diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmavz" "b/data_all_eng_slimpj/shuffled/split2/finalzzmavz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmavz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and Main Results}\n\nThe system of Rotating Shallow Water (RSW) equations is a widely\nadopted 2D approximation of the 3D incompressible Euler equations\nand the Boussinesque equations in the regime of large scale\ngeophysical fluid motion (\\cite{Pedlosky}). It is also regarded as\nan important extension of the compressible Euler equations with\nadditional rotational forcing.\n\nStart with the following formulation,\n\\begin{align}\n\\label{RSWh}\\partial_t h+\\nabla\\!\\!\\cdot\\!(h{\\bf u})&=0,\\\\\n\\label{RSWu} \\partial_t{\\bf u}+{\\bf u}\\!\\cdot\\!\\!\\nabla{\\bf u}+\\nabla h+{\\bf u}^\\perp&=0,\n\\end{align}\nwhere $h=h(t,x_1,x_2)$ and $ {\\bf u}=(u_1(t,x_1,x_2),u_2(t,x_1,x_2))^T$ denote the total height and\nvelocity of the fluids, respectively, and ${\\bf u}^\\perp:=(-u_2, u_1)^T$\ncorresponds to the rotational force. For mathematical convenience,\nall physical parameters are scaled to the unit (cf. \\cite{Majda} for\ndetailed discussion on scaling).\n\nSince $(h, {\\bf u})=(1,0)$ is a steady-state solution of (\\ref{RSWh}), (\\ref{RSWu}), we introduce the perturbations\n$(\\rho,{\\bf u}):=(h-1,{\\bf u})$ and arrive at\n\\begin{align}\n\\label{RSWrho}\\partial_t\\rho+\\nabla\\!\\!\\cdot\\!(\\rho{\\bf u})+\\nabla\\!\\!\\cdot\\!{\\bf u}&=0,\\\\\n\\label{RSWurho} \\partial_t{\\bf u}+{\\bf u}\\!\\cdot\\!\\!\\nabla{\\bf u}+\\nabla\\rho+ {\\bf u}^\\perp&=0,\n\\end{align}\nsubject to initial data\n\\begin{equation}\\label{RSWinit}\\quad\\rho(0,\\cdot)=\\rho_0,\\;{\\bf u}(0,\\cdot)={\\bf u}_0.\\end{equation}\n\nAn important feature of the RSW system is that the relative\nvorticity \\(\\theta:=\\nabla\\times{\\bf u}-\\rho=(\\partial_1 u_2-\\partial_2 u_1)-\\rho\\) is\nconvected by ${\\bf u}$, \\begin{equation}\\label{theta}\\partial_t\\theta+\\nabla\\!\\!\\cdot\\!(\\theta{\\bf u})=0.\\end{equation}\nIndeed, $\\nabla\\times$(\\ref{RSWurho})$-$(\\ref{RSWrho}) readily leads\nto (\\ref{theta}). The linearity of (\\ref{theta}) then suggests that\n$\\theta\\equiv0$ be an invariant with respect to time (as long as\n${\\bf u}\\in C^1$), i.e. \\begin{equation}\\label{inv}\n\\theta_0\\equiv0\\iff\\theta(t,\\cdot)\\equiv0\\iff\\nabla\\times{\\bf u}\\equiv\\rho.\n\\end{equation}\n\nBefore stating the main theorems, we fix some notations. For $1\\leq\np\\leq \\infty$, let $L^p$ denote the standard $L^p$ space on $\\mbox{R}^2$.\nFor $l \\geq 0$ and $s\\geq 0$, define the weighted Sobolev norm\nassociated with the space $H^{l,s}$ as\n\\begin{align}\\label{def:weighted}\n\\|v\\|_{H^{l,s}}:=\\|(1+|x|^2)^{s\/2}(1-\\Delta)^{l\/2} v\\|_{L^2}.\n\\end{align}\nAlso, denote the standard Sobolev space $H^l:=H^{l,0}$.\n\n\\begin{theorem}\\label{mainTh}\nConsider the RSW system (\\ref{RSWrho}), (\\ref{RSWurho}), (\\ref{RSWinit}) with initial data ${\\bf u}_0=(u_{1,0}, u_{2,0})^T\\in H^{k+2,k}$ for $k\\ge52$ and zero relative vorticity,\n\\[\n\\rho_0=\\partial_1 u_{2,0}-\\partial_2 u_{1,0}.\n\\]\nThen, there exists a universal constant $\\delta_0>0$ such that the\nRSW system admits a unique\nclassical solution $(\\rho, {\\bf u})$ for all time, provided that the\ninitial data satisfy\n\\begin{equation*}\\label{Isizeu}\n\\|{\\bf u}_0\\|_{H^{k+2,k}}=\\delta<\\delta_0.\n\\end{equation*}\nMoreover, there exists a free solution ${\\bf u}^+(t,\\cdot)$ such that \n\\[\\|{\\bf u}(t,\\cdot)-{\\bf u}^+(t,\\cdot)\\|_{H^{k-15}}+\\|\\partial_t{\\bf u}(t,\\cdot)-\\partial_t{\\bf u}^+(t,\\cdot)\\|_{H^{k-16}}\\leq C(1+t)^{-1},\\]\nwhere\n${\\bf u}^+(t,\\cdot):=(\\cos(1-\\Delta)^{1\/2}\nt){\\bf u}_0^++\\left((1-\\Delta)^{-1\/2}\\sin((1-\\Delta)^{1\/2}\nt)\\right){\\bf u}_1^+$ for some ${\\bf u}_{0}^+\\in H^{k-15}$ and ${\\bf u}_1^+\\in H^{k-16}$.\n\\end{theorem}\nThe proof is a straightforward combination of Lemma \\ref{symm}, Theorem \\ref{KGsystem} and Theorem \\ref{asymTh} below.\n\nThis result shows fundamentally different lifespan of the classical\nsolutions for the RSW system in comparison with the compressible\nEuler systems. Note that the relative vorticity $\\theta=\\nabla\\times{\\bf u}-\\rho$ in the RSW\nequations plays a very similar role as vorticity $\\nabla\\times{\\bf u}$ in the compressible\nEuler equations. Correspondingly, RSW solutions with zero relative\nvorticity $\\theta$ is an analogue of irrotational solutions for the\ncompressible Euler equations. However, the life span for 2D\ncompressible Euler equations with zero vorticity was proved to be\nbounded from below (Sideris \\cite{Sideris:2D}) and above (Rammaha\n\\cite{Rammaha}) by $O({1}\/{\\delta^2})$. Here, $\\delta$ indicates the\nsize of the initial data. Sideris also showed that the life span in\nthe 3D case is bounded from below by $O(e^{1\/\\delta})$ in\n\\cite{Sideris:3D} and from above by $O(e^{1\/{\\delta^2}})$ in\n\\cite{Sideris:3D:singularity}. Our result for 2D RSW system, on the other hand, is global in time due to the additional rotating\nforce. Consult \\cite{Babin, LiTa:rotation, ChTa:SIAM} for related results on global and long-time existence of classical RSW solutions in various regimes.\n\nA key ingredient of the proof is to treat the RSW system as a system\nof quasilinear Klein-Gordon equations -- cf. Lemma \\ref{symm} for a formal\ndiscussion. Such reformulation allows us to utilize the fruitful\nresults on nonlinear Klein-Gordon equations appearing in recent decades. To\nmention a few, for spatial dimensions $N\\ge5$, Klainerman and\nPonce \\cite{KlainermanP} and Shatah \\cite{Shatahev} showed that the\nKlein-Gordon equation admits a unique, global solution for small\ninitial data and that the solution approaches the free solution of\nthe linear Klein-Gordon equation as $t\\rightarrow \\infty$. The\nproofs in \\cite{KlainermanP, Shatahev} are based on $L^p-L^q$ decay\nof the linear Klein-Gordon equations. The global existence for quasilinear Klein-Gordon equations in dimensions $N=4,3$ was proved\nindependently by Klainerman in \\cite{Klainerman} using the vector\nfields approach and Shatah in \\cite{Shatah} using the normal forms.\nIn the $N=2$ case, global existence of classical solutions become increasingly subtle due\nto the $(1+t)^{-1}$ decay rate of solutions to linear Klein-Gordon equations. Nevertheless, it has been proved by Ozawa et al in\n\\cite{OzawaSL} for semilinear, scalar equations. The authors combined\nthe vector fields approach and the normal form method after partial\nresults in \\cite{GeorgievP, Georgiev, Simon}. The result of global existence on\nquasilinear, scalar Klein-Gordon equations was announced in\n\\cite{OzawaQL}. Recently, Delort et al obtained global existence for\na two dimensional system of two Klein-Gordon equations in\n\\cite{Fang}, where the authors transformed the problem using\nhyperbolic coordinates and then studied it with the vector fields\napproach, which was restricted to compactly supported initial data.\nFor applications of the Klein-Gordon equations in fluid equations,\nwe refer to \\cite{Guo} by Y. Guo on global existence of three\ndimensional Euler-Poisson system. Note that the irrotationality\ncondition used there plays a counterpart of the\nzero-relative-vorticity constraint in our result.\n\nFor general initial data, we have the following theorem on the\nlifespan of classical solutions. Its proof is given in Section 5.\n\\begin{theorem}\\label{perTh}\nConsider the RSW system (\\ref{RSWrho}), (\\ref{RSWurho}), (\\ref{RSWinit}) with initial data $(\\rho_0,{\\bf u}_0)\\in H^{k+1,k}$ for $k\\ge52$. Let $\\delta$ denote the size of the initial data \\begin{equation*}\n\\delta=\\|(\\rho_0,{\\bf u}_0)\\|_{H^{k+1,k}},\n\\end{equation*}\n and $\\varepsilon$ the size of the initial relative vorticity,\n \\[\\varepsilon=\\|(\\partial_1 u_{2,0}-\\partial_2 u_{1,0})-\\rho_0\\|_{H^2}.\\]\nThen, there exists a universal constant $\\delta_0>0$ such that, for any $\\delta\\le\\delta_0$, the\nRSW system admits a unique\nclassical solution $(\\rho, {\\bf u})$ for \\begin{equation}\\label{span:general}t\\in[0,\nC_1\\varepsilon^{-\\frac{1}{1+C_2\\delta}}].\\end{equation}\nHere, $C_1$ and $C_2$ are constants independent of $\\delta$ and $\\varepsilon$. \n\\end{theorem}\n\nThis theorem confirms the key role that relative vorticity plays in the studies of Geophysical Fluid Dynamics (\\cite{Pedlosky}). In fact, having two uncorrelated scales $\\varepsilon$ and $ \\delta$ in (\\ref{span:general})\nallows us to solely let the size of the initial relative vorticity\n$\\varepsilon\\to0$ and achieve a very long lifespan of classical solutions, regardless of the total size of initial data. The proof, given in Section \\ref{sec:general}, treats the full RSW system as perturbation to the zero-relative-vorticity one and utilizes the standard energy methods for symmetric hyperbolic PDE systems. The sharp estimates of Theorem \\ref{mainTh} play a crucial role in controlling the total energy growth. We note that a similar problem of the compressible Euler equations is studied by Sideris in \\cite{Sideris:2D}.\n\nWe also note by passing that the study of hyperbolic PDE systems\nwith small initial data is closely related, if not entirely\nequivalent, to the singular limit problems with large initial data.\nSee \\cite{Majda}, \\cite{Babin} and references therein for results on\nthe particular case of inviscid RSW equations. A recent survey paper\n\\cite{Bresch} contains a collection of open problems and recent\nprogress on viscous Shallow Water Equations and related models.\n\nWe finally comment that all results in this paper should be true for two dimensional compressible Euler equations with rotating force and general pressure law. The proof remains largely the same except for the symmetrization part and associated energy estimates.\n\nThe structure of the rest of the paper is outlined as following. In\nSection 2, we reformulate the RSW system into a symmetric hyperbolic\nsystem of first order PDEs. Under the zero-relative-vorticity\nconstraint, it is further transformed into a system of quasilinear\nKlein-Gordon equations with symmetric quasilinear part. Section 3 is\ndevoted to the local wellposedness of the RSW equation with general\ninitial data and zero-relative-vorticity initial data. Section 4\ncontains the proof of Theorem \\ref{mainTh} in a series of lemmas,\nadapting results from \\cite{Georgiev}, \\cite{Shatah}, \\cite{OzawaSL}.\nThe discussion and proof of Theorem \\ref{perTh} on general initial\ndata can be found in Section 5. The Appendix contains the proof of a\ntechnical proposition used in Section 4.\n\n\\section{Reformulation of the Problem}\n\nIn order to obtain local wellposedness for (\\ref{RSWrho}) and\n(\\ref{RSWurho}), we first symmetrize the system into a symmetric\nhyperbolic system. This will also be used for proving global\nexistence, where we need to reduce (\\ref{RSWrho}) and(\\ref{RSWurho})\nto a symmetric quasilinear Klein-Gordon system.\n\nIntroduce a symmetrizer $m:=2\\left(\\sqrt{1+\\rho}-1\\right)$ such that\n$\\rho=m+{1\\over4}m^2$, then (\\ref{RSWrho}), (\\ref{RSWurho}) are\ntransformed into a symmetric hyperbolic PDE system,\n\\begin{align}\n\\label{RSWm}\\partial_t m+{\\bf u}\\!\\cdot\\!\\!\\nabla m+\\dfrac{1}{2} m\\nabla\\!\\!\\cdot\\!{\\bf u}+\\nabla\\!\\!\\cdot\\!{\\bf u}&=0,\\\\\n\\label{RSWum} \\partial_t{\\bf u}+{\\bf u}\\!\\cdot\\!\\!\\nabla{\\bf u}+\\dfrac{1}{2} m\\nabla m+\\nabla m+{\\bf u}^\\perp&=0.\n\\end{align}\n\nThe following lemma asserts that, under the invariant (\\ref{inv}),\nthe above system amounts to a system of Klein-Gordon equations with\nsymmetric quasilinear terms.\n\n\\begin{lemma}\\label{symm}\nUnder the invariant (\\ref{inv}), $\\nabla\\times{\\bf u}=\\rho$, and\ntransformation $m=2\\left(\\sqrt{\\rho+1}-1\\right)$, the solution to\nthe RSW system (\\ref{RSWrho}), (\\ref{RSWurho}) satisfies the\nfollowing symmetric system of quasilinear Klein-Gordon equations for\n${\\bf U}:=\\left(\\begin{matrix} m\\\\{\\bf u}\\end{matrix}\\right)$, \\begin{equation}\\label{ptt}\n\\partial_{tt}{\\bf U}-\\Delta{\\bf U}+{\\bf U}=\\sum_{i,j=1}^2A_{ij}({\\bf U})\\partial_{ij}{\\bf U} +\n\\sum_{j=1}^2A_{0j}({\\bf U})\\partial_{0j}{\\bf U}+R(\\tilde{\\vU}\\otimes\\tilde{\\vU}), \\end{equation} where\nlinear functions $A_{ij}$ and $A_{0j}$ map $\\mbox{R}^3$ vectors to\n\\emph{symmetric} $3\\times 3$ matrices and satisfy $A_{ij}=A_{ji}$.\nThe remainder term $R$ depends linearly on the tensor product\n$\\tilde{\\vU}\\otimes\\tilde{\\vU}$ with $\\tilde{\\vU}:=({\\bf U}^T,\\partial_t{\\bf U}^T,\\partial_1{\\bf U}^T,\\partial_2{\\bf U}^T)$.\n\\end{lemma}\n\nHere and below, for notational convenience, we use both $\\partial_t$ and\n$\\partial_0$ to denote the time derivatives.\n\\begin{proof}\nRewrite (\\ref{RSWm}), (\\ref{RSWum}) into a matrix-vector form,\n\\begin{equation}\n\\label{RSWmatrix}\\partial_t{\\bf U}+\\sum_{a=1,2}(u_aI+\\dfrac{1}{2} mJ_a)\\partial_a{\\bf U}={\\mathcal L}({\\bf U}),\n\\end{equation}\nwhere\n\\begin{equation}\\label{defJL}\nJ_1:=\\bma0&1&0\\\\1&0&0\\\\0&0&0\\end{matrix}\\right),\\,\\,\\,\\, J_2:=\\bma0&0&1\\\\0&0&0\\\\1&0&0\\end{matrix}\\right),\n\\,\\,\\, \\, \\text{and}\\,\\, {\\mathcal L}({\\bf U}):=-\\left(\\begin{matrix} \\nabla\\!\\!\\cdot\\!{\\bf u}\\\\\n\\nabla m+{\\bf u}^\\perp\\end{matrix}\\right). \\end{equation}\n\nBy taking time derivative on the above system, we have\n\\[\\partial_{tt}{\\bf U}+N({\\bf U})={\\mathcal L}^2({\\bf U}),\\]\nwhere the nonlinear term\n\\begin{equation}\\label{N1N2}\\begin{split}\nN({\\bf U})&=\\partial_t\\sum_{a=1,2}(u_aI+\\dfrac{1}{2}\nmJ_a)\\partial_a{\\bf U}+{\\mathcal L}\\left(\\sum_{a=1,2} (u_aI+\\dfrac{1}{2}\nmJ_a)\\partial_a{\\bf U}\\right)\\\\&:=N_1+N_2.\n\\end{split}\\end{equation}\nThe ${\\mathcal L}^2$ term, using the calculus identity\n$\\nabla(\\nabla\\cdot{\\bf u}) -\\nabla^\\perp(\\nabla\\times{\\bf u})=\\Delta{\\bf u}$,\nis\n\\begin{equation}\\label{N3}\\begin{split}\n{\\mathcal L}^2({\\bf U})&=\\left(\\begin{matrix} \\nabla\\!\\!\\cdot\\!(\\nabla m+{\\bf u}^\\perp)\\\\ \\nabla(\\nabla\\!\\!\\cdot\\!{\\bf u})+\n(\\nabla m+{\\bf u}^\\perp)^\\perp\\end{matrix}\\right)\\\\\n&=\\left(\\begin{matrix}(\\Delta-1)m -(\\nabla\\times{\\bf u}-m)\\\\\n(\\Delta-1){\\bf u}+\\nabla^\\perp(\\nabla\\times{\\bf u}-m)\\end{matrix}\\right)\\\\\n&=(\\Delta-1){\\bf U}+\\left(\\begin{matrix}-(\\nabla\\times{\\bf u}-\\rho+{1\\over4} m^2)\\\\\n\\nabla^\\perp(\\nabla\\times{\\bf u}-\\rho+{1\\over 4} m^2)\\end{matrix}\\right) \\qquad\n\\mbox{since } \\rho=m+{1\\over4}m^2 \\\\\n&=(\\Delta-1){\\bf U}+{1\\over4}\\left(\\begin{matrix}-m^2\\\\\n\\nabla^\\perp(m^2)\\end{matrix}\\right)\\qquad\\mbox{ under (\\ref{inv})}\\\\\n&:=(\\Delta-1){\\bf U}+N_3.\\end{split}\\end{equation}\n\nNow that we've revealed the Klein-Gordon structure of (\\ref{ptt}),\nit suffices to show that the other terms $N_1,N_2,N_3$ can all be\nsplit into a symmetric second order part and a remainder lower order\npart as given in (\\ref{ptt}).\n\\begin{itemize}\n\\item The $N_1$ term in (\\ref{N1N2}). It is easy to see that the lower\norder terms (with less than second order derivatives) in $N_1$ are\nquadratic in $\\tilde{\\vU}$, i.e. linear in $\\tilde{\\vU}\\otimes\\tilde{\\vU}$. The terms\nwith second order derivatives are\n\\[\\sum_{a=1,2}(u_aI+\\dfrac{1}{2} mJ_a)\\partial_t\\partial_a{\\bf U}\\]\nwhere matrices $I$, $J_a$ are all symmetric. \n\\item The $N_2$ term in (\\ref{N1N2}).\nObserve that the linear operator ${\\mathcal L}$ partially comes from a\nlinearization of the nonlinear terms in (\\ref{RSWmatrix}) and it\nindeed can be represented as\n\\[{\\mathcal L}({\\bf U})=\\sum_{a=1,2}J_a\\partial_a{\\bf U}+K{\\bf U}\\]\nwith constant matrix $K$. Thus, manipulate the $N_2$ term,\n\\[\\begin{split}\nN_2&=\\sum_{a=1,2}J_a\\partial_a\\left(\\sum_{b=1,2}(u_bI+\\dfrac{1}{2} mJ_b)\\partial_b{\\bf U}\\right)\n+K\\left(\\sum_{b=1,2}(u_bI+\\dfrac{1}{2} mJ_b)\\partial_b{\\bf U}\\right)\\\\\n&=\\sum_{a=1,2}\\sum_{b=1,2}(u_bJ_{a}+\\dfrac{1}{2} mJ_{a}J_b)\\partial_{a}\\partial_b{\\bf U}+\n\\mbox{ quadratic terms of }\\tilde{\\vU}.\n\\end{split}\\]\nThe quasilinear terms above have the desired symmetric structure\nsince for each index pair $(a,b)$, the coefficient of\n$\\partial_{a}\\partial_b{\\bf U}=\\partial_b\\partial_{a}{\\bf U}$ is\n\\[\n\\frac{1}{2}(u_bJ_{a}+\\dfrac{1}{2}\nmJ_{a}J_b)+\\frac{1}{2}(u_{a}J_{b}+\\dfrac{1}{2} mJ_{b}J_{a})\n\\]\nwhich, by the definition of $J_a$, is symmetric.\n\\item The $N_3$ term in (\\ref{N3}). By\ndefinition, this term has no second order derivatives and is\nquadratic in $\\tilde{\\vU}$.\n\\end{itemize}\n\\end{proof}\n\n\n\\section{Local Wellposedness}\n\nAs in the previous section, let\n$\\partial_0=\\frac{\\partial}{\\partial t}$, $\\partial_1=\\frac{\\partial}{\\partial x_1}$, $\\partial_2=\n\\frac{\\partial}{\\partial x_2}$. Define the vector fields\n\\begin{align}\\label{def:vfields}\n\\Gamma:=\\{\\Gamma_j\\}_{j=1}^6=\\{\\partial_0,\\partial_1,\\partial_2, L_1, L_2,\n\\Omega_{12}\\},\n\\end{align}\nwhere\n\\begin{align*}\nL_j:=x_j\\partial_t+t\\partial_j,\\,\\,j=1,2; \\quad\n\\Omega_{12}:=x_1\\partial_2-x_2\\partial_1.\n\\end{align*}\nWe abbreviate\n\\begin{align*}\n\\partial^{\\alpha}=\\partial_t^{\\alpha_1}\\partial_1^{\\alpha_2}\\partial_2^{\\alpha_3}\n\\quad\\mbox{ for } \\alpha=(\\alpha_1,\\alpha_2,\\alpha_3),\n\\end{align*}\nand\n\\begin{align*}\n\\Gamma^{\\beta}=\\Gamma_1^{\\beta_1}\\cdots\\Gamma_6^{\\beta_6}\n\\quad\\mbox{ for } \\beta=(\\beta_1,\\cdots, \\beta_6).\n\\end{align*}\n\nThe local wellposedness of (\\ref{RSWm}) and (\\ref{RSWum})\nand their regularity are contained in the following theorem.\n\\begin{theorem}\\label{LocalTh}\n\\begin{itemize}\n\\item[(i)] Let $(m_0,{\\bf u}_0)\\in H^n$ with $n\\ge3$. Then, there exists a $T>0$\ndepending only on $\\|(m_0,{\\bf u}_0)\\|_{H^3}$ such that (\\ref{RSWm}), (\\ref{RSWum})\nadmit a unique solution ${\\bf U}=\\left(\\begin{matrix} m \\\\ {\\bf u}\\end{matrix}\\right) $ on $[0, T]$ satisfying\n\\begin{align}\n{\\bf U}\\in \\bigcap_{j=0}^{n} C^j([0,T], H^{n-j}).\n\\end{align}\n\\item[(ii)] Under the assumptions in (i), also assume\n$\\rho_0=\\partial_1 u_{2,0}-\\partial_2 u_{1,0}$. Then,\n\\begin{align}\\label{Zerovor2}\n\\rho(t,\\cdot)=\\partial_1 u_2(t,\\cdot)-\\partial_2 u_1(t,\\cdot)\\,\\, \\text{for}\\,\\, t\\in[0,T],\n\\end{align}\nand ${\\bf U}=\\left(\\begin{matrix} m\\\\{\\bf u}\\end{matrix}\\right)=\\bma2\\left(\\sqrt{\\rho+1}-1\\right)\\\\{\\bf u}\\end{matrix}\\right)$\nsatisfies the Klein-Gordon system (\\ref{ptt}). If the initial data\nbelong to the weighted Sobolev space (cf. definition\n(\\ref{def:weighted})) such that ${\\bf U}_0\\in H^{k+1,k}$ with $k\\geq\n3$, then the above solution ${\\bf U}$ satisfies\n\\begin{align}\\label{locvec}\n\\Gamma^{\\alpha}{\\bf U},\\,\\Gamma^{\\alpha}\\partial{\\bf U}\\in C([0,T], L^2),\\qquad(1+|x|)\\Gamma^{\\beta}{\\bf U}\\in C([0,T],L^\\infty),\n\\end{align}\nfor any multi-indices $|\\alpha|\\leq k$ and $|\\beta|\\leq k-3$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nThe proof of (i) follows from the standard local wellposedness and\nregularity theory for symmetric hyperbolic system, cf. \\cite{Kato,\nMajda84}.\n\nFor part (ii), (\\ref{Zerovor2}) comes from the derivation of\n(\\ref{inv}). Then, it follows from Lemma \\ref{symm} that ${\\bf U}$\nsolves (\\ref{ptt}). Finally, the proof of (\\ref{locvec}) is based on\nthe arguments in \\cite{KlainermanP, Shatahev}. Note that, using\n(\\ref{RSWm}), (\\ref{RSWum}), one has\n\\begin{align*}\n {\\bf U}(0,\\cdot)\\in H^{k+1,k} \\text{ and } \\partial_t^l {\\bf U}(0,\\cdot)\\in H^{k+1-l,k}\n \\end{align*}\nas long as ${\\bf u}_0\\in H^{k+2,k}$ and $\\rho_0=\\nabla\\times{\\bf u}_0$.\n\\end{proof}\n\n\\section{Global Existence and Asymptotic Behavior with Zero Relative Vorticity}\n\nThroughout this section, we focus on the solutions with zero\nrelative vorticity -- cf. (\\ref{inv}). Theorem \\ref{LocalTh}, part\n(ii), suggests that the RSW system be treated as a system of\nquasilinear Klein-Gorden equations. To this end, it is convenient to\nintroduce the following generalized Sobolev norms associated with\nvector fields $\\Gamma$ defined in (\\ref{def:vfields}).\n\\begin{align*}\n|{\\bf U}\\RGN{l,d}(t):=&\n\\sum_{|\\alpha|\\leq l }|(1+t+|x|)^{-d}\\Gamma^{\\alpha} {\\bf U}(t,x)|_{L^{\\infty}_x},\\\\\n\\|{\\bf U}\\RGNN{l}(t):=&\\sum_{|\\alpha|\\leq l}\n\\|\\Gamma^{\\alpha} {\\bf U}(t,x) \\|_{L^2_x}.\n\\end{align*}\n\nTo extend the local solution of (\\ref{RSWm}) and (\\ref{RSWum})\nglobally in time, we need to derive the decay and energy estimates\nof solutions to (\\ref{ptt}). We start with\ndefining a functional (see e.g. \\cite{OzawaSL}) measuring the size of the solution at time\n$t\\ge0$,\\begin{equation}\\label{def:X}\\begin{split}\nX(t):=&\\sup_{s\\in[0,t]}\\left\\{|{\\bf U}\\RGN{k-25,-1}(s)+\\|{\\bf U}\\RGNN{k-9}(s)+\\|{\\partial} {\\bf U}\\RGNN{k-9}(s)\\right.\\\\\n&\\left.+(1+s)^{-\\sigma}\\|{\\bf U}\\RGNN{k}(s)+(1+s)^{-\\sigma}\\|{\\partial}\n{\\bf U}\\RGNN{k}(s)\\right\\},\n\\end{split}\\end{equation}\nhere, pick any fixed $\\sigma\\in(0,1\/2)$ and $k\\ge52$.\n\nWe then state and prove the following global existence result regarding any symmetric quasilinear system of Klein-Gordon equations in 2D. Two key lemmas used in the proof will be discussed immediately after this.\n\\begin{theorem}\\label{KGsystem}Consider a two dimensional $n$ by $n$ system (\\ref{ptt}) satisfying the conclusion of Lemma \\ref{symm}. Then, for any $k\\ge52$, there exists a universal constant $\\delta_0$ such that the system admits a unique classical solution for all times if\n\\begin{equation*}\n\\|{\\bf U}_0\\|_{H^{k+1,k}}=\\delta<\\delta_0.\n\\end{equation*}\nIn particular, $X(t)\\le C\\delta$ uniformly for all positive times.\n\\end{theorem}\n\\begin{proof}By the definition of $X(t)$ and local existence (\\ref{locvec}) of Theorem \\ref{LocalTh}, there exists $T$ such that\n\\begin{equation}\\label{XT}\n X(T)\\leq 4C_1\\delta.\n \\end{equation}\n Here, we choose constant $C_1$ to be greater than all constants appearing in Lemma \\ref{lemL} and \\ref{lemenergy} below. Then, choose $\\delta$ to be sufficiently small so that the assumptions of Lemma \\ref{lemL} and \\ref{lemenergy} are satisfied, which in turn implies\n\\[\nX(T)\\leq 2C_1\\delta+32C_1^3\\delta^2.\n\\]\nImpose one more smallness condition on $\\delta$ so that $X(T)\\le 3C_1\\delta$ in the above estimate. Finally, by the continuity argument, we can extend $T$ in (\\ref{XT}) to infinity, i.e., have $X(t)\\le 4C_1\\delta$ uniformly for all positive times. \n\\end{proof}\n\nThe following lemmas provide estimates on the lower order norms and highest order norms of $X(t)$ respectively. The quadratic term $X^2(t)$, rather than linear term, on the RHS of these estimates guarantees that we can extend such estimates to global times as long as $X(t)$ stays sufficiently small.\n\\begin{lemma}\\label{lemL}\nAssume\n$\\|{\\bf U}_0\\|_{H^{k+1,k}}\\le1$, $X(t)\\le1$. Then, the solution ${\\bf U}$\nof (\\ref{ptt}) satisfies \\begin{equation}\\label{Lestimate}\n|{\\bf U}\\RGN{k-25,-1}(t)+\\|{\\bf U}\\RGNN{k-9}(t)+\\|\\partial{\\bf U}\\RGNN{k-9}(t)\\leq\nC(\\|{\\bf U}_0\\|_{H^{k+1,k}}+ X^2(t)). \\end{equation} as long as the solution\nexists. Here, constant $C$ is independent of $\\delta$ and $t$.\n\\end{lemma}\n\\begin{proof}\nWe start the proof with defining the bilinear form associated with\nkernel $Q(y,z)$,\n\\\n\\begin{split}\n[G,Q,H](x):=&\\int_{\\mbox{R}^2\\times\\mbox{R}^2}G^T(y)Q(x-y,x-z)H(z)\\,dydz\\\\\n=&{1\\over(2\\pi)^4}\\int_{\\mbox{R}^2\\times\\mbox{R}^2}e^{i(\\xi+\\eta)\\cdot\nx}\\hat{G}^T(\\xi)\\hat{Q}(\\xi,\\eta)\\hat{H}(\\eta)\\,d\\xi\nd\\eta.\n\\end{split}\n\\]\nHere, $G(\\cdot)$, $H(\\cdot)$ are any $(2\\times1)$-vector-valued\nfunctions defined on $\\mbox{R}^2$ and $Q(\\cdot,\\cdot)$ is $(2\\times\n2)$-matrix-valued distribution defined on $\\mbox{R}^2\\times\\mbox{R}^2$.\nFourier transform is denoted with $\\hat{\\;\\;}$ for both $\\mbox{R}^2$ and\n$\\mbox{R}^2\\times\\mbox{R}^2$.\n\nThe same notation will be used for scalars\n\\[\n[g,q,h](x):=\\int_{\\mbox{R}^2\\times\\mbox{R}^2}g(y)q(x-y,x-z)h(z)\\,dydz.\n\\]\n\nThen, we follow the construction of \\cite{Shatah} to transform\n(\\ref{ptt}) in terms of the new variable\n\\begin{equation}\\label{def:V}{\\bf V}=(V_1,V_2,V_3)^T={\\bf U}+{\\bf W}={\\bf U}+(W_1,W_2,W_3)^T\\end{equation}\nwhere\n\\begin{equation}\\label{def:vi}\nW_k:=\\sum_{i,j=1}^3\\left[\\left(\\begin{matrix} U_i\\\\\n\\partial_t U_i\\end{matrix}\\right),Q^{ij}_k,\\left(\\begin{matrix} U_j\\\\\n\\partial_t U_j\\end{matrix}\\right)\\right],\\quad k=1,2,3.\n\\end{equation}\nThe kernels $Q^{ij}_k$ are to be determined later so that the new\nvariable ${\\bf V}$ satisfies a Klein-Gordon system with \\emph{cubic}\nnonlinearity for which estimate (\\ref{Lestimate}) will be proved\nusing techniques from \\cite{OzawaSL}, \\cite{Georgiev}.\n\nWithout loss of generality, we will demonstrate the proof using\n$V_1,U_1,W_1,Q^{ij}_1$ associated with the mass equation. From now\non, the subscript ``1'' is neglected for simplicity.\n\n\\emph{Step 1.} We claim that there exists kernels $Q^{ij}$ in\n(\\ref{def:vi}) such that $V=U+W$ satisfies the following\nKlein-Gordon equation with cubic and quadruple nonlinearity,\n\\begin{equation}\\label{cubicKG}(\\partial_{tt}-\\Delta +1)V=S\\end{equation} where the\nRHS\n\\begin{equation}\\label{cubicRHS}\n\\begin{split}S:=&\\sum_{|\\alpha|+|\\beta|+|\\gamma|\\le4\\atop\n\\max\\{|\\alpha|,|\\beta|,|\\gamma|\\}\\le3}\\sum_{a,b,c=1}^3[\\partial^\\alpha\nU_a\\partial^\\beta U_b,q_{\\alpha\\beta\\gamma}^{abc},\\partial^\\gamma\nU_c]\\\\&+\\sum_{|\\alpha|+|\\beta|+|\\gamma|+|\\zeta|\\le4\\atop\n\\max\\{|\\alpha|,|\\beta|,|\\gamma|,|\\zeta|\\}\\le3}\\sum_{a,b,c,d=1}^3[\\partial^\\alpha\nU_a\\partial^\\beta U_b,q_{\\alpha\\beta\\gamma\\zeta}^{abcd},\\partial^\\gamma\nU_c\\partial^\\zeta U_d] \\end{split}\\end{equation} with $q_{\\alpha\\beta\\gamma}^{abc}$,\n$q_{\\alpha\\beta\\gamma\\zeta}^{abcd}$ being linear combinations of the\nentries of all $Q_k^{ij}$'s. Moreover, all the $Q_k^{ij}$'s satisfy\nthe growth\ncondition\\begin{equation}\\label{growth:Q}\\left|D^N\\hat{Q}(\\xi,\\eta)\\right|\\le\nC_N(1+|\\xi|^4+|\\eta|^4)\\end{equation} for any nonnegative integer N.\n\nIndeed, substitute $V$ on the LHS of (\\ref{cubicKG}) with $V=U+W$\nwhere $W$ is defined in (\\ref{def:vi}) for $k=1$ and $U$ satisfies\nthe first equation of (\\ref{ptt}),\n\\begin{equation}\\label{form:0}(\\partial_{tt}-\\Delta+1)V=(\\partial_{tt}-\\Delta+1)U+(\\partial_{tt}-\\Delta+1)W.\\end{equation}\n\n\nThen, apply the normal form transform on each term of the RHS of the\nabove equation. By (\\ref{ptt}), the\n$U$ term amounts to\n\\begin{equation}\\label{form:1}(\\partial_{tt}-\\Delta+1)U=\\sum_{i,j=1}^3\\left[\\left(\\begin{matrix} U_i\\\\\\partial_t\nU_i\\end{matrix}\\right),P^{ij},\\left(\\begin{matrix} U_j\\\\\\partial_t U_j\\end{matrix}\\right)\\right]\\end{equation} with\n$\\hat{P}^{ij}(\\xi,\\eta)$ being $2\\times2$ matrices of polynomials\nwith degree less than or equal to 2. By (\\ref{def:vi}), the $W$ term amounts to\n\\begin{equation}\\label{form:2}(\\partial_{tt}-\\Delta+1)W=\\sum_{i,j=1}^3\\left[\\left(\\begin{matrix} U_i\\\\\\partial_t\nU_i\\end{matrix}\\right),{\\mathcal A} Q^{ij},\\left(\\begin{matrix} U_j\\\\\\partial_t U_j\\end{matrix}\\right)\\right]+S\\end{equation} with $S$\nsatisfying (\\ref{cubicRHS}) and linear transform ${\\mathcal A}$ defined as\n\\begin{equation}\\label{form:3}\\widehat{{\\mathcal A}\nQ}(\\xi,\\eta):=\\bma0&|\\xi|^2+1\\\\-1&0\\end{matrix}\\right)\n\\hat{Q}\\bma0&-1\\\\|\\eta|^2+1&0\\end{matrix}\\right)+(2\\xi\\cdot\\eta-1)\\hat{Q}\\end{equation}\n\nCombining (\\ref{form:0}) --- (\\ref{form:3}), we find that, for\nproving (\\ref{cubicKG}) --- (\\ref{growth:Q}), it suffices to show\nthat there exist solutions $\\hat{Q}^{ij}(\\xi,\\eta)$ to\n\\begin{equation}\\label{eq:Q}\\hat{P}^{ij}(\\xi,\\eta)+\\widehat{{\\mathcal A}\n{Q}}^{ij}(\\xi,\\eta)\\equiv0\\end{equation} that satisfy the growth condition\n(\\ref{growth:Q}). This part of the calculation only involves basic\nLinear Algebra and Calculus so we neglect the details.\n\n\\emph{Step 2.} We apply the decay estimate of Georgiev in\n\\cite{Georgiev} to obtain the $L^\\infty$ estimate for $V$ and\ntherefore $U$ with $(1+t)^{-1}$ decay in time.\n\\begin{theorem}[{\\cite[Theorem 1]{Georgiev}}]\\label{thm:Georgiev}Suppose $u(t,x)$\nis a solution of\\[(\\partial_{tt} -\\Delta +1)u=F(t,x).\\]Then, for $t\\ge0$, we\nhave\n\\[\n\\begin{split}|(1+t+|x|)u(t,x)|\\le& C\\sum_{n=0}^\\infty\\sum_{|\\alpha|\\le4}\n\\sup_{s\\in(0,t)}\\phi_n(s)\\|(1+s+|y|)\\Gamma^{\\alpha} f(s,y)\\|_{L^2_y}\\\\\n&+C\\sum_{n=0}^\\infty\\sum_{|\\alpha|\\le5}\\|(1+|y|)\\phi_n(y)\\Gamma^{\\alpha}\nu(0,y)\\|_{L^2_y}.\n\\end{split}\n\\]\nHere, $\\{\\phi_n\\}_{n=0}^\\infty$ is a Littlewood-Paley partition of\nunity,\n\\[\\sum_{n=0}^\\infty\\phi_n(s)=1,\\quad s\\ge0;\\quad\\phi_n\\in C^\\infty_0(\\mbox{R}),\n\\quad\\phi_n\\ge0\\quad\\mbox{for all }n\\ge0\\]\n\\[\\mbox{supp}\\,\\phi_n=[2^{n-1},2^{n+1}]\\quad\\mbox{for }n\\ge1,\n\\quad\\mbox{supp}\\,\\phi_0\\cap\\mbox{R}_+=(0,2].\\]\n\\end{theorem}\n\nApply $\\Gamma^{\\alpha}$ on the Klein-Gordon equation (\\ref{cubicKG}) and use the\ncommutation properties of the vector fields to obtain \\((\\partial_{tt}\n-\\Delta +1)\\Gamma^{\\alpha} V=\\Gamma^{\\alpha} S\\) so that by the above theorem,\n\\[\n\\begin{split}|(1+t+|x|)\\Gamma^{\\alpha} V(t,x)|\\le& C\\sum_{n=0}^\\infty\n\\sum_{|\\beta|\\le|\\alpha|+4}\n\\sup_{s\\in(0,t)}\\phi_n(s)\\|(1+s+|y|)\\Gamma^{\\beta} S(s,y)\\|_{L^2_y}\\\\\n&+C\\sum_{n=0}^\\infty\\sum_{|\\beta|\\le|\\alpha|+5}\n\\|(1+|y|)\\phi_n(y)\\Gamma^{\\beta} V(0,y)\\|_{L^2_y}.\n\\end{split}\n\\]\nBy definition $V=U+W$, we immediately have estimates for\n$(1+t+|x|)\\Gamma^{\\alpha} U$. After taking summation over all $\\alpha$'s with\n$|\\alpha|\\le k-25$, we arrive at \\begin{equation}\\label{estimate:infty}\n\\begin{split}| U\\RGN{k-25,-1}(t)\\le& | W\\RGN{k-25,-1}+\nC\\left(\\sum_{n=0}^\\infty\\sum_{|\\beta|\\le k-21}\n\\sup_{s\\in(0,t)}\\phi_n(s)\\|(1+s+|y|)\\Gamma^{\\beta} S(s,y)\\|_{L^2_y}\\right.\\\\\n&\\left.+\\|U(0,y)\\|_{H^{k+1,k}}+\\sum_{|\\beta|\\le k-20}\\|(1+|y|)\\Gamma^{\\beta}\nW(0,y)\\|_{L^2_y}\\right).\\end{split} \\end{equation}\n\nTo obtain estimate on each term, we use the following proposition,\nthe proof of which is given in the Appendix.\n\\begin{proposition}\\label{prop:est}\nLet the scalar function $q:\\mbox{R}^2\\times\\mbox{R}^2\\mapsto\\mbox{R}$ satisfy the\ngrowth condition (\\ref{growth:Q}) in terms of its Fourier transform.\nLet $f(t,x)$, $g(t,x)$, $h(t,x)$ be functions with sufficient\nregularity. Consider $p=\\infty$ (respectively $p=2$). Then, at each\n$t\\ge0$, for $a=|\\beta|+8$ (respectively $a=|\\beta|+6$) and\n$b=\\left\\lceil {|\\beta|\\over2}\\right\\rceil+7$,\n\\begin{align}\\label{Estimate1prop}\n\\left\\|(1+t+|x|)\\Gamma^{\\beta}[f,q,g]\\right\\|_{L^p_x}\\le& C\\left(\\|\nf\\RGNN{a}| g\\RGN{b,-1}+| f\\RGN{b,-1}\\| g\\RGNN{a}\\right),\n\\end{align}\n\\begin{align}\\label{Estimate2prop}\n\\begin{split}\n\\left\\|(1+t+|x|)\\Gamma^{\\beta}[f,q,gh]\\right\\|_{L^p_x}\\le &C(1+t)^{-1}\n\\left(\\| f\\RGNN{a}| g\\RGN{b,-1}| h\\RGN{b,-1}\\right.\\\\\n&+\\left.| f\\RGN{b,-1}\\left(\\| g\\RGNN{a}|\nh\\RGN{\\lceil{a\\over2}\\rceil,-1} +|\ng\\RGN{\\lceil{a\\over2}\\rceil,-1}\\| h\\RGNN{a}\\right)\\right).\n\\end{split}\n\\end{align}\n\\end{proposition}\n\nApply this proposition (with $p=\\infty$) on the first term, and\n(with $p=2$) on the second and fourth terms of the RHS of\n(\\ref{estimate:infty}) and use the definition of $W$ in\n(\\ref{def:vi}) and $S$ in (\\ref{cubicKG}),\n\\[\\begin{split}| U\\RGN{k-25,-1}(t)\\le& CX^2(t)+C\\sum_{n=0}^\\infty\n\\sup_{s\\in(0,t)}\\phi_n(s)(1+s)^{-1}(X^3(t)+X^4(t))\\\\\n&+ C\\left(\\|U_0\\|_{H^{k+1,k}}+\\|{\\bf U}_0\\|^2_{H^{k+1,k}}\\right).\n\\end{split}\\]\n\nWe finish the $L^\\infty$ estimate part of this Lemma\nusing the fact that \\[\\sum_{n=0}^\\infty\\sup_{s\\in\n(0,t)}\\phi_n(s)(1+s)^{-1}<{5\\over2}\\]and the assumptions\n$\\|{\\bf U}_0\\|_{H^{k+1,k}}\\le1$, $X(t)\\le1$.\n\n\\emph{Step 3.} We obtain the $L^2$ estimate part regarding the terms\n\\(\\|{\\bf U}\\RGNN{k-9}(t)+\\|{\\partial} {\\bf U}\\RGNN{k-9}(t)\\) using a very\nsimilar approach as in Step 2. In fact, apply $\\Gamma^{\\alpha}$ on the\nKlein-Gordon equation (\\ref{cubicKG}) and use the commutation\nproperties of vector fields to obtain \\((\\partial_{tt} -\\Delta +1)\\Gamma^{\\alpha} V=\\Gamma^{\\alpha}\nS\\). Then, we take the inner product of this equation with $\\partial_t\\Gamma^{\\alpha}\nV$, sum over all $\\alpha$ with $|\\alpha|\\le k-9$ to obtain\n\\[\n\\begin{split}\\| V(t,x)\\RGNN{k-9}+&\\| \\partial V(t,x)\\RGNN{k-9}\\le\nC\\int_0^t\\| S(s,x)\\RGNN{k-9}ds\\\\&+C\\left(\\| V(0,x)\\RGNN{k-9}\n+\\| \\partial V(0,x)\\RGNN{k-9}\\right).\\end{split}\n\\]\nSince $V=U+W$, we have\n\\[\n\\begin{split}\\| U(t,x)\\RGNN{k-9}+&\\| \\partial U(t,x)\\RGNN{k-9}\n\\le\\left(\\| W(t,x)\\RGNN{k-9}+\\| \\partial W(t,x)\\RGNN{k-9}\\right)\\\\&\n+C\\int_0^t\\| S(s,x)\\RGNN{k-9}ds\\\\&+ C\\left(\\|\nU(0,x)\\RGNN{k-9}+\\| \\partial U(0,x)\\RGNN{k-9}+\n\\| W(0,x)\\RGNN{k-9}+\\| \\partial W(0,x)\\RGNN{k-9}\\right)\\\\\n&=:I+II+III.\\end{split}\n\\]\n\nThe estimate of the $I,II,III$ terms above follows closely to that of\n(\\ref{estimate:infty}), which evokes Proposition \\ref{prop:est} repeatedly,\n\\[\\begin{split}I\\le&(1+t)^{-1}\\sum_{|\\beta|\\le k-9}\\left\\|(1+t+|x|)\\Gamma^{\\beta} W(t,x)\n\\right\\|_{L^2_x}\\\\\n\\le&C(1+t)^{-1}\\| {\\bf U}(t,x)\\RGNN{k}| {\\bf U}(t,x)\\RGN{k-25,-1}\n\\quad\\mbox{ by (\\ref{def:vi}) and Prop. \\ref{prop:est}}\\\\\n\\le&C(1+t)^{-1+\\sigma}X^2(t),\\\\\nII\\le&\\int_0^t(1+s)^{-1}\\sum_{|\\beta|\\le k-9}\\left\\|(1+s+|x|)\\Gamma^{\\beta}\nS(s,x)\n\\right\\|_{L^2_x}ds\\\\\n\\le&C\\int_0^t(1+s)^{-2}\\| {\\bf U}(s,x)\\RGNN{k} \\left(|\n{\\bf U}(s,x)\\RGN{k-25,-1}^2\n+| {\\bf U}(s,x)\\RGN{k-25,-1}^3\\right)\\\\\n&\\phantom{C\\int_0^t(1+s)^{-2+\\sigma}\\left(X^3(s)+X^4(s)\\right)\\,ds}\n\\quad\\mbox{ by (\\ref{cubicRHS}) and Prop. \\ref{prop:est}}\\\\\n\\le&C\\int_0^t(1+s)^{-2+\\sigma}\\left(X^3(s)+X^4(s)\\right)\\,ds,\\\\\nIII\\le&C\\left(\\|U(0,y)\\|_{H^{k+1,k}}+\\|{\\bf U}(0,y)\\|^2_{H^{k+1,k}}\\right),\n\\end{split}\n\\]\nwhere we use $k\\geq 52$ and that $S$ contains at most third order\nderivatives of ${\\bf U}$. These estimates shall finish the proof of\nLemma \\ref{lemL} given assumptions \n$\\|{\\bf U}_0\\|_{H^{k+1,k}}\\le1$ and $X(t)\\le1$.\n\\end{proof}\n\nNote that in the above estimates for $I$ and $II$, we use the $k-th$\norder norms to bound all lower order norms. In order to get the\nglobal a priori estimate, we have to close the estimates for the\nhighest order norms. For the RSW system, this is achieved by the\nenergy estimates on the highest order Sobolev norms\n$\\|\\cdot\\RGNN{k}$,\n where its symmetric structure shown in Lemma\n\\ref{symm} plays a crucial role.\n\\begin{lemma}\\label{lemenergy}\nAssume $\\|A_{ij}({\\bf U})\\|_{L^\\infty}\\leq 1\/4$.\nThen, the solution ${\\bf U}$ of (\\ref{ptt}) satisfies\n\\begin{align*}\\label{Enestimate}\n(1+t)^{-\\sigma}(\\|{\\bf U}\\RGNN{k}(t)+\\|{\\partial} {\\bf U}\\RGNN{k}(t))\\leq\nC(\\|{\\bf U}_0\\|_{H^{k+1,k}}+X^2(t))\n\\end{align*}\nas long as the solution exists. Here, constant $C$ is independent of $\\delta$ and $t$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof of this lemma combines the ideas in \\cite{Hormander,\nOzawaSL, Fang} for energy estimates for the Klein-Gordon equations\ntogether.\n\nDefine an energy functional\n\\begin{align*} F(t):=&\\frac{1}{2}\\sum_{|\\alpha|\\leq k} (\\|\\partial_t\\Gamma^{\\alpha}\n{\\bf U}\\|_{L^2}^2+\\|\\nabla \\Gamma^{\\alpha} {\\bf U}\\|_{L^2}^2+\\|\\Gamma^{\\alpha}{\\bf U}\\|_{L^2}^2)(t)\\\\\n&+\\frac{1}{2}\\sum_{|\\alpha|\\leq k}\n\\sum_{i,j=1}^2\\left\\langle A_{ij}({\\bf U})\\partial_i\\Gamma^{\\alpha} {\\bf U}, \\partial_j\\Gamma^{\\alpha}{\\bf u}\\right\\rangle(t),\n\\end{align*} where $\\langle, \\rangle$ is the $L^2$ inner product defined by\n$\\left\\langle f, g\\right\\rangle=\\int_{\\mbox{R}^2}f^Tgdx$.\nClearly, by the commutation property of $\\Gamma$, $\\partial$ and the assumption $\\|A_{ij}({\\bf U})\\|_{L^\\infty}\\leq\n\\frac{1}{4}$, we have \\begin{equation}\\label{MEE} C_1\\sqrt{F(t)}\\leq \\|{\\bf U}\\RGNN{k}(t)+\\|{\\partial} {\\bf U}\\RGNN{k}(t)\\leq C_2\\sqrt{F(t)}, \\end{equation} which ensures the equivalence\nof $\\sqrt{F(t)}$ and $\\|{\\bf U}\\RGNN{k}(t)+\\|{\\partial} {\\bf U}\\RGNN{k}(t)$.\n\nApplying $\\Gamma^{\\alpha}$ to (\\ref{ptt}) and taking the $L^2$\ninner product on the resulting system with $\\partial_t\\Gamma^{\\alpha}{\\bf U}$\ni.e. $\\partial_0\\Gamma^{\\alpha}{\\bf U}$ , it follows from Leibniz's rule that\n\\begin{align*}\n\\left\\langle(\\partial_{tt}-\\Delta+1)\\Gamma^{\\alpha}{\\bf U}, \\partial_t\\Gamma^{\\alpha}{\\bf U}\\right\\rangle=\n&\\sum_{i,j=1}^2\\left\\langle\nA_{ij}({\\bf U})\\partial_{ij}\\Gamma^{\\alpha}{\\bf U},\n\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle \\\\+\\sum_{j=1}^2\\left\\langle A_{0j}\n\\partial_{0j}\\Gamma^{\\alpha}{\\bf U},\n\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle&+\\sum_{|\\beta|+|\\gamma|\\le |\\alpha|}\\left\\langle\nR^{\\beta\\gamma}(\\Gamma^{\\beta}\\tilde{\\vU}\\otimes\\Gamma^{\\gamma}\\tilde{\\vU}),\n\\partial_0\\Gamma^{\\alpha}{\\bf U} \\right\\rangle,\n\\end{align*}\nwhere all $R^{\\beta\\gamma}$'s are linear functions. Here\n$\\tilde{\\vU}=({\\bf U}^T,\\partial_t{\\bf U}^T,\\partial_1{\\bf U}^T,\\partial_2{\\bf U}^T)$.\n\nUpon integrating by parts, one has\n\\begin{equation}\\label{Highenergyest}\n\\begin{split}\n&\\frac{1}{2}\\partial_t(\\|\\partial_t\\Gamma^{\\alpha} {\\bf U}\\|_{L^2}^2+\\|\\nabla \\Gamma^{\\alpha} {\\bf U}\\|_{L^2}^2\n+\\|\\Gamma^{\\alpha}{\\bf U}\\|_{L^2}^2)\\\\\n=&-\\sum_{i,j=1}^2 \\left\\langle\nA_{ij}\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_0\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle -\\left\\langle\n\\partial_jA_{ij}({\\bf U})\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle\\\\\n& -\\frac{1}{2}\\sum_{j=1}^2\\left\\langle\\partial_j\nA_{0j}\\partial_0\\Gamma^{\\alpha}{\\bf U},\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle +\\sum_{|\\beta|+|\\gamma|\\le\n|\\alpha|}\\left\\langle R^{\\beta\\gamma}(\\Gamma^{\\beta}\\tilde{\\vU}\\otimes\\Gamma^{\\gamma}\\tilde{\\vU}),\n\\partial_0\\Gamma^{\\alpha}{\\bf U}\\right\\rangle\n\\end{split}\n\\end{equation}\nand the first terms on the RHS above\\begin{align*} \\left\\langle\nA_{ij}\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_0\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle=&-\\frac{1}{2}\\partial_t\\left\\langle\nA_{ij}\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle+\\frac{1}{2}\\left\\langle \\partial_0A_{ij}\n\\partial_i\\Gamma^{\\alpha}{\\bf U},\\partial_j\\Gamma^{\\alpha}{\\bf U}\\right\\rangle.\n\\end{align*}\nThe equations above hold because $A_{ij}$ and $A_{0j}$ are symmetric\nmatrices. Summing for all indices $\\alpha$ with $|\\alpha|\\leq k$ in\nthe above equations and using the fact that $\\min\\{|\\beta|,|\\gamma|\\}\\le k\/20$ this function has two,\nand only two, real roots. This is an encouraging result \\ consistent with\nour expectations [21] that there should be two apparent horizons $r_{AH}^{-}$\nand $r_{AH}^{+}$ associated with the black hole and the cosmological\nconstant, respectively.\n\nThe case of the $r^{n-1}$ turning points at $r=0$ suggests the existence of\nother roots for the function $f\\left( r\\right) $. Such roots are actually\ncomplex and would suggest the existence of {\\it ghost horizons} (at least\nfor an observer in our four dimensional spacetime). I will defer, for now,\nfurther speculation of their significance and other issues about the turning\npoints for a future discussion. It is, however, interesting to note that\nsuch {\\it ghost horizons }only{\\it \\ }start to appear at five dimensions $%\n\\left( n=2\\right) $ and persist for higher dimensions.\n\nTo put equation (3.3) in a more manageable form, it is useful to institute a\nchange of variables. Thus setting\\ \n\\begin{equation}\nr=k\\xi , \\eqnum{3.5}\n\\end{equation}\nwhere $k=\\sqrt{\\frac{\\left( n+1\\right) \\left( n+2\\right) }{2\\Lambda }}$\nequation (3.3) can be cast in a form \n\\begin{equation}\n\\xi ^{n+2}-\\xi ^{n}+\\beta \\left( n\\right) =0, \\eqnum{3.6}\n\\end{equation}\nwhere \n\\begin{equation}\n\\beta \\left( n\\right) =\\frac{2G_{N}m\\left( v\\right) }{n}\\left[ \\frac{%\n2\\Lambda }{\\left( n+1\\right) \\left( n+2\\right) }\\right] ^{\\frac{n}{2}}. \n\\eqnum{3.7}\n\\end{equation}\nOne notes that since in our model $\\Lambda >0$, and so $\\beta \\left(\nn\\right) >0$, then (3.6), along with the necessary positive reality of $r$,\nimply that $0<\\xi <1$ and $0<\\beta <1$.\n\nNow set $\\xi =1-x$, where $02$. The\nsolutions for the 5-dimensional $\\left( n=2\\right) $ case are actually\nexact. Further, the solutions have the right limits when $n=1$ and for this\ncase one does recover the exact Mallett's cubic equation [21] for the\n4-dimensional case from either of the equations (3.3) or 3.6). In general\nthis approximation is good in the limit $x\\rightarrow 0$, $\\xi \\rightarrow 1$%\n. It, however, breaks down when $x\\rightarrow 1$, $\\xi \\rightarrow 0$. In\nthis latter limit the solution to (3.6) simply goes to $\\xi =\\left[ \\beta\n\\left( n\\right) \\right] ^{\\frac{1}{n}}\\Longrightarrow r=\\left( \\frac{2Gm}{n}%\n\\right) ^{\\frac{1}{n}}$and the geometry loses knowledge of the cosmological\nconstant.\n\nEquation (3.9) has a resolvent cubic equation that can\\ be that can be\nwritten in the form \n\\begin{equation}\ny^{3}+py+q=0 \\eqnum{3.11}\n\\end{equation}\nwhere $p=\\frac{1}{3}\\left[ \\left( 3ac-b^{2}\\right) -12d\\right] $ and$\\;q=%\n\\frac{1}{27}\\left[ 9abc-2b^{3}-27c^{2}-9\\left( 3a^{2}+b\\right) d\\right] $.\nEquation (3.11) admits three solutions. One such solution that is real for\nthe parameters $p$, $q$ as defined above can be written as\n\n\\begin{equation}\ny_{1}=2\\sqrt{\\frac{-p}{3}}\\cos \\frac{1}{3}\\varphi , \\eqnum{3.12}\n\\end{equation}\nwhere $\\frac{\\pi }{2}<\\varphi <\\pi $\\ is given by $\\varphi =\\arccos \\left( \n\\frac{3q}{2\\sqrt{\\frac{-p^{3}}{3}}}\\right) $.\n\nThere are four solutions to the quartic equation (3.9) in $x$. The only\nphysically interesting solutions to equation (3.9) consistent with $r=k\\xi\n=k\\left( 1-x\\right) $ should be real and satisfy $x<1$. There are two such\nsolutions obtained by letting \n\\[\nR=\\sqrt{\\frac{a^{2}}{4}-b+\\left( 2\\sqrt{\\frac{-p}{3}}\\right) \\cos \\frac{1}{3}%\n\\varphi } \n\\]\nand \n\\begin{equation}\nD=\\sqrt{\\frac{3a^{2}}{4}-R^{2}\\left( p,\\varphi \\right) -2b+\\frac{4ab-8c-a^{3}%\n}{4R\\left( p,\\varphi \\right) }} \\eqnum{3.13}\n\\end{equation}\nThese are: \n\\begin{equation}\nx_{\\pm }=-\\frac{1}{4}a+\\frac{1}{2}R\\left( p,\\varphi \\right) \\pm \\frac{1}{2}%\nD\\left( p,\\varphi \\right) . \\eqnum{3.14}\n\\end{equation}\nNote that all the time dependency of $R$ and so of $D$ is expressed in $p$\nand $\\varphi $ via $d\\left( \\beta \\left( m\\left( v\\right) \\right) \\right) $.\n\nUsing the solutions in equation (3.14) and recalling that $\\xi =1-x$ \\ we\nobtain two values $\\xi _{1}$ and $\\xi _{2}$.\\ On applying these results to\nequation (3.5) i.e. $r=k\\xi =k\\left( 1-x\\right) $ we find two solutions $%\nr_{1}=r_{AH}^{-}\\left( v\\right) $ and $r_{2}=r_{AH}^{+}\\left( v\\right) $\nsuch that\n\n\\begin{equation}\nr_{AH}^{-}\\left( v\\right) =r_{1}\\simeq k\\left\\{ 1-\\frac{1}{4}\\left[ 2D\\left(\np,\\varphi \\right) +2R\\left( p,\\varphi \\right) -a\\right] \\right\\} , \n\\eqnum{3.15}\n\\end{equation}\nand \n\\begin{equation}\nr_{AH}^{+}\\left( v\\right) =r_{2}\\simeq k\\left\\{ 1+\\frac{1}{4}\\left[ 2D\\left(\np,\\varphi \\right) -2R\\left( p,\\varphi \\right) +a\\right] \\right\\} \n\\eqnum{3.16}\n\\end{equation}\n\\ \n\nIn the limit $n\\rightarrow 1$ one recovers the well known solutions [21].\nThus \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{AH}^{-}\\left( v\\right) =-\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi +\\frac{1}{3}\\pi \\right) \\eqnum{3.17}\n\\end{equation}\nand \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{AH}^{+}\\left( v\\right) =\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi \\right) \\eqnum{3.18}\n\\end{equation}\nwhere now $\\left( \\frac{\\pi }{2}<\\Psi \\left( v\\right) <\\pi \\right) =\\arccos %\n\\left[ -3m\\left( v\\right) \\sqrt{\\Lambda }\\right] $. (Note, however, that $%\n\\varphi $ is not simply related to $\\Psi $ by taking the limit $n\\rightarrow\n1$). Further, in the limit $\\Lambda \\longrightarrow 0$ and $n\\rightarrow 1$\nwe recover the black hole apparent horizon $r_{AH}^{-}\\left( v\\right)\n\\rightarrow 2m\\left( v\\right) $ and in the limit $m\\rightarrow 0$ and $%\nn\\rightarrow 1$ we find, as one expects that $r_{AH}^{+}\\left( v\\right)\n\\rightarrow \\sqrt{\\frac{\\Lambda }{3}}$. Hence our solutions reduce to all\nthe well known solutions [21] in a four dimensional spacetime. Consequently,\nwe identify the locus of $r_{AH}^{-}\\left( v\\right) $\\ in equation (3.15) as\nthe black hole apparent horizon $\\left( AH^{-}\\right) $ while we identify\nthe locus of $r_{AH}^{+}\\left( v\\right) $ in equation\\ (3.16) as the de\nSitter apparent horizon $\\left( AH^{+}\\right) $, for a black hole in an\nN-dimensional background with a cosmological constant.\n\n\\subsection{The Event Horizons}\n\nThe event horizons are null surfaces. \\ To $O(L_{0})$\\ the evolution of\nthese surfaces can be determined from the second of the York conditions that \n$\\frac{d\\theta }{dv}\\simeq 0$. We first show that for a black hole imbedded\nin an N-dimensional de Sitter background this condition is satisfied. We\nshall then solve the resulting equations to locate the event horizons and\nstudy their structure.\n\nThe surface gravity $\\kappa $ in this spacetime is given from (2.11) by \n\\begin{equation}\n\\kappa =\\frac{G_{N}m\\left( v\\right) }{r^{n+1}}-\\frac{2\\Lambda }{\\left(\nn+1\\right) \\left( n+2\\right) }r. \\eqnum{3.17}\n\\end{equation}\n\\bigskip Consider now the acceleration of the geodesics $\\frac{d^{2}r}{dv^{2}%\n}$ \\ in our space-time, parametrized by $v$. Since from the line element\n(2.1) we have that \n\\begin{equation}\n\\frac{dr}{dv}=\\frac{1}{2}\\left[ 1-\\frac{2G_{N}m\\left( v\\right) }{nr^{n}}-%\n\\frac{2\\Lambda }{\\left( n+1\\right) \\left( n+2\\right) }r^{2}\\right] \n\\eqnum{3.18}\n\\end{equation}\nthen \n\\begin{equation}\n\\frac{d^{2}r}{dv^{2}}=\\frac{G_{N}L_{0}}{nr^{n}}+\\kappa \\frac{dr}{dv} \n\\eqnum{3.19}\n\\end{equation}\nEquations (3.2) and (3.18) in (3.19) give \n\\begin{equation}\n\\frac{d^{2}r}{dv^{2}}=\\frac{G_{N}L_{0}}{nr^{n}}+\\kappa \\theta \\frac{r}{n+1}.\n\\eqnum{3.20}\n\\end{equation}\nBut the event horizon is a null surface and satisfies the general\nrequirement that null-geodesic congruencies have a vanishing acceleration. \\\nThus at the event horizon (3.20) takes the form \n\\begin{equation}\n\\kappa \\theta _{EH}+\\frac{n+1}{n}\\frac{G_{N}L_{0}}{\\left( r_{EH}\\right)\n^{n+1}}=0. \\eqnum{3.21}\n\\end{equation}\n\nNow the Einstein field equations for the $N(N=n+3)$-dimensional space-time\nare [4]\n\n\\begin{equation}\nR_{ab}=8\\pi G_{N}\\left[ T_{ab}-\\frac{1}{n+1}g_{ab}T_{c}^{c}\\right] +\\frac{%\n2\\Lambda }{n+1}g_{ab}. \\eqnum{3.22}\n\\end{equation}\nUsing equations $\\left( 2.3\\right) $, $\\left( 2.6\\right) $, and $\\left(\n2.8\\right) $ with $\\left( 3.22\\right) $ one finds that \n\\begin{equation}\nR_{ab}l^{a}l^{b}=\\frac{\\left( n+1\\right) }{n}\\frac{G_{N}}{r^{n+1}}\\dot{m}%\n\\left( v\\right) \\eqnum{3.23}\n\\end{equation}\n\nFor a spherically symmetric irrotational space-time, such as under\nconsideration, the vorticity $\\omega $\\ and the shear $\\sigma $ vanish and\nthe Raychauduri equation (2.10) reduces to \n\\begin{equation}\n\\frac{d\\theta }{dv}=\\kappa \\theta -R_{ab}l^{a}l^{b}-\\left( \\gamma\n_{c}^{c}\\right) ^{-1}\\theta ^{2}. \\eqnum{3.24}\n\\end{equation}\nEquations (3.21) and (3.23) when substituted in (3.24) show (on neglecting\nthe term in $\\theta ^{2}$) that, indeed, the York condition for the event\nhorizon is satisfied and we have \n\\begin{equation}\n\\left( \\frac{d\\theta }{dv}\\right) _{EH}\\simeq 0. \\eqnum{3.25}\n\\end{equation}\nThe event horizons $(EHs)$ in our problem are therefore located by equation\n(3.25). Using equation (3.2) equation (3.25) can be written in the form \n\\begin{equation}\nr^{n+2}-\\frac{\\left( n+1\\right) \\left( n+2\\right) }{2\\Lambda }r^{n}+\\frac{%\n\\left( n+1\\right) \\left( n+2\\right) }{n}\\frac{G_{N}m^{\\ast }\\left( v\\right) \n}{\\Lambda }=0. \\eqnum{3.26}\n\\end{equation}\nwhere $m^{\\ast }$ is some effective mass given by \n\\begin{equation}\nm^{\\ast }\\left( v\\right) =m\\left( v\\right) -\\frac{L_{0}}{\\kappa } \n\\eqnum{3.27}\n\\end{equation}\n\nEquation (3.26)\\ for the location of event horizons is exactly of the same\nform as its counterpart equation (3.3) for location of the apparent horizons\nwith the mass $m$ replaced by the effective mass $m^{\\ast }$ as defined in\nequation (3.27). Hence borrowing from our previous techniques in solving\nequation (3.3) we can immediately write down the solutions to 3.26 as \n\\begin{equation}\nr_{EH}^{-}\\left( v\\right) \\simeq k\\left\\{ 1-\\frac{1}{4}\\left[ 2D^{\\ast\n}\\left( p,\\varphi \\right) +2R^{\\ast }\\left( p,\\varphi \\right) -a^{\\ast }%\n\\right] \\right\\} , \\eqnum{3.28}\n\\end{equation}\nand \n\\begin{equation}\nr_{EH}^{+}\\left( v\\right) \\simeq k\\left\\{ 1+\\frac{1}{4}\\left[ 2D^{\\ast\n}\\left( p,\\varphi \\right) -2R^{\\ast }\\left( p,\\varphi \\right) +a^{\\ast }%\n\\right] \\right\\} , \\eqnum{3.29}\n\\end{equation}\nwhere $\\ast $\\ means $m\\left( v\\right) \\rightarrow m^{\\ast }\\left( v\\right)\n=m\\left( v\\right) -\\frac{L_{0}}{\\kappa }$ and $\\frac{1}{2}\\pi <\\varphi\n^{\\ast }<\\pi $. In the limit $n\\rightarrow 1$ equation (3.28) reduces to, \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{EH}^{-}\\left( v\\right) =-\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi ^{\\ast }+\\frac{1}{3}\\pi \\right) \\eqnum{3.30}\n\\end{equation}\nwhile \n\\begin{equation}\n\\mathrel{\\mathop{\\lim }\\limits_{n=1}}%\nr_{EH}^{+}\\left( v\\right) =\\left( \\frac{2}{\\sqrt{\\Lambda }}\\right) \\cos\n\\left( \\frac{1}{3}\\Psi ^{\\ast }\\right) \\eqnum{3.31}\n\\end{equation}\nwhere $\\left( \\frac{\\pi }{2}<\\Psi ^{\\ast }\\left( v\\right) <\\pi \\right)\n=\\arccos \\left[ -3m^{\\ast }\\left( v\\right) \\sqrt{\\Lambda }\\right] $. These\nlimiting cases then reproduce exactly the known equations [21] for locations\nof the event horizons in such a four dimensional spacetime. Further, as one\nswitches $\\Lambda $ off one finds from equations (3.17) and (3.3) that the\nsurface gravity of the black hole measured at the apparent horizon becomes\n\n\\begin{equation}\n\\kappa =\\frac{n}{2}\\left[ \\frac{2G_{N}m\\left( v\\right) }{n}\\right] ^{-\\frac{1%\n}{n}} \\eqnum{3.32}\n\\end{equation}\nEquation (3.26) along with equation (3.32) imply that in the limit $\\Lambda\n\\rightarrow 0$, then \n\\begin{equation}\nr_{EH}^{-}\\left( v\\right) \\rightarrow \\left( \\frac{2G_{N}}{n}\\right) ^{\\frac{%\n1}{n}}\\left[ m\\left( v\\right) -\\frac{2}{n}\\left( \\frac{2G_{N}m\\left(\nv\\right) }{n}\\right) ^{\\frac{1}{n}}L_{0}\\right] \\eqnum{3.33}\n\\end{equation}\nEquation (3.33) locates the event horizon of a radiating black hole imbedded\nin a higher dimensional \\ Schwarzschild spacetime. And as the dimensionality\nis reduced to four, it is clear that \n\\begin{equation}\nr_{EH}^{-}\\left( v\\right) \\rightarrow 2Gm\\left( v\\right) \\left(\n1-4GL_{0}\\right) . \\eqnum{3.34}\n\\end{equation}\nThis is the exact result originally obtained by York [26] and later verified\nby Mallett [18]. It follows then that in equation (3.28) $r_{EH}^{-}\\left(\nv\\right) $\\ does indeed represent the locus of the black hole event horizon\nin a higher dimensional spacetime with a cosmological constant. Further, the \n$r_{EH}^{+}\\left( v\\right) $ in equation (3.29) is seen to give the locus of\nthe cosmological event horizon \\ $r_{EH}^{+}\\left( v\\right) $.\n\nFor an observer positioned at $r_{AH}^{-}\\left( v\\right)