (\\ref{sp20}), \\eqref{va1}, \\eqref{w3},\nand (\\ref{qq1}). The proof of Lemma \\ref{sq90} is completed.\n\\end{proof}\n\n\\begin{lemma}\\la{sq91} The following estimate holds:\n\t\\be \\la{egg17}\\sup_{ 0\\le t\\le T}\\si^2 \\left(\\|\\na^2\\te\\|_{H^2}+\\| \\te_t\\|_{H^2}+\\|\\n^{1\/2}\\te_{tt}\\|_{L^2} \\right)\n\t+\\int_0^T\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2 dt\\le C.\\ee\n\\end{lemma}\n\n\\begin{proof}\nFirst, differentiating $(\\ref{eg1})$ with respect to $t$ yields\n\\be\\la{eg2}\\ba \\begin{cases}\n\\n\\te_{ttt}-\\frac{\\ka(\\ga-1)}{R}\\Delta \\te_{tt}\\\\\n=-\\n u\n\\cdot\\na\\te_{tt}+ 2\\div(\\n u)\\te_{tt} - \\n_{tt}\\left(\\te_t+ u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)\\\\\n\\quad - 2\\n_t\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)_t\\\\\n\\quad - \\n\\left(u_{tt}\\cdot\\na \\te+2u_t\\cdot\\na\\te_t+(\\ga-1)(\\te\\div u)_{tt}\\right)\\\\\n\\quad +\\frac{\\ga-1}{R}\\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2 \\right)_{tt}, \\\\\n\\na\\te_{tt}\\cdot n|_{\\p\\O\\times(0,T)}=0 .\n\\end{cases}\\ea\\ee\nMultiplying (\\ref{eg2})$_1$ by $\\te_{tt}$ and integrating the resulting equality over $\\Omega$ yield that\n\\be\\la{eg3}\\ba\n&\\frac{1}{2}\\frac{d}{dt}\\int\\n|\\te_{tt}|^2dx +\\frac{\\ka(\\ga-1)}{R}\\int|\\na \\te_{tt}|^2dx\\\\\n&=-4\\int \\te_{tt}\\n u\\cdot\\na\\te_{tt}dx -\\int \\n_{tt}\\left(\\te_t\n+ u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)\\te_{tt}dx\\\\\n&\\quad - 2\\int\\n_t\\left(u\\cdot\\na \\te+(\\ga-1)\\te\\div u\\right)_t\\te_{tt}dx\\\\\n& \\quad - \\int\\n\\left(u_{tt}\\cdot\\na \\te+2u_t\\cdot\\na\\te_t\n+(\\ga-1)(\\te\\div u)_{tt}\\right)\\te_{tt}dx\\\\\n& \\quad +\\frac{\\ga-1}{R}\\int \\left(\\lambda (\\div u)^2+2\\mu |\\mathfrak{D}(u)|^2\\right)_{tt}\\te_{tt}dx\n\\triangleq \\sum_{i=1}^5K_i.\\ea\\ee\n\nIt follows from\nLemmas \\ref{le11}--\\ref{pe1}, \\ref{sq90}, \\eqref{kk}, (\\ref{eg10}), and \\eqref{va1} that\n \\be \\la{eg4} \\ba\n\\si^4|K_1|&\\le C\\si^4\\|\\n^{1\/2}\\te_{tt}\\|_{L^2}\\|\\na \\te_{tt}\\|_{L^2}\\|u\\|_{L^\\infty}\\\\\n&\\le \\de \\si^4\\|\\na \\te_{tt}\\|_{L^2}^2+C(\\de) \\si^4\\|\\n^{1\/2}\\te_{tt}\\|^2_{L^2} ,\\ea\\ee\n\\be \\la{eg16}\\ba\n\\si^4|K_2|&\\le C \\si^4\\|\\n_{tt}\\|_{L^2}\\|\\te_{tt}\\|_{L^6} \\left( \\|\\te_t\\|_{H^1}\n+\\|\\na\\te\\|_{L^3}+\\|\\na u\\|_{L^6}\\|\\te\\|_{L^6}\\right) \\\\\n&\\le C\\si^2 (\\|\\na \\te_{tt}\\|_{L^2}+\\|\\n^{1\/2} \\te_{tt}\\|_{L^2})\\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2+C(\\de)(\\si^4\\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2+1),\\ea\\ee\n\\be \\la{eg7}\\ba\n\\si^4|K_4|&\\le C\\si^4\\|\\te_{tt}\\|_{L^6}\n\\left( \\|\\na\\te\\|_{L^3}\\|\\n u_{tt}\\|_{L^2}\n+\\|\\na\\te_t\\|_{L^2}\\|u_t\\|_{L^3}\\right)\\\\\n&\\quad+ C\\si^4\\|\\te_{tt}\\|_{L^6} \\left( \\|\\na u\\|_{L^3}\\|\\n \\te_{tt}\\|_{L^2}\n+ \\|\\na u_t\\|_{L^2}\\| \\te_t\\|_{L^3}\\right)\\\\\n&\\quad+C\\si^4\\|\\te\\|_{L^\\infty}\\|\\n\\te_{tt}\\|_{L^2} \\|\\na u_{tt}\\|_{L^2} \\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2\n+C(\\de)\\left(\\si^4 \\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2+\\si^3\\|\\na u_{tt}\\|_{L^2}^2 \\right)+C(\\de),\\ea\\ee\n\\be \\la{eg8}\\ba\n\\si^4|K_5|&\\le C\\si^4\\|\\te_{tt}\\|_{L^6}\n\\left( \\|\\na u_t\\|_{L^2}^{3\/2}\\|\\na u_{t}\\|_{L^6}^{1\/2}+\\|\\na u\\|_{L^3}\\|\\na u_{tt}\\|_{L^2}\\right) \\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2\n+C(\\de)\\si^4\\left(\\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2+\\|\\na u_{tt}\\|_{L^2}^2 \\right) +C(\\de),\\ea\\ee\nand\n\\be \\la{eg6}\\ba\n\\si^4|K_3|&\\le C \\si^4\\|\\n_t\\|_{L^3} \\|\\te_{tt}\\|_{L^6}\n\\left( \\si^{-1\/2}\\|\\na u_t\\|_{L^2} +\\|\\rho^{1\/2} \\theta_t\\|_{L^2} +\\|\\na\\te_{t}\\|_{L^2} \\right) \\\\\n&\\le \\de\\si^4\\|\\na \\te_{tt}\\|_{L^2}^2+ C \\si^4 \\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2 +C(\\de),\\ea\\ee\nwhere in the last inequality we have used (\\ref{eg12}).\n\nThen, multiplying (\\ref{eg3}) by $\\si^4,$ substituting (\\ref{eg4})--(\\ref{eg6}) into the resulting equality and choosing $\\de$ suitably small, one obtains\n\\bnn \\ba\n& \\frac{d}{dt}\\int\\si^4\\n|\\te_{tt}|^2dx +\\frac{\\ka(\\ga-1)}{R}\\int\\si^4|\\na \\te_{tt}|^2dx\\\\\n& \\le C\\si^2\\left(\\|\\n^{1\/2} \\te_{tt}\\|_{L^2}^2\n+\\|\\na u_{tt}\\|_{L^2}^2 \\right)+C,\\ea\\enn\nwhich together with (\\ref{eg10}) gives\n\\be\\la{eg13} \\sup_{ 0\\le t\\le T}\\si^4\\int \\n |\\te_{tt}|^2dx\n+\\int_{0}^T\\si^4\\int_{ } |\\nabla \\te_{tt}|^2 dxdt\\le C.\\ee\n\nFinally, applying the standard $L^2$-estimate to (\\ref{eg1}), one obtains after using Lemmas \\ref{le11}--\\ref{pe1}, \\ref{sq90}, (\\ref{va1}), and (\\ref{eg13}) that\n\\be\\la{eg14}\\ba\n&\\sup_{0\\le t\\le T}\\si^2\\|\\na^2\\te_t\\|_{L^2}\\\\\n&\\le C\\sup_{0\\le t\\le T}\\si^2\\left(\\|\\n\\te_{tt}\\|_{L^2}\n+ \\|\\n_t\\|_{L^3}\\|\\te_t\\|_{L^6}+\\|\\n_t\\|_{L^6} \\left(\\|\\na\\te\\|_{L^3}+\\|\\te\\|_{L^6}\\|\\na u\\|_{L^6}\\right)\\right)\\\\\n& \\quad +C\\sup_{0\\le t\\le T}\\si^2\\left(\\|\\n^{1\/2} \\te_t\\|_{L^2}+\\|\\na\\te_t\\|_{L^2}+ (1+\\|\\na^2\\te\\|_{L^2}) \\|\\na u_t\\|_{L^2}+\n \\|\\na u_t\\|_{L^6}\\right)\\\\\n& \\le C.\\ea\\ee\nMoreover, it follows from the standard $H^2$-estimate to\n$(\\ref{3.29})$, (\\ref{hs}), \\eqref{nq1}, and Lemma \\ref{le11} that\n\\bnn\\ba \\|\\na^2\\te\\|_{H^2}\n&\\le C\\left(\\|\\n\\te_t\\|_{H^2}+\\|\\n u\\cdot\\na\\te\\|_{H^2}\n+\\|\\n\\te\\div u\\|_{H^2}+\\||\\na u|^2\\|_{H^2}\\right)\\\\\n&\\le C\\left( \\|\\n\\|_{H^2} \\|\\te_t\\|_{H^2}\n+\\|\\n\\|_{H^2} \\| u\\|_{H^2}\\|\\na\\te\\|_{H^2}\\right)\\\\\n&\\quad+C\\|\\n\\|_{H^2} \\|\\te\\|_{H^2} \\| \\div u\\|_{H^2}+C\\|\\na u\\|^2_{H^2}+C\\\\\n&\\le C\\si^{-1}+ C\\| \\na^3\\te \\|_{L^2}+C\\| \\te_t\\|_{H^2}.\\ea\\enn\nCombining this with (\\ref{eg17}), (\\ref{eg14}), and (\\ref{eg13}) shows (\\ref{egg17}).\nThe proof of Lemma \\ref{sq91} is completed.\n\\end{proof}\n\n\n\n\n\n\n\\section{\\la{se5}Proof of Theorems \\ref{th1} and \\ref{th2}}\n\nWith all the a priori estimates in Sections \\ref{se3} and \\ref{se4}\nat hand, we are ready to prove the main results of this paper in\nthis section.\n\n\n\n\\begin{pro} \\la{pro2}\n\n For given numbers $M>0$ (not necessarily small),\n $\\on> 2,$ and $\\bt>1,$ assume that $(\\rho_0,u_0,\\te_0)$ satisfies (\\ref{2.1}), (\\ref{3.1}),\nand (\\ref{z01}). Then there exists a unique classical solution $(\\rho,u,\\te) $ of problem (\\ref{a1})--(\\ref{h1})\n in $\\Omega\\times (0,\\infty)$ satisfying (\\ref{mn5})--(\\ref{mn2}) with $T_0$ replaced by any $T\\in (0,\\infty).$\n Moreover, (\\ref{zs2}), (\\ref{a2.112}), (\\ref{ae3.7}), and (\\ref{vu15}) hold for any $T\\in (0,\\infty)$ and (\\ref{h22}) holds for any $t\\geq 1$.\n\n\n\n \n\n \\end{pro}\n\n\n\n\n\n\n\\begin{proof}\nFirst, by the standard local existence result (Lemma \\ref{th0}), there exists a $T_0>0$ which may depend on\n$\\inf\\limits_{x\\in \\Omega}\\n_0(x), $ such that the problem\n (\\ref{a1})--(\\ref{h1}) with initial data $(\\n_0 ,u_0,\\te_0 )$\n has a unique classical solution $(\\n,u,\\te)$ on $\\O\\times(0,T_0]$ satisfyinng (\\ref{mn6})--(\\ref{mn2}).\nIt follows from (\\ref{As1})--(\\ref{3.1}) and (\\ref{z01}) that\n\\bnn A_1(0)\\le M^2,\\quad A_2(0)\\le C_0^{1\/4},\\quad A_3(0)=0, \\quad \\n_0<\n \\hat{\\rho},\\quad \\te_0\\le \\bt,\\enn which implies there exists a\n$T_1\\in(0,T_0]$ such that (\\ref{z1}) holds for $T=T_1.$\n We set \\bnn \\notag T^* =\\sup\\left\\{T\\,\\left|\\, \\sup_{t\\in [0,T]}\\|(\\n,u,\\te)\\|_{H^3}<\\infty\\right\\},\\right.\\enn and \\be \\la{s1}T_*=\\sup\\{T\\le T^* \\,|\\,{\\rm (\\ref{z1}) \\\nholds}\\}.\\ee Then $ T^*\\ge T_* \\geq T_1>0.$\n Next, we claim that\n \\be \\la{s2} T_*=\\infty.\\ee Otherwise, $T_*<\\infty.$\nProposition \\ref{pr1} shows (\\ref{zs2}) holds for all $00,\\quad\\inf_{x\\in \\Omega}\\te(x,T_*)>0.\\enn\n Thus, Lemma \\ref{th0} implies that there exists some $T^{**}>T_*,$ such that\n(\\ref{z1}) holds for $T=T^{**},$ which contradicts (\\ref{s1}).\nHence, (\\ref{s2}) holds. This along with Lemmas \\ref{th0}, \\ref{a13.1}, \\ref{le8}, and Proposition \\ref{pr1}, thus\nfinishes the proof of Proposition \\ref{pro2}.\n\n\n Finally, it remains to prove (\\ref{y12}). Using $(\\ref{a1})_3$ and (\\ref{mn6}), we can define\n \\bnn\n \\theta_t(\\cdot,0)\\triangleq - u_0 \\cdot\\na \\te_0 + \\frac{\\ga-1}{R} \\rho_0^{-1}\n\\left(\\ka\\Delta\\te_0-R\\rho_0 \\theta_0 \\div u_0+\\lambda (\\div u_0)^2+2\\mu |\\mathfrak{D}(u_0)|^2\\right),\n \\enn\n which along with (\\ref{2.1}) gives\n \\be \\la{ssp91}\\|\\theta_t(\\cdot,0)\\|_{L^2}\\le \\tilde{C}.\\ee\n Thus, one deduces from (\\ref{3.99}), \\eqref{2.1}, \\eqref{ssp91}, and Lemma \\ref{le11} that\n\\be \\ba \\la{a51}\n\\sup\\limits_{0\\le t\\le T} \\int \\n|\\dot\\te|^2dx+\\int_0^T \\|\\na\\dot\\te\\|_{L^2}^2dt \\le \\tilde{C},\n\\ea\\ee\nwhich together with (\\ref{lop4}) and Lemma \\ref{le11} yields\n\\be\\la{sp211}\n\\sup\\limits_{0\\le t\\le T}\\|\\na^2 \\theta\\|_{L^2} \\le \\tilde{C}.\\ee\nUsing $(\\ref{a1})_2$ and (\\ref{mn6}), we can define\n\\bnn\nu_t(\\cdot,0) \\triangleq -u_0\\cdot\\na u_0+\\n_0^{-1}\\left( \\mu \\Delta u_0 + (\\mu+\\lambda) \\na \\div u_0 - R\\na (\\n_0\\te_0)\\right),\n\\enn\nwhich along with (\\ref{2.1}) gives\n\\be \\la{ssp9}\\|\\na u_t(\\cdot,0)\\|_{L^2}\\le \\tilde{C}.\\ee\nThus, it follows from Lemmas \\ref{le11}, \\ref{le9-1}, (\\ref{4.052}), (\\ref{s4}), (\\ref{a51})--(\\ref{ssp9}), and Gr\\\"{o}nwall's inequality that\n\\be \\la{ssp1}\n\\sup_{0\\le t\\le T}\\|\\na u_t\\|_{L^2}+\\int_0^T\\int \\n |u_{tt}|^2dxdt\\le \\tilde{C},\n\\ee\nwhich as well as (\\ref{va2}), \\eqref{sp211}, and (\\ref{qq1}) yields\n\\be\\la{sp221} \\sup\\limits_{0\\le t\\le T}\\|u\\|_{H^3} \\le \\tilde{C}.\\ee\nCombining this with Lemma \\ref{le11}, \\eqref{ex4}, \\eqref{nt4}, (\\ref{a51}), (\\ref{sp211}), (\\ref{ssp1}), and (\\ref{sp221}) gives\n\\be\\la{ssp24} \\ia\\left(\\|\\na^3\\te\\|_{L^2}^2+ \\|\\nabla u_t\\|_{H^1}^2\\right)dt\\le \\tilde{C} . \\ee\nThen, applying (\\ref{rmk1}), \\eqref{hs}, \\eqref{sp211}, \\eqref{ssp1}, \\eqref{sp221}, and Lemma \\ref{le11}, one has\n\\be \\notag\\ba \\|\\na^2 u\\|_{H^2}\n&\\le\\tilde{C}\\left(\\| \\n \\dot u \\|_{H^2}+\\|\\na P\\|_{H^2} + \\|\\na u\\|_{L^2}\\right)\\\\\n&\\le \\tilde{C}\\left(\\| \\n \\|_{H^2}\\| u_t \\|_{H^2}+\\| \\n \\|_{H^2}\\| u \\|_{H^2}\\| \\na u \\|_{H^2}\\right)\\\\\n&\\quad+\\tilde C\\left(\\|\\na\\n \\|_{H^2}\\|\\te\\|_{H^2}+\\|\\n\\|_{H^2}\\|\\na \\te\\|_{H^2}+1\\right)\\\\\n& \\le \\tilde{C} (1+ \\|\\na^2 u_t\\|_{L^2}+\\|\\na^3 \\n \\|_{L^2}+\\|\\na^3 \\te \\|_{L^2}),\\ea\\ee\n\nwhich along with some standard calculations leads to\n \\bnn\\la{sp134}\\ba & \\left(\\|\\na^3 \\n\\|_{L^2} \\right)_t \\\\\n&\\le \\tilde{C}\\left(\\| |\\na^3u| |\\na \\n| \\|_{L^2}+ \\||\\na^2u||\\na^2\n \\n|\\|_{L^2}+ \\||\\na u||\\na^3 \\n|\\|_{L^2} +\\| \\na^4u \\|_{L^2} \\right)\\\\\n&\\le \\tilde{C}\\left(\\| \\na^3 u\\|_{L^2}\\|\\na \\n \\|_{H^2}+ \\| \\na^2u\\|_{L^3}\\|\\na^2 \\n \\|_{L^6}\n +\\|\\na u\\|_{L^\\infty}\\|\\na^3 \\n\\|_{L^2} +\\| \\na^4u \\|_{L^2}\\right) \\\\\n&\\le \\tilde{C}(1+ \\| \\na^3\\n \\|_{L^2}+ \\| \\na^2 u_t\\|^2_{L^2}+ \\|\\na^3\\te\\|^2_{L^2}), \\ea\\enn where we have used (\\ref{sp221}) and Lemma \\ref{le11}.\n Combining this with (\\ref{ssp24}) and\nGr\\\"{o}nwall's inequality yields \\bnn\\la{sp26} \\sup\\limits_{0\\le t\\le\nT}\\|\\nabla^3 \\n\\|_{L^2} \\le \\tilde{C},\\enn which together with\n(\\ref{qq1}) gives (\\ref{y12}).\nThe proof of Proposition \\ref{pro2} is completed.\n\\end{proof}\n\nWith Proposition \\ref{pro2} at hand, we are now in a position to prove Theorem \\ref{th1}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th1}]\n Let $(\\n_0,u_0,\\te_0)$ satisfying (\\ref{co3})--(\\ref{co2}) be the initial data in Theorem \\ref{th1}. Assume that $C_0$ satisfies (\\ref{co14}) with\n\\be\\la{xia}\\ve\\triangleq \\ve_0\/2,\\ee\nwhere $\\ve_0$ is given in Proposition \\ref{pr1}.\n\nFirst, we construct the approximate initial data $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$ as follows. For constants\n\\be\\la{5d0}\nm \\in \\mathbb{Z}^+,\\ \\ \\eta \\in \\left(0, \\eta_0 \\right),\\ \\ \\eta_0\\triangleq \\min\\xl\\{1,\\frac{1}{2}(\\on-\\sup\\limits_{x\\in \\O}\\n_0(x)) \\xr\\},\n\\ee\nwe define\n\\begin{align*}\n\\n_0^{m,\\eta} = \\n_0^{m}+ \\eta,\\ \\ u_0^{m,\\eta}=\\frac{u_0^m }{1+\\eta},\\ \\ \\te_0^{m,\\eta}= \\frac{\\te_0^{m} + \\eta}{1+2\\eta},\n\\end{align*}\nwhere $\\n_0^{m}$ satisfies\n\\begin{align*}\n0 \\le \\n_0^{m} \\in C^{\\infty},\\ \\ \\lim_{m \\to \\infty} \\|\\n_0^{m} -\\rho_0\\|_{W^{2,q}}=0,\n\\end{align*}\n $u_0^m$ is the unique smooth solution to the following elliptic equation:\n\\be\\notag\\begin{cases}\n\t\\Delta u_0^m=\\Delta \\tilde{u}_0^m,&\\text{in}\\,\\, \\O,\\\\\n\tu_0^m\\cdot n=0 ,\\,\\,\\curl u_0^m\\times n=0,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nwith $\\tilde{u}_0^m \\in C^{\\infty}$ satisfying $\\lim_{m \\to \\infty}\\| \\tilde{u}_0^m -{u}_0\\|_{H^2}=0$, and $\\te_0^m$ satisfying $\\int_\\O\\te_0^m dx=\\int_\\O\\te_0 dx$ is the unique smooth solution to the following Poisson equation:\n\\be\\notag\\begin{cases}\n\t\\Delta \\te_0^m=\\Delta \\tilde{\\te}_0^m- \\overline{\\Delta \\tilde{\\te}_0^m},&\\text{in}\\,\\,\\O,\\\\\n\t\\na \\te_0^m\\cdot n=0 ,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nwith $0\\le \\tilde{\\te}_0^m \\in C^{\\infty}$ satisfying $\\lim\\limits_{m \\to \\infty}\\| \\tilde{\\te}_0^m -{\\te}_0\\|_{H^2}=0$.\n\nThen for any $\\eta\\in (0, \\eta_0)$, there exists $m_1(\\eta)\\ge 1$ such that for $m \\ge m_1(\\eta)$, the approximate initial data\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$ satisfies\n\\be \\la{de3}\\begin{cases}(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})\\in C^\\infty ,\\\\\n\t\\dis \\eta\\le \\n_0^{m,\\eta} <\\hat\\n,~~\\, \\frac{\\eta}{4}\\le \\te_0^{m,\\eta} \\le \\hat \\te,~~\\,\\|\\na u_0^{m,\\eta}\\|_{L^2} \\le M, \\\\\n\t\\dis u_0^{m,\\eta}\\cdot n=0,~~\\,\\curl u_0^{m,\\eta}\\times n=0,~~\\,\\na \\te_0^{m,\\eta}\\cdot n=0\\,\\, \\text{on}\\,\\p\\O,\\end{cases}\n\\ee\nand\n \\be \\la{de03}\n\\lim\\limits_{\\eta\\rightarrow 0} \\lim\\limits_{m\\rightarrow \\infty}\n\\left(\\| \\n_0^{m,\\eta} - \\n_0 \\|_ {W^{2,q}}+\\| u_0^{m,\\eta}-u_0\\|_{H^2}+\\| \\te_0^{m,\\eta}- \\te_0 \\|_{H^2}\\right)=0.\n\\ee\nMoreover, the initial norm $C_0^{m,\\eta}$\nfor $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$ which is defined by the right-hand side of (\\ref{e})\nwith $(\\n_0,u_0,\\te_0)$ replaced by\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$\nsatisfies \\bnn \\lim\\limits_{\\eta\\rightarrow 0} \\lim\\limits_{m\\rightarrow \\infty} C_0^{m,\\eta}=C_0.\\enn\nTherefore, there exists an $\\eta_1\\in(0, \\eta_0) $\nsuch that, for any $\\eta\\in(0,\\eta_1),$ we can find some $m_2(\\eta)\\geq m_1(\\eta)$ such that \\be \\la{de1} C_0^{m,\\eta}\\le C_0+\\ve_0\/2\\le \\ve_0 , \\ee\nprovided that\\be \\la{de7}0<\\eta<\\eta_1 ,\\,\\, m\\geq m_2(\\eta).\\ee\n\n\n\n \\iffalse\n First, we choose $0\\le\\n^m_0\\in C^\\infty$ satisfying\n\\be\\la{5d1}\\n^m_0\\rightarrow\\n_0\\quad \\text{in} \\,\\,W^{2,q}\\quad \\text{as}\\,\\,m\\rightarrow \\infty.\\ee\n Combining this with Sobolev's inequality yields that there exists some $m_0(\\eta)\\geq1$ such that for any $m\\geq m_0(\\eta)$,\n\\be\\la{5d2} \\|\\n_0^m-\\n_0\\|_{L^\\infty}\\le \\min\\xl\\{\\frac{1}{4}(\\hat\\n-\\sup\\limits_{x\\in \\O}\\n_0(x)),\\eta\\xr\\}\\ee\nwith fixed $\\eta \\in (0, \\eta_0)$. We set $\\n_0^{m,\\eta}=\\n_0^m+\\eta$ .\n\nNext, for the initial velocity $u_0$,\nthere exists a sequence $v_0^m\\in C^\\infty$ such that $v_0^m\\rightarrow u_0$ in $ H^2$ as $m\\rightarrow \\infty$.\nFor fixed $v_0^m$, the Lame's system\n\\be\\notag\\begin{cases}\n-\\mu\\Delta w_0^m-(\\mu+\\lambda)\\na \\div w_0^m=0,&\\text{in}\\,\\, \\O,\\\\\nw_0^n\\cdot n=v_0^m\\cdot n,\\,\\,\\curl w_0^m\\times n=\\curl v_0^m\\times n,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nadmits a unique smooth solution $w_0^m$. Then, it follows from the standard $H^2$-estimate to the above elliptic system and Sobolev's inequality that\n $w_0^m\\rightarrow 0$ in $H^2$ as $m\\rightarrow \\infty$.\nDenote $u_0^m=v_0^m-w_0^m$, it holds that $u_0^m\\in C^\\infty$ satisfy\n\\be\\la{5d3} u_0^m\\rightarrow u_0\\,\\,\\text{in}\\,\\,H^2~\\mbox{as}~m\\rightarrow \\infty,~~~\\,\\,u_0^m\\cdot n=\\curl u_0^m\\times n=0\\,\\,\\text{on}\\,\\,\\p\\O.\\ee\nHence, for any fixed $\\eta \\in (0, \\eta_0),$ there exists $m_1(\\eta)\\geq m_0(\\eta)$\nsuch that for any $m\\geq m_1(\\eta)$\n\\be\\la{5d4}\\|\\na u_0^m\\|_{L^2}\\le(1+\\eta)\\|\\na u_0\\|_{L^2}.\\ee\n We set $u_0^{m,\\eta}= \\frac{1}{1+\\eta}u_0^m$.\n\n Similarly, we will construct $\\te_0^{m,\\eta}$.\nIt's easy to find a sequence $0\\le f_0^m\\in C^\\infty$ such that $f_0^m\\rightarrow \\te_0$ in $ H^2$ as $m\\rightarrow \\infty$.\n For any fixed $f_0^m$, considering the unique smooth solution $h_0^m$ to the Poisson's equation\n\\be\\notag\\begin{cases}\n\\Delta h_0^m=\\overline{\\Delta f_0^m},&\\text{in}\\,\\,\\O,\\\\\n\\na h_0^m\\cdot n=\\na f_0^m\\cdot n,&\\text{on}\\,\\,\\p\\O,\n\\end{cases}\\ee\nwith $\\overline{h_0^m}=0,$ it is easy to deduce that $h_0^m\\rightarrow 0$ in $H^2$ as $m\\rightarrow \\infty$.\nDenote $\\te_0^{m}= f_0^m-h_0^m$, it holds that $\\te_0^m\\in C^\\infty$ satisfy\n\\be\\la{5d5} \\te_0^m\\rightarrow \\te_0\\,\\,\\text{in}\\,\\,H^2~\\mbox{as}~m\\rightarrow \\infty,~~~\\,\\,\\na \\te_0^m\\cdot n=0\\,\\,\\rm{on}\\,\\,\\p\\O.\\ee\nHence, for any $\\eta \\in (0, \\eta_0),$ there exists $m_2(\\eta)\\geq m_1(\\eta)$\nsuch that for any $m\\geq m_2(\\eta),$\n\\be\\la{5d6}\\|\\te_0^m-\\te_0\\|_{L^\\infty}\\le\\frac{\\eta}{4}.\\ee\nWe finally set $\\te_0^{m,\\eta}= \\frac{1}{1+\\eta}(\\te_0^m+\\frac{\\eta}{2})$.\n\nTherefore, one deduces from \\eqref{co4} and \\eqref{5d0}-\\eqref{5d6} that\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$ satisfies\n\\be \\la{de3}\\begin{cases}(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})\\in C^\\infty ,\\\\\n\\dis \\eta\\le \\n_0^{m,\\eta} <\\hat\\n,~~\\, \\frac{\\eta}{8}\\le \\te_0^{m,\\eta} \\le \\hat \\te,~~\\,\\|\\na u_0^{m,\\eta}\\|_{L^2} \\le M, \\\\\n\\dis u_0^{m,\\eta}\\cdot n =\\curl u_0^{m,\\eta}\\times n=\\na \\te_0^{m,\\eta}\\cdot n=0\\,\\, \\text{on}\\,\\p\\O,\\end{cases}\n\\ee\nprovided that $\\eta \\in (0, \\eta_0)$, $m\\geq m_2(\\eta)$, and \\be \\la{de03}\n\\lim\\limits_{m\\rightarrow \\infty} \\lim\\limits_{\\eta\\rightarrow 0}\n\\left(\\| \\n_0^{m,\\eta} - \\n_0 \\|_ {W^{2,q}}+\\| u_0^{m,\\eta}-u_0\\|_{H^2}+\\| \\te_0^{m,\\eta}- \\te_0 \\|_{H^2}\\right)=0.\n\\ee\n Moreover, the initial norm $C_0^{m,\\eta}$\nfor $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$ the right hand side of (\\ref{e})\nwith $(\\n_0,u_0,\\te_0)$ replaced by\n$(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta}),$\nsatisfy \\bnn \\lim\\limits_{m\\rightarrow \\infty} \\lim\\limits_{\\eta\\rightarrow 0}C_0^{m,\\eta}=C_0.\\enn\nTherefore, there exists an $\\eta_1\\in(0, \\eta_0) $\nsuch that, for any $\\eta\\in(0,\\eta_1),$ we can find some $m_3(\\eta)\\geq m_2(\\eta)$ such that \\be \\la{de1} C_0^{m,\\eta}\\le C_0+\\ve_0\/3\\le \\ve_0 , \\ee\nprovided that\\be \\la{de7}0<\\eta<\\eta_1 ,\\,\\, m\\geq m_3(\\eta).\\ee\n\\fi\n\n We assume that $m,\\eta$ satisfy (\\ref{de7}).\n Proposition \\ref{pro2} together with (\\ref{de1}) and (\\ref{de3}) thus yields that\n there exists a smooth solution $(\\n^{m,\\eta},u^{m,\\eta}, \\te^{m,\\eta}) $\n of problem (\\ref{a1})--(\\ref{h1}) with initial data $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$\n on $\\Omega\\times (0,T] $ for all $T>0. $\n Moreover, one has (\\ref{h8}), \\eqref{zs2}, \\eqref{h22}, (\\ref{a2.112}), \\eqref{ae3.7}, and (\\ref{vu15}) with $(\\n,u,\\te)$ being replaced by $(\\n^{m,\\eta},u^{m,\\eta}, \\te^{m,\\eta}).$\n\n\n Next, for the initial data $(\\n_0^{m,\\eta},u_0^{m,\\eta}, \\te_0^{m,\\eta})$, the function $\\tilde g$ in (\\ref{co12}) is\n \\be \\la{co5}\\ba \\tilde g & \\triangleq(\\n_0^{m,\\eta})^{-1\/2}\\left(-\\mu \\Delta u_0^{m,\\eta}-(\\mu+\\lambda)\\na\\div\n u_0^{m,\\eta}+R\\na (\\n_0^{m,\\eta}\\te^{m,\\eta}_0)\\right)\\\\\n& = (\\n_0^{m,\\eta})^{-1\/2}\\sqrt{\\n_0}g+\\mu(\\n_0^{m,\\eta})^{-1\/2}\\Delta(u_0-u_0^{m,\\eta})\\\\\n&\\quad+(\\mu+\\lambda) (\\n_0^{m,\\eta})^{-1\/2} \\na \\div(u_0-u_0^{m,\\eta})+ R(\\n_0^{m,\\eta})^{-1\/2} \\na(\\n_0^{m,\\eta}\\te_0^{m,\\eta}-\\n_0\\te_0),\\ea\\ee\nwhere in the second equality we have used (\\ref{co2}).\nSince $g \\in L^2,$ one deduces from (\\ref{co5}), (\\ref{de3}), (\\ref{de03}), and (\\ref{co3}) that for any $\\eta\\in(0,\\eta_1),$ there exist some $m_3(\\eta)\\geq m_2(\\eta)$ and a positive constant $C$ independent of $m$ and $\\eta$ such that\n \\be\\la{de4}\n \t\\|\\tilde g\\|_{L^2}\\le \\|g\\|_{L^2}+C\\eta^{-1\/2}\\de(m) + C\\eta^{1\/2},\n \t\\ee\nwith $0\\le\\de(m) \\rightarrow 0$ as\n$m \\rightarrow \\infty.$ Hence, for any $\\eta\\in(0,\\eta_1),$ there exists some $m_4(\\eta)\\geq m_3(\\eta)$ such that for any $ m\\geq m_4(\\eta)$,\n \\be \\la{de9}\\de(m) <\\eta.\\ee We thus obtain from (\\ref{de4}) and (\\ref{de9}) that\nthere exists some positive constant $C$ independent of $m$ and $\\eta$ such that \\be\\la{de14} \\|\\tilde g \\|_{L^2}\\le \\|g \\|_{L^2}+C,\\ee provided that\\be \\la{de10} 0<\\eta<\\eta_1,\\,\\, m\\geq m_4(\\eta).\\ee\n\n\n Now, we assume that $m,$ $\\eta$ satisfy (\\ref{de10}).\n It thus follows from (\\ref{de3})--(\\ref{de1}), (\\ref{de14}), Proposition \\ref{pr1},\n and Lemmas \\ref{le8}, \\ref{le11}--\\ref{sq91} that for any $T>0,$\n there exists some positive constant $C$ independent of $m$ and $\\eta$ such that\n (\\ref{h8}), (\\ref{zs2}), (\\ref{a2.112}), \\eqref{ae3.7}, (\\ref{vu15}), \\eqref{lee2}, \\eqref{qq1}, (\\ref{va5}), (\\ref{vva5}), (\\ref{y2}), (\\ref{eg17}),\n and (\\ref{egg17}) hold for $(\\n^{m,\\eta},u^{m,\\eta}, \\te^{m,\\eta}) .$\n Then passing to the limit first $m\\rightarrow \\infty,$ then $\\eta\\rightarrow 0,$\n together with standard arguments yields that there exists a solution $(\\n,u,\\te)$ of the problem (\\ref{a1})--(\\ref{h1})\n on $\\Omega\\times (0,T]$ for all $T>0$, such that the solution $(\\n,u,\\te)$\n satisfies (\\ref{h8}), (\\ref{a2.112}), \\eqref{ae3.7}, (\\ref{vu15}), \\eqref{lee2}, \\eqref{qq1}, (\\ref{va5}), (\\ref{vva5}), (\\ref{y2}), (\\ref{eg17}), (\\ref{egg17}),\n and the estimates of $A_i(T)\\,(i=1,2,3)$ in\n (\\ref{zs2}). Hence, $(\\n,u,\\te)$ satisfying (\\ref{h8}) and \\eqref{h9} refers to \\cite{H-L} for the detailed proof. Moreover, one deduces from Proposition \\ref{pr1} that the desired exponential decay property \\eqref{h11}.\n\n\n\\iffalse\n $(\\ref{h9})_2,$ $(\\ref{h9})_3 ,$ and for any $0<\\tau2\/m\\}$ and $u_0^{m}$ satisfies\n\\be\\notag u_0^m\\in C^\\infty\\cap H^1_\\omega\\ \\ \\text{and}\\ \\ \\lim_{m\\rightarrow \\infty}\\|u_0^{m}-u_0\\|_{H^1}=0.\\ee\n Then for any $\\eta\\in (0, \\eta_0)$ with $\\eta_0$ as in (\\ref{5d0}), there exists $m(\\eta)>1$ such that for $m \\ge m(\\eta)$, the approximate initial data\n$(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta}, \\hat\\te_0^{m,\\eta})$ satisfies\n\\be \\la{dee3}\\begin{cases}(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta}, \\hat\\te_0^{m,\\eta})\\in C^\\infty ,\\\\\n\t\\dis \\eta\\le \\hat \\n_0^{m,\\eta} <\\hat\\n,~~\\, \\frac{\\eta}{\\hat\\n+\\eta}\\le \\hat\\te_0^{m,\\eta} \\le \\hat \\te,~~\\,\\|\\na \\hat u_0^{m,\\eta}\\|_{L^2} \\le M, \\\\\n\\dis \\hat u_0^{m,\\eta}\\cdot n=0,~~\\,\\curl \\hat u_0^{m,\\eta}\\times n=0,~~\\,\\na \\hat \\te_0^{m,\\eta}\\cdot n=0,\\,\\,\\,\\, \\text{on}\\,\\p\\O,\\end{cases}\n\\ee\nand for any $p\\geq1$,\n \\be \\la{dee03}\n\\lim\\limits_{\\eta\\rightarrow 0} \\lim\\limits_{m\\rightarrow \\infty}\n\\left(\\| \\hat\\n_0^{m,\\eta} - \\n_0 \\|_ {L^p}+\\|\\hat u_0^{m,\\eta}-u_0\\|_{H^1}+\\|\\hat\\n_0^{m,\\eta}\\hat\\te_0^{m,\\eta}-\\n_0\\te_0 \\|_{L^2}\\right)=0\n\\ee\nowing to (\\ref{co4}) and (\\ref{co14}).\n\nNow, we claim that the initial norm $\\hat C_0^{m,\\eta}$\nfor $(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta},\\hat\\te_0^{m,\\eta}),$ i.e., the right hand side of\n(\\ref{e}) with $(\\n_0,u_0,\\te_0)$ replaced by\n$(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta},\\hat\\te_0^{m,\\eta}),$ satisfies\n\\be \\la{uv9} \\lim\\limits_{\\eta\\rightarrow 0}\n\\lim\\limits_{m\\rightarrow \\infty}\\hat C_0^{m,\\eta}\\le C_0,\\ee\nwhich leads to that there exists an $\\hat\\eta\\in(0,\\eta_0)$ such that, for any $\\eta\\in\n(0,\\hat\\eta ),$ there exists some $\\hat m (\\eta)\\geq m(\\eta)$ such that\n\\be \\la{uv8}\\hat C_0^{m,\\eta}\\le C_0+\\ve_0\/2\\le \\ve_0 , \\ee\nprovided \\be\\la{uv01}\n0<\\eta<\\hat\\eta , \\quad m\\geq\\hat m (\\eta).\\ee\nThen if we assume (\\ref{uv01}) holds, it directly follows from Proposition \\ref{pro2}, (\\ref{dee3}) and (\\ref{uv8}) that there exists a classical solution $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta}) $ of problem (\\ref{a1})--(\\ref{h1}) with initial data $(\\hat\\n_0^{m,\\eta},\\hat u_0^{m,\\eta},\\hat\\te_0^{m,\\eta})$ on $\\O\\times(0,T]$ for all $T>0$. Furthermore, $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta}) $ satisfies \\eqref{h8}, (\\ref{zs2}), (\\ref{a2.112}), \\eqref{key}, (\\ref{ae3.7}), (\\ref{vu15}), and \\eqref{h22} respectively for any $T>0$ and $t\\geq1$ with $(\\n ,u ,\\te )$ replaced by $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta})$.\n\n\nIt remains to prove (\\ref{uv9}). Indeed, we just need to infer\n\\be\\la{uv10}\\lim_{\\eta\\rightarrow 0}\\lim_{m\\rightarrow \\infty}\\int\\hat\\n_0^{m,\\eta}\\left(\\hat\\te_0^{m,\\eta}- \\log\n \\hat\\te_0^{m,\\eta} -1 \\right)dx\\le \\int\\n_0\\left(\\te_0- \\log\n \\te_0 -1 \\right)dx ,\\ee since the other terms in (\\ref{uv9}) can be proved in a similar and even simpler way.\nNote that\n\\bnn\\ba\n&\\hat\\n_0^{m,\\eta}\\left(\\hat\\te_0^{m,\\eta}- \\log \\hat\\te_0^{m,\\eta} -1\\right)\\\\\n&=\\hat\\n_0^{m,\\eta}(\\hat\\te_0^{m,\\eta}-1)^2 \\int_0^1\\frac{\\al}{\\al(\\hat\\te_0^{m,\\eta}-1)+1}d\\al\\\\\n&= \\frac{\\hat\\n_0^{m,\\eta}(j_{m^{-1}}*(\\n_0(\\te_0-1)1_{\\O_m}))^2}{j_{m^{-1}}* (\\n_01_{\\O_m} )+\\eta}\\\\\n&\\quad\\cdot\\int_0^1\\frac{\\al}{\\al j_{m^{-1}}*(\\n_0(\\te_0-1)1_{\\O_m})+j_{m^{-1}}* (\\n_01_{\\O_m} )+\\eta}d\\al \\\\\n&\\in \\left[0, \\, \\hat\\n\\eta^{-2}(j_{m^{-1}}*(\\n_0(\\te_0-1)1_{\\O_m}))^2 \\right],\n\\ea\\enn\nwhich combined with Lebesgue's dominated convergence theorem yields that\n\\be \\notag\\ba\n&\\lim_{m\\rightarrow \\infty}\\int\\hat\\n_0^{m,\\eta}\\left(\\hat\\te_0^{m,\\eta}- \\log\n \\hat\\te_0^{m,\\eta} -1 \\right)dx\\\\\n&=\\int(\\n_0+\\eta)\\left(\\frac{\\n_0\\te_0+\\eta}{\\n_0+\\eta}\n - \\log\\frac{\\n_0\\te_0+\\eta}{\\n_0+\\eta}-1 \\right)dx\\\\\n&=\\int\\left(\\n_0\\te_0 -\\n_0+(\\n_0+\\eta)\\log(\\n_0+\\eta)\\right)dx\\\\\n &\\quad-\\int\\left(\\n_0\\log(\\n_0\\te_0+\\eta)+\\eta\\log(\\n_0\\te_0+\\eta)\\right)dx\\\\\n &\\le \\int\\left(\\n_0\\te_0 -\\n_0+(\\n_0+\\eta)\\log(\\n_0+\\eta)\\right)dx-\\int\\left(\\n_0\\log(\\n_0\\te_0)+\\eta\\log\\eta\\right)dx\\\\\n & \\rightarrow \\int \\n_0\\left(\\te_0- \\log\\te_0 -1 \\right)dx,\\quad \\mbox{ as }\\eta\\rightarrow 0.\n\\ea\\ee\nIt thus gives \\eqref{uv10}.\n\n\n\n{\\it Step 2. Compactness results.}\n With the approximate solutions $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta}) $ obtained in the previous step at hand, we can derive the global existence of weak solutions by passing to the limit first $m\\rightarrow \\infty,$ then $\\eta\\rightarrow 0 .$ Since the two steps are similar, we will only sketch the arguments for $m\\rightarrow \\infty.$ For any fixed $\\eta\\in (0,\\hat\\eta)$, we simply denote $(\\hat\\n^{m,\\eta},\\hat u^{m,\\eta},\\hat\\te^{m,\\eta}) $\n by $(\\n^m ,u^m ,\\te^m ).$ Then the combination of Aubin-Lions Lemma with (\\ref{zs2}), (\\ref{a2.112}), \\eqref{key}, (\\ref{vu15}), and Lemma \\ref{le4} yields that there exists some appropriate subsequence $ m_j \\rightarrow \\infty$ of $m\\rightarrow \\infty$ such that, for any $0<\\tau