\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 39, pp. 1--23.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/39\hfil Existence and uniqueness of global solutions] {Existence and uniqueness of global solutions \\ to a model for the flow of an incompressible, barotropic fluid with capillary effects} \author[D. L. Denny\hfil EJDE-2007/39\hfilneg] {Diane L. Denny} \address{Diane Denny \newline Department of Mathematics and Statistics\\ Texas A\&M University - Corpus Christi\\ Corpus Christi, TX 78412, USA} \email{diane.denny@tamucc.edu} \thanks{Submitted August 10, 2006. Published March 6, 2007.} \subjclass[2000]{35A05} \keywords{Existence; uniqueness; capillary; incompressible; \hfill\break\indent inviscid fluid} \begin{abstract} We study the initial-value problem for a system of nonlinear equations that models the flow of an inviscid, incompressible, barotropic fluid with capillary stress effects. We prove the global-in-time existence of a unique, classical solution to this system of equations, with a small initial velocity gradient. The key to the proof lies in using an $L^2$ estimate for the density $\rho$, and using the smallness of the initial velocity gradient, to obtain uniqueness for the density. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} In this paper, we consider equations which arise from a model of the multi-dimensional flow of an incompressible, barotropic fluid with capillary stresses. When viscosity is neglected, these equations reduce to the following system, written in terms of the density $\rho$, the pressure $p$, and velocity $\mathbf{v}$: \begin{gather} \frac{D\mathbf{v}}{Dt} + \rho^{-1} \nabla p = c\nabla \Delta \rho, \label{e1.1} \\ \nabla\cdot \mathbf{v} = 0. \label{e1.2} \end{gather} Here $c$ is a coefficient of capillarity which is a small, positive constant, and the material derivative $D/{Dt}= \partial/{\partial t}+\mathbf{v}\cdot\nabla$. The term $c\nabla \Delta \rho$ arises from capillary stresses, as described in the theory of Korteweg-type materials developed by Dunn and Serrin \cite{d2}. The fluid's thermodynamic state is determined by the density $\rho$. The pressure $p$ is determined from the density by an equation of state $p=\hat{p}(\rho)$. In related work, the existence of a solution to a similar system of equations for the case of viscous fluid flow, which also includes a hyperbolic equation for density and a parabolic equation for temperature, has been proven by Hattori and Li \cite{h1,h2}, and by Bresch, Desjardins, and Lin \cite{b1}. Anderson, McFadden and Wheeler \cite{a1} have given a review of related theories and applications to diffuse-interface modelling. In this model, it will be shown that $\partial \rho/\partial t$, $\nabla \rho$, and $\nabla \mathbf{v}$ are small, for suitable initial data. Although the conservation of mass equation is only approximately satisfied, the model equations \eqref{e1.1}, \eqref{e1.2} might be useful as an approximation in the case of almost constant density, and nearly incompressible fluid flow with a small velocity gradient. We write equation \eqref{e1.1} equivalently as $$\frac{D\mathbf{v}}{Dt} + \rho^{-1} \hat{p}' (\rho)\nabla \rho = c\nabla \Delta \rho \label{e1.3}$$ The purpose of this paper is to prove the existence of a unique, global-in-time, classical solution $\mathbf{v}$, $\rho$ to equations \eqref{e1.2}, \eqref{e1.3} with suitable initial velocity data $\mathbf{v}_0 \in H^s$, under periodic boundary conditions. That is, we choose for our domain the N-dimensional torus $\mathbb{T}^N$, where $N=2$ or $N=3$. The proof of the existence theorem is based on the method of successive approximations, in which an iteration scheme, based on solving a linearized version of the equations, is designed and convergence of the sequence of approximating solutions to a unique solution satisfying the nonlinear equations is sought. The framework of the proof follows one used, for example, by A. Majda for proving the existence of a solution to a system of conservation laws \cite{m1}. Embid \cite{e1} also uses the same general framework to prove the existence of a solution to equations for zero Mach number combustion. Under this framework, the convergence proof is presented in two steps. In the first step, we prove uniform boundedness of the approximating sequence of solutions in a high Sobolev norm. The second step is to prove contraction of the sequence in a low Sobolev norm. Standard compactness arguments will be used to finish the proof. The key to the proof lies in using an $L^2$ estimate for the density $\rho$, and using the smallness of the initial velocity gradient, to obtain uniqueness for the density. \section{A priori estimates} The main tools utilized in the existence proof are a priori estimates. We will work with the Sobolev space $H^s(\Omega )$ (where $s\geq 0$ is an integer) of real-valued functions in $L^2(\Omega )$ whose distribution derivatives up to order $s$ are in $L^2(\Omega )$, with norm given by $\|f\|_s^2=\sum_{|\alpha |\leq s}\int_\Omega |D^\alpha f|^2dx$ and inner product $(f,g)_s=\sum_{|\alpha |\leq s}\int_\Omega (D^\alpha f)\cdot (D^\alpha g)dx$. Here, we adopt the standard multi-index notation. For convenience, we will denote derivatives by $f_\alpha =D^\alpha f$. We will let $Df$ denote the gradient of $f$. Also, we will denote the $L^2$ inner product by $(f,g)=\int_\Omega f\cdot g$ $dx$. We will use standard function spaces. $L^\infty ([0,T],H^s)$ is the space of bounded measurable functions from $[0,T]$ into $H^s(\Omega)$, with the norm $\|f\| _{s,T}^2=\mathop{\rm ess\,sup}_{0\leq t\leq T}\|f(t)\|_s^2$. $C([0,T],H^s)$ is the space of continuous functions from $[0,T]$ into $H^s(\Omega )$. The following technical lemmas will be needed for the proof of the existence of a classical solution to the initial-value problem for the system \eqref{e1.2}, \eqref{e1.3}. \begin{lemma}[Standard Calculus Inequalities] \label{lem2.1} \quad \begin{itemize} \item[(a)] If $f\in H^{s_1}(\Omega)$, $g\in H^{s_2}(\Omega)$ and $s_3=\min\{s_1,s_2, s_1+s_2-s_{0}\}\geq 0$, where $s_0=[\frac N2] +1$, then $fg\in H^{s_3}(\Omega)$, and $\|fg\|_{s_3}\leq C\|f\|_{s_1}\|g\|_{s_2}$. We note that $s_0=2$ for $N=2$ or $N=3$. \item[(b)] If $f\in H^s(\Omega)$, $g\in H^{s-1}(\Omega)\cap L^\infty(\Omega)$, $Df\in L^\infty(\Omega)$, and $| \alpha | \leq s$, then $\| D^\alpha (fg)-fD^{\alpha}g\|_0\leq C(|Df|_{L^\infty} \|g\|_{s-1}+|g|_{L^\infty}\|Df\|_{s-1})$. \end{itemize} \end{lemma} In $(a)$ the constant $C$ depends on $s_1$, $s_2$, and $\Omega$, while in $(b)$ the constant $C$ depends on $s$ and $\Omega$. These inequalities are well known. Proofs may be found, for example, in \cite{k1,m2}. \begin{lemma}[Low-Norm Commutator Estimate]\label{lem2.2} If $Df\in H^{r_1}(\Omega )$, $g\in H^{r-1}(\Omega )$, where $r_1=\max\{r-1,s_0\}$, $s_0=[ \frac N2]+1$, then for any $r\geq 1$, $f,g$ satisfy the estimate $\|D^\alpha (fg)-fD^\alpha g\|_0\leq C\|Df\|_{r_1}\|g\|_{r-1}$, where $r=|\alpha |$, and the constant $C$ depends on $r$, $\Omega$. \end{lemma} \begin{proof} The proof is based on the Sobolev calculus inequalities from Lemma \ref{lem2.1}. We consider separately the cases $r-10$, and where $\Omega = \mathbb{T}^N$, $N=2,3$, and $r \geq 1$, we obtain the following estimates for $\|\Delta \rho \|_0^2+\|\nabla\rho\|_0^2$ and for $\|\nabla\rho\|_{r+1}^2$ \begin{gather*} \|\Delta \rho \| _0^2+\|\nabla\rho\|_0^2\leq C|D \mathbf{v}|_{L^{\infty}}^2\|D\mathbf{u}\|_{0}^2 +C\|\mathbf{f}\|_{0}^2, \\ \begin{aligned} \|\nabla\rho\|_{r+1}^2 &\leq C( \|\Delta \rho \|_r^2+\|\nabla\rho\|_r^2)\\ &\leq C\| Da\|_{r_1}^2 \|\nabla \rho \|_{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2+C\|D\mathbf{v}\| _{r_2+1}^2\|D\mathbf{u}\|_{r-1}^2 \end{aligned} \end{gather*} where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$, with $s_0=[\frac N2 ]+1$, and $N=2,3$. \end{lemma} \begin{proof} First, we obtain an $L^2$ estimate. Integrating equation \eqref{e2.9} by parts with $\bar{\rho}$, where $\bar{\rho}=\rho-\frac 1{|\Omega |}\int_\Omega \rho d \mathbf{x}$, yields \begin{align*} (\Delta \rho ,\Delta \rho ) &=(\Delta ^2\rho ,\bar{\rho} )= \frac 1c(\nabla \cdot (a\nabla \rho),\bar{\rho} )+\frac 1c (\nabla \cdot (\mathbf{v}\cdot \nabla \mathbf{u}),\bar{\rho} ) - \frac 1c(\nabla \cdot \mathbf{f},\bar{\rho}) \\ &=-\frac 1c(a\nabla \rho,\nabla \rho )+\frac 1c (\nabla \mathbf{v}^T: \nabla \mathbf{u} ,\bar{\rho} )+ \frac 1c(\mathbf{f},\nabla \rho) \\ &\leq - \frac {c_1}{c} \|\nabla \rho\|_0^2 +C(\epsilon)|D \mathbf{v}|_{L^\infty }^2\| D\mathbf{u}\|_0^2+\epsilon\|\bar{\rho} \|_0^2 +C(\epsilon)\|\mathbf{f}\|_0^2+\epsilon\|\nabla \rho \|_0^2 \\ &\leq - \frac {c_1}{c} \| \nabla \rho\|_0^2 +C(\epsilon)|D \mathbf{v}| _{L^\infty }^2\|D\mathbf{u}\|_0^2+\epsilon\| \nabla \rho \|_0^2 +C(\epsilon)\|\mathbf{f}\|_0^2+\epsilon\|\nabla \rho \|_0^2 \end{align*} where we used Cauchy's inequality with $\epsilon$, and where we used the fact that $a(\mathbf{x},t) \geq c_1$ where $c_1>0$. We also used Poincar\'{e}'s inequality to estimate $\|\bar{\rho} \|_0^2\leq C\|\nabla \rho \|_0^2$. And we used the fact that $\nabla \cdot (\mathbf{v}\cdot \nabla \mathbf{u})=\nabla \mathbf{v}^T: \nabla \mathbf{u}$, because $\nabla \cdot \mathbf{u}=0$. Moving the $\|\nabla \rho \|_0^2$ terms to the left-hand side yields the estimate $$\label{e2.10} \|\Delta \rho \| _0^2+\|\nabla\rho\|_0^2\leq C|D \mathbf{v}|_{L^{\infty}}^2\|D\mathbf{u}\|_{0}^2 +C\|\mathbf{f}\|_{0}^2$$ Next, after applying $D^\alpha$ to the equation \eqref{e2.9}, we obtain the equation: $$\label{e2.11} \Delta ^2\rho _\alpha =\frac 1c\nabla \cdot (a\nabla \rho_\alpha)+\frac 1c \nabla \cdot(\mathbf{v}\cdot \nabla \mathbf{u})_\alpha +F_\alpha$$ where $F_\alpha =-\frac 1c\nabla \cdot \mathbf{f}_{\alpha}+\frac 1c[\nabla \cdot (a\nabla \rho)_\alpha -\nabla \cdot (a\nabla \rho_\alpha )]$. Integrating equation \eqref{e2.11} by parts with $\rho _\alpha$, when $|\alpha|\geq 1$, yields \label{e2.12} \begin{aligned} (\Delta \rho _\alpha ,\Delta \rho _\alpha ) &=(\Delta ^2\rho_\alpha ,\rho _\alpha )\\ &=\frac 1c(\nabla \cdot(a\nabla \rho_\alpha), \rho _\alpha )+\frac 1c(\nabla \cdot(\mathbf{v}\cdot \nabla \mathbf{u})_{\alpha} , \rho _{\alpha} )+(F_\alpha ,\rho _\alpha ) \\ &= -\frac 1c(a\nabla \rho_\alpha, \nabla \rho _\alpha )+\frac 1c((\nabla \mathbf{v}^T: \nabla \mathbf{u})_{\alpha} , \rho _{\alpha}) +(F_{\alpha} ,\rho _{\alpha} ) \\ &= -\frac 1c(a\nabla \rho_\alpha, \nabla \rho _\alpha )-\frac 1c((\nabla \mathbf{v}^T: \nabla \mathbf{u})_{\alpha-\gamma} , \rho _{\alpha+\gamma}) +(F_{\alpha} ,\rho _{\alpha} ) \\ &\leq -\frac {c_1}{c}(\nabla \rho_\alpha ,\nabla \rho _\alpha) +C\|(\nabla \mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma }\|_0\|\rho_{\alpha +\gamma }\|_0 +| (F_{\alpha} ,\rho_{\alpha} )| \\ &\leq -\frac {c_1}{c}(\nabla \rho_\alpha ,\nabla \rho _\alpha) +C(\epsilon) \|(\nabla \mathbf{v}^T: \cdot\nabla \mathbf{u})_{\alpha -\gamma }\|_0^2 +\epsilon \|\rho_{\alpha +\gamma }\|_0^2+|(F_{\alpha} ,\rho_{\alpha} )| \end{aligned} where $|\gamma| = 1$. Here, we used Cauchy's inequality with $\epsilon$, and we used the fact that $a(\mathbf{x},t) \geq c_1$ where $c_1>0$. Next, we estimate the terms on the right-hand side of the above inequality. First, we estimate $\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma }\|_0^2$. When $|\alpha|=1$, we choose $\gamma =\alpha$ and obtain $$\label{e2.13} \|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma }\|_0^2=\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})\|_0^2 \leq C|D\mathbf{v}|_{L^{\infty}}^2\|D\mathbf{u}\|_0^2$$ Next, we use the triangle inequality and the commutator estimate from Lemma \ref{lem2.2} to estimate $\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma }\|_0^2$, when $|\alpha|>|\gamma|=1$, obtaining \label{e2.14} \begin{aligned} \|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma }\|_0^2 &\leq C( \|\nabla \mathbf{v}^T: \nabla \mathbf{u}_{\alpha -\gamma }\|_0^2 +\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma }-\nabla\mathbf{v}^T: \nabla \mathbf{u}_{\alpha -\gamma }\|_0^2) \\ &\leq C|D\mathbf{v}|_{L^{\infty}}^2\| D\mathbf{u}_{\alpha -\gamma} \|_0^2 +C\|D^{2} \mathbf{v}\|_{k_2}^2\|D\mathbf{u}\|_{k-2}^2 \\ &\leq C\|D\mathbf{v}\|_{k_2+1}^2\| D\mathbf{u}\|_{k-1}^2, \end{aligned} Here $|\gamma|=1$, $k=|\alpha |$, and $k_2=\max \{k-2,s_0\}$, with $s_0=[\frac N2 ]+1$. Here, we also used the Sobolev inequality $|h|_{L^{\infty}} \leq C\|h\|_{s_0}$. Next, we estimate the following term from the right-hand side of \eqref{e2.12}, obtaining $$\label{e2.15} \epsilon \|\rho_{\alpha +\gamma }\|_0^2 \leq \epsilon C \|\nabla \rho\|_k^2$$ where $|\gamma| = 1$ and $|\alpha|=k$. Next, we use the commutator estimate from Lemma \ref{lem2.2} to estimate $|(F_{\alpha} ,\rho _{\alpha} )|$ from the right-hand side of \eqref{e2.12}, obtaining \label{e2.16} \begin{aligned} |(F_{\alpha} ,\rho _{\alpha} )| &\leq |\frac 1c(\nabla \cdot \mathbf{f}_{\alpha},\rho_{\alpha})| +|\frac 1c([\nabla \cdot (a\nabla \rho)_{\alpha} -\nabla \cdot (a\nabla \rho_{\alpha} )],\rho _{\alpha} )| \\ &= |\frac 1c(\mathbf{f}_{\alpha-\gamma},\nabla \rho_{\alpha+\gamma})|+|\frac 1c([(a\nabla \rho)_{\alpha} -a\nabla \rho_{\alpha} ],\nabla \rho _{\alpha} )| \\ &\leq C\|\mathbf{f}\|_{k-1}\|\nabla \rho\|_{k+1}+ C\|Da\|_{k_1}\|\nabla \rho \|_{k-1}\|\nabla \rho \|_k \\ &\leq C(\epsilon)\|\mathbf{f}\|_{k-1}^2+\epsilon\| \nabla \rho \|_{k+1}^2+ C(\epsilon)\|Da\| _{k_1}^2\|\nabla \rho \|_{k-1}^2+\epsilon\| \nabla \rho \|_k^2 \end{aligned} where $|\gamma| = 1$, $k=|\alpha |$, and $k_1=\max \{k-1,s_0\}$, with $s_0=[\frac N2 ] +1$. Again, we used Cauchy's inequality with $\epsilon$. Substituting \eqref{e2.13}-\eqref{e2.16} into \eqref{e2.12}, and adding \eqref{e2.12} over $|\alpha |=k\leq r$, including the $L^2$ estimate \eqref{e2.10}, we obtain for $r\geq 1$ the estimate \label{e2.17} \begin{aligned} \|\Delta \rho \|_r^2 +\|\nabla \rho \|_r^2 &\leq C\|Da\|_{r_1}^2 \|\nabla \rho \|_{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\| D\mathbf{v}\|_{r_2+1}^2\|D\mathbf{u}\|_{r-1}^2 \\ &\quad +\epsilon C\|\nabla \rho \|_{r+1}^2+\epsilon C\|\nabla \rho\|_{r}^2, \end{aligned} where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$, with $s_0=[\frac N2 ]+1$. Here, we also used the Sobolev inequality $|h|_{L^{\infty}} \leq C\|h\|_{s_0}$. From Lemma \ref{lem2.5}, we have $\epsilon \|\nabla \rho \| _{r+1}^2\leq \epsilon C\|\Delta \rho \|_{r}^2$. After substituting this estimate into the right-hand side of \eqref{e2.17}, we obtain the estimate $$\label{e2.18} \|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2 \leq C\|Da\|_{r_1}^2 \|\nabla\rho \| _{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\| D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2,$$ where we have moved the terms $\epsilon C\|\Delta \rho \|_r^2$ and $\epsilon C\|\nabla \rho \|_r^2$ to the left-hand side. Finally, using the estimate for $\| \nabla \rho \|_{r+1}^2$ from Lemma \ref{lem2.5}, we obtain from \eqref{e2.18} the estimate \begin{align*} \|\nabla \rho \|_{r+1}^2 &\leq C(\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2) \\ &\leq C\| Da\|_{r_1}^2 \|\nabla\rho \|_{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\|D\mathbf{v}\| _{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2 \end{align*} \end{proof} \begin{lemma}\label{lem2.7} If $\mathbf{u}$, $\mathbf{v}$, $a$, $\mathbf{f}$, and $\rho$ are sufficiently smooth in $$\label{e2.19} \Delta ^2\rho =\frac 1c\nabla \cdot (a\nabla \rho) +\frac1c\nabla \cdot (\mathbf{v}\cdot \nabla \mathbf{u}) -\frac 1c \nabla \cdot \mathbf{f}$$ where $c$ is a positive constant, $\nabla \cdot \mathbf{u}=0$, $\nabla \cdot \mathbf{v}=0$, $a(\mathbf{x},t) \geq c_1$, with $c_1>0$, and $\Omega = \mathbb{T}^N$, $N=2,3$, then for $r \geq 1$ , $\rho$ satisfies the following estimate \begin{align*} \|\nabla \rho \|_{r+1}^2 &\leq C\left(\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2 \right) \\ &\leq C\Big[1+\sum_{j=1}^{r}\|Da\| _{r_1}^{2j}\Big](\|\mathbf{f}\|_{r-1}^2 +\| D\mathbf{v}\|_{r_2+1}^2 \|D \mathbf{u}\|_{r-1}^2 ) \end{align*} where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$, with $s_0=[\frac N2 ]+1$, and $C$ depends on $r$. \end{lemma} \begin{proof} From Lemma \ref{lem2.6} applied to equation \eqref{e2.19}, we have the estimate $$\label{e2.20} \|\Delta \rho \|_s^2+\|\nabla \rho \|_s^2 \leq C\|Da\|_{s_1}^2 \|\nabla\rho \| _{s-1}^2 +C\|\mathbf{f}\|_{s-1}^2 +C\| D\mathbf{v}\|_{s_2+1}^2\|D \mathbf{u}\|_{s-1}^2$$ where $s \geq 1$, and where $s_1=\max\{s-1,s_0\}$, $s_2=\max \{s-2,s_0\}$, with $s_0=[\frac N2 ] +1$. Letting $s=r$ in the estimate \eqref{e2.20} yields $$\label{e2.21} \|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2 \leq C\|Da\|_{r_1}^2 \|\nabla\rho \| _{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\| D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2$$ Applying the estimate \eqref{e2.20}, letting $s=r-1$, to the term $C\|Da\|_{r_1}^2 \|\nabla\rho \| _{r-1}^2$ which appears on the right-hand side of \eqref{e2.21} yields \label{e2.22} \begin{aligned} \|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2 &\leq C\|Da\|_{r_1}^2\Big[\|Da\|_{r_2}^2 \|\nabla\rho \|_{r-2}^2 +\|\mathbf{f}\|_{r-2}^2 +\|D\mathbf{v}\| _{r_3+1}^2\|D\mathbf{u}\|_{r-2}^2 \Big] \\ &\quad +C\|\mathbf{f}\|_{r-1}^2 +C\|D\mathbf{v}\| _{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2 \\ &\leq C\|Da\|_{r_1}^4 \|\nabla\rho \|_{r-2}^2 +C\|Da\|_{r_1}^2 \Big(\|\mathbf{f}\|_{r-2}^2 +\| D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\| _{r-2}^2\Big) \\ &\quad+C\|\mathbf{f}\|_{r-1}^2 +C\| D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2 \end{aligned} where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$, and $r_3=\max \{r-3,s_0\}$, $r_3\leq r_2 \leq r_1$, with $s_0=[\frac N2 ]+1=2$ for $N=2,3$. Similarly, by repeatedly applying the estimate \eqref{e2.20}, letting $s=r-j$, to the term $C\|Da\|_{r_1}^{2j}\|\nabla\rho \| _{r-j}^2$, for $j=2,3,\dots ,r-1$, which will appear on the right-hand side of \eqref{e2.22} yields \label{e2.23} \begin{aligned} \|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2 &\leq C\sum_{j=1}^{r-1}\|Da\|_{r_1}^{2j} \Big(\|\mathbf{f}\|_{r-1-j}^2 +\| D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\| _{r-1-j}^2 \Big) \\ &\quad +C\|Da\|_{r_1}^{2r}\|\nabla\rho \|_{0}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\|D\mathbf{v}\|_{r_2+1}^2 \|D \mathbf{u}\|_{r-1}^2 \\ &\leq C\Big[1+\sum_{j=1}^{r-1}\|Da\| _{r_1}^{2j}\Big]\Big(\|\mathbf{f}\|_{r-1}^2 +\| D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2\Big) \\ &\quad +C\|Da\|_{r_1}^{2r}\|\nabla\rho \|_{0}^2 \end{aligned} Substituting the estimate for $\|\nabla \rho\|_0^2$ from Lemma \ref{lem2.6} into the right-hand side of \eqref{e2.23} yields \begin{align*} \|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2 &\leq C\big[1+\sum_{j=1}^{r-1}\|Da\|_{r_1}^{2j}\Big] \Big(\|\mathbf{f}\|_{r-1}^2 +\| D\mathbf{v}\|_{r_2+1}^2 \|D \mathbf{u}\|_{r-1}^2 \Big) \\ &\quad +C\|Da\|_{r_1}^{2r}\Big(\|\mathbf{f}\|_0^2 +| D\mathbf{v}|_{L^\infty}^2\|D\mathbf{u}\|_{0}^2 \Big) \\ &\leq C\Big[1+\sum_{j=1}^{r}\|Da\|_{r_1}^{2j}\Big] \Big(\|\mathbf{f}\|_{r-1}^2 +\| D\mathbf{v}\|_{r_2+1}^2 \|D \mathbf{u}\|_{r-1}^2\Big) \end{align*} This completes the proof. \end{proof} \section{Existence theorem} In this section, we prove the existence of a unique classical solution to the initial-value problem for equations \eqref{e1.2}, \eqref{e1.3}, on any given time interval $0\leq t\leq T$, with periodic boundary conditions, for sufficiently small initial velocity gradient. \begin{theorem}\label{thm3.1} Suppose $s>\frac N2+3$ and $\Omega=\mathbb{T}^N$, $N=2,3$. For any given time interval $0 \leq t\leq T$, equations \eqref{e1.2}, \eqref{e1.3} have a unique classical solution $\rho$, $\mathbf{v}$, for initial data $\mathbf{v}_0(\mathbf{x})\in H^{s}(\Omega )$, $\nabla \cdot \mathbf{v}_0=0$, and for given data $\rho^0(\mathbf{x},t)$ and $\mathbf{x}_0$, provided $D \mathbf{v}_0$ is sufficiently small. Here $\rho^0(\mathbf{x},t)$ is a given positive function with sufficiently small gradient $\nabla \rho^0$, and $\mathbf{x}_0$ is a point in the domain $\Omega$. The regularity of the solution is \begin{gather*} \rho \in C([0,T],C^5)\cap L^\infty([0,T],H^{s+2}), \\ \mathbf{v} \in C([0,T],C^3)\cap L^\infty([0,T],H^s), \\ \frac{\partial \mathbf{v}}{\partial t} \in C([0,T],C^2)\cap L^\infty([0,T],H^{s-1}). \end{gather*} and $\rho(\mathbf{x},t) \geq c_1$, and $\rho^{-1} \hat{p}' (\rho)(\mathbf{x},t) \geq c_1$, for some positive constant $c_1$, for $\mathbf{x}\in \Omega$, and $0\leq t \leq T$. \end{theorem} \begin{proof} We will construct the solution of the problem for \eqref{e1.2}, \eqref{e1.3}, with $\Omega = \mathbb{T}^N$, through an iteration scheme. To define the iteration scheme, we will let the sequence of approximate solutions be $\mathbf{v}^k$ and $\rho^k$. Set $\mathbf{v}^0(\mathbf{x},t)=\mathbf{v}_0(\mathbf{x})$, the initial velocity data, and set $\rho^0(\mathbf{x},t)\in C([0,T],C^5)\cap L^\infty([0,T],H^{s+2})$ to be a positive function satisfying $\|\nabla \rho ^{0}\|_{s+1,T} \leq \|D \mathbf{v}_0\|_{s-1}$. For $k=0,1,2,\dots$, construct $\mathbf{v}^{k+1}$, $\rho^{k+1}$ from the previous iterates $\mathbf{v}^k$, $\rho ^k$ by solving the linear system of equations \begin{gather} \frac{D^k\mathbf{v}^{k+1}}{Dt}+(\rho^k)^{-1} \hat{p}' (\rho^k)\nabla \rho ^{k+1} = c\nabla \Delta \rho ^{k+1}, \label{e3.1} \\ \nabla \cdot \mathbf{v}^{k+1} =0, \label{e3.2} \end{gather} where $D^k/{Dt}= \partial/{\partial t}+\mathbf{v}^k \cdot\nabla$, and with initial data $\mathbf{v}^{k+1}(\mathbf{x}, 0)=\mathbf{v}_0(\mathbf{x})$. Since the solution $\rho ^{k+1}$ is unique up to an arbitrary function of $t$, we specify that the solution $\rho ^{k+1}$ satisfy $\rho ^{k+1}(\mathbf{x}_0,t) =\rho ^{k}(\mathbf{x}_0,t)$ at a single, fixed point $\mathbf{x}_0 \in \Omega$, for all $k\geq 0$. Hence $\rho ^{k+1}(\mathbf{x}_0,t) = \rho ^{0}(\mathbf{x}_0,t)$. Existence of a sufficiently smooth solution to equations \eqref{e3.1}, \eqref{e3.2} for fixed $k$ follows from the results stated in Section 4. We proceed now to prove convergence of the iterates as $k\to \infty$ to a unique classical solution of \eqref{e1.2}, \eqref{e1.3}. We assume that $p$ is a sufficiently smooth function of the thermodynamic state variable $\rho$ in an open interval $G\subset \mathbb{R}$. We fix connected, bounded open sets $G_0$ and $G_1$ such that $\bar{G}_0\subset G_1$ and $\bar{G}_1\subset G$, and we require that the initial iterate $\rho^0$ satisfies $\rho^0 \in G_0$. We fix $\delta =\hat{\delta}(G_0,G_1)$ so that $0 < \delta < \mbox{dist}(\bar{G}_0,\partial{G_1})$; therefore if $|\rho^k -\rho ^0|_{L^\infty } \leq \delta$, then $\rho^{k}(\mathbf{x},t)\in G_1$ for all $\mathbf{x}\in \mathbb{T}^N$, and for $t \in [0,T]$. The values of $\rho^k \in G_1$ are assumed to be strictly positive, bounded, and bounded away from zero. And the values of $(\rho^k)^{-1} \hat{p}' (\rho^k)$ for $\rho^k \in G_1$ are also assumed to be strictly positive, bounded, and bounded away from zero. Using a proof by induction on $k$, we assume that $\rho^k \in G_1$, and then later we will show that $\rho^{k+1} \in G_1$. First, we proceed with the proof of uniform boundedness of the approximating sequence in a high Sobolev norm. \begin{proposition}\label{prop3.1} Assume that the hypotheses of Theorem \ref{thm3.1} hold. Let $\rho^0$, $\mathbf{v}_0$ satisfy $C_0\|\rho^0\|_{0 }^2+C_0\|\nabla \rho^0 \|_{2 }^2 \leq L_0^2$ and $e^{2T}\|\mathbf{v}_0\|_{0 }^2\leq L_0^2$, where $L_0$ is a constant and where $C_0$ is the constant from Lemma \ref{lem2.4}. There are constants $\epsilon_0$, $L_1$, $L_2$, $L_3$, where $0<\epsilon_0<1$, such that the following estimates hold for $k=1,2,3\dots$, provided $\|D\mathbf{v}_0\|_{s-1}$ is sufficiently small: \begin{itemize} \item[(a)] $\|\nabla \rho^k\|_{s+1,T}^2 \leq \epsilon_0$, $\|\rho^k\|_{s+2,T} \leq L_1$, \item[(b)] $\|D\mathbf{v} ^k\|_{s-1,T}^2\leq \epsilon_0$, $\|\mathbf{v} ^k\|_{s,T}\leq L_2$, \item[(c)] $\|{\partial \mathbf{v}^k}/{\partial t}\|_{s-1,T}^2 \leq \epsilon_0 L_3$. \end{itemize} \end{proposition} \begin{proof} The proof is by induction on $k$. We show only the inductive step. We will derive estimates for $\rho ^{k+1}$ and $\mathbf{v}^{k+1}$, and then use these estimates to prescribe $\epsilon_0$, $L_1$, $L_2$, $L_3$ a priori, independent of $k$, so that if $\rho ^k$ and $\mathbf{v}^k$ satisfy the estimates in (a)-(c), then $\rho ^{k+1}$ and $\mathbf{v} ^{k+1}$ also satisfy the same estimates. And we will show that $\epsilon_0$ will be small if $\|D\mathbf{v} _0\|_{s-1}$ is sufficiently small. In the estimates below, we use $C$ to denote a generic constant whose value may change from one instance to the next, but is independent of $\epsilon_0$, $L_1$, $L_2$, and $L_3$. \smallskip \noindent\textbf{Estimate for $\nabla\rho ^{k+1}$:} Applying the divergence operator to equation \eqref{e3.1}, we obtain $$\label{e3.3} c\Delta ^2\rho ^{k+1}-\nabla \cdot ((\rho ^k)^{-1}\hat{p}' (\rho^k)\nabla \rho ^{k+1})=\nabla \cdot (\mathbf{v}^k\cdot \nabla \mathbf{v} ^{k+1})$$ where we used the fact that $\nabla \cdot \mathbf{v}^{k+1}=0$, and where $c$ is a positive constant. Applying Lemma \ref{lem2.7} we obtain from the above equation the following estimate for $\nabla \rho ^{k+1}$, for $s>\frac N2+3$ \label{e3.4} \begin{aligned} \|\nabla \rho ^{k+1}\|_{s+1}^2 &\leq C(\| \Delta \rho^{k+1}\|_s^2+\|\nabla \rho ^{k+1}\|_s^2) \\ &\leq C[1+\sum_{j=1}^s\|D((\rho ^k)^{-1}\hat{p}'(\rho ^k))\|_{s_1}^{2j}] \|D\mathbf{v}^k\|_{s_2+1}^2\| D\mathbf{v}^{k+1}\|_{s-1}^2 \\ &\leq C_1\|D\mathbf{v}^k\|_{s-1}^2\|D \mathbf{v}^{k+1}\|_{s-1}^2 \end{aligned} where $C_1=\hat{C}_1(L_1)$. Here, $s_1=\max\{s-1,s_0\}=s-1$, $s_2=\max\{s-2,s_0\}=s-2$, where $s_0=[ \frac N2]+1=2$, and $s>\frac N2+3$, so $s\geq 5$ for $N=2,3$. And we used the induction hypothesis for $\rho ^k$, $\mathbf{v}^k$. \smallskip \noindent\textbf{Estimate for $\rho ^{k+1}$:} To obtain an $L^2$ estimate for $\rho ^{k+1}$, we apply Lemma \ref{lem2.4}, where we define $u =\rho ^{k+1}$ and we define $w=\rho ^0$ , to be used for the functions $u ,w$ appearing in Lemma \ref{lem2.4}. Note that since by hypothesis we have $\rho ^{k+1}( \mathbf{x}_0,t)=\rho ^0(\mathbf{x}_0,t)$ at a single fixed point $\mathbf{x}_0\in \Omega$, the hypotheses of Lemma \ref{lem2.4} are satisfied, and so we obtain the estimate \label{e3.5} \begin{aligned} \|\rho ^{k+1}\|_0^2 &\leq C_0\|\rho ^0\|_0^2+C_0\|\nabla \rho ^0\|_{2 }^2+C_0\|\nabla \rho ^{k+1}\|_{2}^2 \\ &\leq L_0^2+C_2\|D\mathbf{v}^k\|_{s_0+1}^2\|D \mathbf{v}^{k+1}\|_1^2 \\ &\leq L_0^2+C_2\|D\mathbf{v}^k\|_{s-1}^2\|D \mathbf{v}^{k+1}\|_1^2 \end{aligned} where $C_0$ is the constant from Lemma \ref{lem2.4} (recall $C_0$ depends only on $\Omega$), and where $C_2=\hat{C}_2(L_1)$. Here we used the estimate for $\|\Delta \rho ^{k+1}\|_2^2+\| \nabla \rho ^{k+1}\|_2^2$ from applying Lemma \ref{lem2.7} to \eqref{e3.3} with $r=2$. We also used the hypothesis that $C_0 \|\rho ^0\|_0^2+C_0\|\nabla \rho ^0\|_2^2\leq L_0^2$. Adding the estimates \eqref{e3.4}, \eqref{e3.5} yields the following estimate for $\rho ^{k+1}$, \label{e3.6} \begin{aligned} \|\rho ^{k+1}\|_{s+2}^2 &\leq \|\rho^{k+1}\| _0^2+C\|\nabla \rho ^{k+1}\|_{s+1}^2 \\ &\leq L_0^2+C_3\|D\mathbf{v}^k\|_{s-1}^2\|D \mathbf{v}^{k+1}\|_{s-1}^2 \end{aligned} where $C_3=\hat{C}_3(L_1)$. \smallskip \noindent\textbf{Estimate for $\mathbf{v}^{k+1}$:} Applying Lemma \ref{lem2.3} to equations \eqref{e3.1}, \eqref{e3.2}, we obtain for $D\mathbf{v}^{k+1}$ the estimate \label{e3.7} \begin{aligned} \|D \mathbf{v}^{k+1}\|_{s-1}^2 & \leq Ce^{2T}(1+Te^{2T}e^{\beta (T)}\|D\mathbf{v}^k\| _{s-1,T}^2) (\|D\mathbf{v}_0\|_{s-1}^2 \\ &\quad +\int_0^t (\|D ((\rho^k)^{-1} \hat{p}'(\rho^k))\|_{s_1}^2 \|\nabla\rho^{k+1} \|_{s-1}^2 )d\tau) \\ &\leq Ce^{2T}(1+ Te^{2T}e^{ Te^{2T}})( \|D\mathbf{v}_0\|_{s-1}^2 +C_4\int_0^t \| \nabla\rho^{k+1} \|_{s-1}^2 d\tau ) \\ &\leq C_5\|D\mathbf{v}_0\|_{s-1}^2+C_5\int_0^t \| D\mathbf{v}^k\|_{s-1,T}^2\|D \mathbf{v}^{k+1} \|_{s-1}^2 d\tau \\ &\leq C_5\|D\mathbf{v}_0\|_{s-1}^2+C_5\int_0^t \|D \mathbf{v}^{k+1} \|_{s-1}^2 d\tau \end{aligned} where $\beta (T)=Te^{2T}\|D\mathbf{v}^k\|_{s-1,T}^2$. Here, we used \eqref{e3.4} to estimate $\|\nabla \rho ^{k+1}\| _{s-1}^2$, and we used the fact that by the induction hypothesis, $\|D \mathbf{v}^{k}\| _{s-1,T}^2 \leq \epsilon_0$, where $0<\epsilon_0 <1$. And here $C_{4}=\hat{C}_{4}(L_1,T)$, $C_{5}=\hat{C}_{5}(L_1,T)$. Applying Gronwall's inequality yields the estimate $$\label{e3.8} \|D \mathbf{v}^{k+1}\|_{s-1}^2 \leq C_6\|D\mathbf{v}_0\|_{s-1}^2$$ where $C_{6}=\hat{C}_{6}(L_1,T)$. We now choose $\epsilon_0$ to satisfy $\epsilon_0 =C_6\|D\mathbf{v}_0\|_{s-1}^2$. It follows that $\|D \mathbf{v}^{k+1}\|_{s-1,T}^2 \leq \epsilon_0$. Substituting the estimate \eqref{e3.8} for $\|D\mathbf{v}^{k+1}\|_{s-1}^2$ into the right-hand side of \eqref{e3.4}, \eqref{e3.6} yields the following estimates for $\|\nabla \rho ^{k+1}\|_{s+1}^2$ and $\|\rho^{k+1}\|_{s+2}^2$: \begin{gather*} \|\nabla \rho ^{k+1}\|_{s+1}^2 \leq C_1\|D\mathbf{v}^k\|_{s-1}^2\|D \mathbf{v} ^{k+1}\|_{s-1}^2\leq \epsilon_0 C_1C_6\|D\mathbf{v}_0\|_{s-1}^2, \\ \|\rho ^{k+1}\|_{s+2}^2 \leq L_0^2 +C_3\|D\mathbf{v}^k\|_{s-1}^2\|D\mathbf{v}^{k+1}\|_{s-1}^2 \leq L_0^2+\epsilon_0 C_3C_6\|D \mathbf{v}_0\|_{s-1}^2 \end{gather*} Therefore, we have $\|\rho ^{k+1}\|_{s+2,T}\leq L_1$ and we have $\|\nabla \rho ^{k+1}\|_{s+1,T}^2\leq \epsilon_0$, provided that we choose $L_1$ large enough so that $\frac 12 L_1^2>L_0^2$, and provided that we choose $\| D\mathbf{v}_0\|_{s-1}$ small enough so that $\epsilon_0 C_3 C_6\|D \mathbf{v}_0\|_{s-1}^2<\frac 12 L_1^2$, and provided that we choose $\|D \mathbf{v}_0\|_{s-1}$ small enough so that $C_1 C_6\|D\mathbf{v}_0\|_{s-1}^2<1$. We also choose $\|D\mathbf{v}_0\|_{s-1}$ small enough so that $C_6\|D\mathbf{v}_0\|_{s-1}^2<1$; therefore, we have $0<\epsilon_0<1$, since $\epsilon_0=C_6\|D\mathbf{v}_0\|_{s-1}^2$. (Note that $\epsilon_0$ will be small if $\|D\mathbf{v} _0\| _{s-1}$ is sufficiently small.) This completes the proof of part (a). Next, by applying Lemma \ref{lem2.3} to \eqref{e3.1}, \eqref{e3.2}, and using the estimates \eqref{e3.4}, \eqref{e3.7}, \eqref{e3.8}, we have the estimate \begin{align*} \|\mathbf{v}^{k+1}\|_s^2 & \leq e^{2T}\|\mathbf{v}_0\|_0^2 +Ce^{2T}(1+Te^{2T}e^{\beta (T)}\|D\mathbf{v}^k\|_{s-1,T}^2)(\|D\mathbf{v}_0\|_{s-1}^2 \\ &\quad +\int_0^t (\|D ((\rho^k)^{-1} \hat{p}' (\rho^k))\|_{s_1}^2\|\nabla\rho^{k+1} \|_{s-1}^2 )d\tau) \\ &\leq L_0^2 + C_7\|D\mathbf{v}_0\|_{s-1}^2 +C_7\int_0^t \|D \mathbf{v}^{k+1} \| _{s-1}^2 d\tau \\ &\leq L_0^2 + C_7\|D\mathbf{v}_0\|_{s-1}^2 + C_8 \|D \mathbf{v}_0 \|_{s-1}^2 \end{align*} with $\beta (T)=Te^{2T}\|D\mathbf{v}^k\|_{s-1,T}^2$. Here $s_1=\max\{s-1,s_0\}=s-1$, $s>\frac N2 +3$, and $s_0=[ \frac N2]+1=2$ for $N=2,3$. We also used the hypothesis that $e^{2T}\|\mathbf{v}_0\|_0^2 \leq L_0^2$, and that $\|D \mathbf{v}^{k}\|_{s-1,T}^2 \leq \epsilon_0$, where $0<\epsilon_0<1$. And here $C_{7}=\hat{C}_{7}(L_1,T)$, $C_{8}=\hat{C}_{8}(L_1,T)$. Therefore, we have $\|\mathbf{v}^{k+1}\|_{s,T}\leq L_2$ provided that we choose $L_2$ large enough so that $\frac 12L_2^2>L_0^2$, and so that $\frac 12 L_2^{2}>(C_7+ C_8)\|D\mathbf{v} _0\|_{s-1}^2$. This completes the proof of part (b). We next consider part (c). \smallskip \noindent\textbf{Estimate for $\mathbf{v}_t^{k+1}$:} Directly from equation \eqref{e3.1} for $\mathbf{v}_t^{k+1}$, and the estimates already derived for $\rho^{k+1}$ and $\mathbf{v}^{k+1}$, we deduce that $\|\mathbf{v}_t^{k+1}\|_{s-1}^2\leq C_{9}(\|D\mathbf{v}^{k+1}\|_{s-1}^2 +\| \nabla\rho^{k+1} \|_{s+1}^2)\leq 2\epsilon_0 C_9$, where $C_{9}= \hat{C}_{9}(L_1,L_2,T)$. Thus $\|\mathbf{v}_t^{k+1}\|_{s-1,T}^2 \leq \epsilon_0 L_3$ provided we choose $L_3$ sufficiently large so that $L_3 \geq 2 C_9$. Summarizing, if we choose $\|D\mathbf{v}_0\|_{s-1}$ to be sufficiently small, then $\rho ^k$ and $\mathbf{v}^k$ satisfy the estimates in (a)-(c) for all $k\geq 1$. This completes the proof of Proposition \ref{prop3.1}. \end{proof} Next, we give the proof of contraction in low Sobolev norm. \begin{proposition}\label{prop3.2} Assume that the hypotheses of Theorem \ref{thm3.1} hold. Then we have $\|\rho ^{k+1}-\rho ^k\|_{2,T}^2+\| \mathbf{v}^{k+1}-\mathbf{v}^k\|_{2,T}^2 \leq \zeta \Big(\|\rho ^{k}-\rho ^{k-1}\|_{2,T}^2+ \|\mathbf{v}^k-\mathbf{v}^{k-1}\|_{2,T}^2 \Big)$ for $k=1,2,3\dots$. Here $\zeta$ is a constant such that $0<\zeta<1$. And $|\rho^{k+1} - \rho^0|_{L^{\infty}} \leq \delta$; hence, $\rho^k \in G_1$, for all $k$. \end{proposition} \begin{proof} Subtracting the equations \eqref{e3.1}, \eqref{e3.2} for $\rho^k$ and $\mathbf{v}^k$ from the equations \eqref{e3.1}, \eqref{e3.2} for $\rho^{k+1}$ and $\mathbf{v }^{k+1}$ yields equations which we write in the form \begin{gather} \label{e3.9} \begin{aligned} &\frac{D^k(\mathbf{v}^{k+1}-\mathbf{v}^k)}{Dt}+ (\rho^k)^{-1} \hat{p} ' (\rho^k)\nabla (\rho ^{k+1}-\rho ^k)\\ & =c\nabla \Delta (\rho ^{k+1}-\rho ^k) -(\mathbf{v}^k-\mathbf{v}^{k-1})\cdot \nabla \mathbf{v}^k\\ &\quad -(\big((\rho^k)^{-1} \hat{p}' (\rho^k)-(\rho^{k-1})^{-1} \hat{p}' (\rho^{k-1})\big)\nabla\rho^k , \end{aligned}\\ \nabla \cdot (\mathbf{v}^{k+1}-\mathbf{v}^k) =0, \label{e3.10} \end{gather} where $D^k/{Dt}= \partial/{\partial t}+\mathbf{v}^k \cdot\nabla$, and where $(\mathbf{v}^{k+1}-\mathbf{v}^k)(\mathbf{x},0)=0$. Using a proof by induction on $k$, we assume that $\rho^k \in G_1$, and then we will show that $\rho^{k+1} \in G_1$. First, we obtain estimates for $\rho ^{k+1}-\rho ^k$ and $\mathbf{v}^{k+1}-\mathbf{v}^k$. \smallskip \noindent\textbf{Estimate for $\mathbf{\rho }^{k+1}-\rho^{k}$:} Applying the divergence operator to \eqref{e3.9} yields \label{e3.11} \begin{aligned} &c\Delta ^2(\rho ^{k+1}-\rho ^k)-\nabla \cdot ((\rho ^k)^{-1}p'(\rho ^k)\nabla (\rho ^{k+1}-\rho ^k)) \\ &=\nabla \cdot (\mathbf{v}^k\cdot \nabla (\mathbf{v}^{k+1} -\mathbf{v}^k)) +\nabla \cdot ((\mathbf{v}^k-\mathbf{v}^{k-1}) \cdot \nabla \mathbf{v}^k) \\ &\quad +\nabla \cdot (((\rho ^k)^{-1}p'(\rho ^k)-(\rho ^{k-1})^{-1}p'(\rho ^{k-1}))\nabla \rho ^k) \end{aligned} Applying Lemma \ref{lem2.7} to equation \eqref{e3.11} with $r=2$ yields the estimate \label{e3.12} \begin{aligned} &\|\Delta (\rho ^{k+1}-\rho ^k)\|_2^2+\|\nabla (\rho ^{k+1}-\rho ^k\|_2^2 \\ &\leq C[1+\sum_{j=1}^2\|D((\rho ^k)^{-1}p'(\rho ^k))\|_2^{2j}](\|D\mathbf{v}^k\| _3^2\| D(\mathbf{v}^{k+1}-\mathbf{v}^k)\|_1^2 \\ &\quad +\|(\mathbf{v}^k-\mathbf{v} ^{k-1})\cdot \nabla \mathbf{v}^k\|_1^2 +\|((\rho ^k)^{-1}p'(\rho ^k)-(\rho ^{k-1})^{-1}p'(\rho ^{k-1}))\nabla \rho ^k\|_1^2) \\ &\leq C_1\|D\mathbf{v}^k\|_3^2\|\mathbf{v}^{k+1}-\mathbf{ v}^k\|_2^2+C_1\|D \mathbf{v}^k\|_2^2\|\mathbf{v}^k- \mathbf{v}^{k-1}\|_2^2+C_1\|\nabla \rho ^k\| _2^2\|\rho ^k-\rho ^{k-1}\|_2^2 \end{aligned} where $C_1=\hat{C}_1(L_1)$ from Proposition \ref{prop3.1}. Here we used the Sobolev calculus inequality $\|f g \|_r \leq C\| f\|_r\|g\|_r$ for $r=2>\frac N2$. To obtain an $L^2$ estimate for $\rho ^{k+1}$, we apply Lemma \ref{lem2.4}, where we define $u =\rho ^{k+1}$ and we define $w=\rho ^k$ , to be used for the functions $u ,w$ appearing in Lemma \ref{lem2.4}. Note that since by hypothesis we have $\rho ^{k+1}(\mathbf{x}_0,t)=\rho ^k(\mathbf{x}_0,t)$ for all $k\geq 0$, at a single fixed point $\mathbf{x}_0\in \Omega$, the hypotheses of Lemma \ref{lem2.4} are satisfied, and so we obtain the estimate \label{e3.13} \begin{aligned} \|\rho ^{k+1}-\rho ^k\|_0^2 &\leq C\|\nabla (\rho^{k+1}-\rho ^k)\|_2^2 \\ &\leq C_2\|D\mathbf{v}^k\|_3^2\|\mathbf{v}^{k+1}-\mathbf{ v}^k\|_2^2+C_2\|D\mathbf{v}^k\|_2^2\|\mathbf{v}^k- \mathbf{v}^{k-1}\|_2^2 \\ &\quad +C_2\|\nabla \rho ^k\|_2^2\|\rho ^k-\rho^{k-1}\|_2^2 \end{aligned} where we used the estimate for $\|\Delta (\rho ^{k+1}-\rho ^k)\|_2^2+\|\nabla (\rho ^{k+1} -\rho ^k)\|_2^2$ from \eqref{e3.12}. Here $C_2=\hat{C}_2(L_1)$ from Proposition \ref{prop3.1}. Now, adding \eqref{e3.12}, \eqref{e3.13}, we obtain the estimate \label{e3.14} \begin{aligned} \|\rho ^{k+1}-\rho ^k\|_2^2 &\leq \|\rho ^{k+1}-\rho ^k\|_0^2+C\|\nabla (\rho ^{k+1}-\rho ^k)\|_1^2 \\ &\leq \|\rho ^{k+1}-\rho ^k\|_0^2+C(\|\nabla (\rho ^{k+1}-\rho ^k)\|_2^2+\|\Delta (\rho ^{k+1}-\rho ^k)\|_2^2) \\ &\leq C_3\|D\mathbf{v}^k\|_3^2\|\mathbf{v}^{k+1}- \mathbf{v}^k\|_2^2+C_3 \|D\mathbf{v}^k\|_2^2\|\mathbf{v} ^k-\mathbf{v}^{k-1}\|_2^2 \\ &\quad +C_3\|\nabla \rho ^k\|_2^2\|\rho ^k-\rho ^{k-1}\|_2^2 \\ &\leq C_3\left(\|\mathbf{v}^{k+1}-\mathbf{v}^k\| _2^2+\epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\| _2^2+\epsilon_0 \|\rho ^k-\rho ^{k-1}\|_2^2\right) \end{aligned} where $C_3=\hat{C}_3(L_1)$. Here, we used the Proposition \ref{prop3.1} estimates $\|D\mathbf{v}^k\|_2^2\leq \|D\mathbf{v}^k\|_3^2 \leq \|D \mathbf{v}^k\|_{s-1,T}^2\leq \epsilon_0$, and $\|\nabla \rho ^k\|_2^2\leq \|\nabla \rho ^k\|_{s+1,T}^2\leq \epsilon_0$, where $0<\epsilon_0<1$, and where $\epsilon_0$ is as small as we like if $\|D \mathbf{v}_0\|_{s-1}$ is sufficiently small. \noindent\textbf{Estimate for $\mathbf{v}^{k+1}\mathbf{-v}^{k}$:} After applying Lemma \ref{lem2.3} to \eqref{e3.9}, using $r=2$, we obtain \label{e3.15} \begin{aligned} \|\mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2 &\leq C_4\int_0^t\|D\mathbf{v}^k\|_2^2\|\mathbf{v}^k- \mathbf{v}^{k-1}\|_2^2d\tau +C_4\int_0^t\|\nabla (\rho^{k+1}-\rho ^k)\|_1^2d\tau \\ &\quad +C_4\int_0^t\|\nabla \rho ^k\|_2^2\|(\rho ^k)^{-1}p'(\rho ^k)-(\rho ^{k-1})^{-1}p'(\rho ^{k-1})\|_2^2d\tau \\ &\leq C_5\int_0^t\left(\| \mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2+\epsilon_0 \| \mathbf{v}^k-\mathbf{v}^{k-1}\|_2^2+\epsilon_0 \|\rho ^k-\rho ^{k-1}\|_2^2\right) d\tau \end{aligned} with $C_4=\hat{C}_4(L_1,L_2,T)$ and $C_5=\hat{C}_5(L_1,L_2,T)$, and where we substituted the estimate for $\|\nabla (\rho ^{k+1}-\rho ^k)\|_1^2$ from \eqref{e3.14}. Here, we used the estimates $\|D\mathbf{v}^k\|_2^2\leq \|D\mathbf{v} ^k\|_{s-1,T}^2\leq \epsilon_0$, and $\|\nabla \rho^k\|_2^2\leq \|\nabla \rho ^k\|_{s+1,T}^2\leq \epsilon_0$ from Proposition \ref{prop3.1}. Next, we apply Gronwall's inequality to \eqref{e3.15}, which yields \label{e3.16} \begin{aligned} \|\mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2 &\leq C_6\int_0^t(\epsilon_0\|\mathbf{v}^k-\mathbf{v }^{k-1}\|_2^2+\epsilon_0\|\rho ^k-\rho ^{k-1}\| _2^2)d\tau \\ &\leq C_7\Big( \epsilon_0 \| \mathbf{v}^k-\mathbf{v}^{k-1}\|_{2,T}^2+\epsilon_0 \| \rho ^k-\rho ^{k-1}\|_{2,T}^2\Big) \end{aligned} where $C_6=\hat{C}_6(L_1,L_2,T)$, $C_7=\hat{C}_7(L_1,L_2,T)$. Substituting \eqref{e3.16} into the right-hand side of \eqref{e3.14} yields \begin{align*} \|\rho ^{k+1}-\rho ^k\|_2^2 &\leq C_3 C_7\left(\epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\| _{2,T}^2+\epsilon_0 \| \rho ^k-\rho ^{k-1}\|_{2,T}^2 \right) \\ &\quad +C_3\left(\epsilon_0 \| \mathbf{v}^k-\mathbf{v}^{k-1}\|_2^2+\epsilon_0 \|\rho ^k-\rho ^{k-1}\|_2^2\right) \end{align*} Adding the above estimate for $\|\rho ^{k+1}-\rho ^k\| _2^2$ to the estimate \eqref{e3.16} yields \begin{align*} \|\rho ^{k+1}-\rho ^k\|_2^2+\| \mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2 &\leq C_3 C_7\left( \epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\| _{2,T}^2+\epsilon_0 \| \rho ^k-\rho ^{k-1}\|_{2,T}^2 \right) \\ &\quad +C_3\left(\epsilon_0 \| \mathbf{v}^k-\mathbf{v}^{k-1}\|_2^2+\epsilon_0 \|\rho ^k-\rho ^{k-1}\|_2^2\right) \\ &\quad + C_7( \epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\| _{2,T}^2+\epsilon_0 \|\rho ^k-\rho ^{k-1}\| _{2,T}^2) \end{align*} Finally, after choosing $\epsilon_0$ sufficiently small, we get $$\label{e3.17} \|\rho ^{k+1}-\rho ^{k}\|_{2,T}^2+\|\mathbf{v} ^{k+1}- \mathbf{v} ^{k}\|_{2,T}^2 \leq \zeta \left(\| \rho^k-\rho^{k-1}\|_{2,T}^2 + \|\mathbf{v}^{k}-\mathbf{v} ^{k-1}\|_{2,T}^2 \right)$$ where $0<\zeta<1$. This completes the proof of the first part of Proposition \ref{prop3.2}. It only remains to prove that $|\rho^{k+1} - \rho^0|_{L^{\infty}} \leq \delta$, so that $\rho^{k+1} \in G_1$. \smallskip \noindent\textbf{Proof that $|\rho ^{k+1} -\rho ^{0}|_{L^{\infty}}\leq \delta$:} Repeated application of the estimate \eqref{e3.17} yields $$\label{e3.18} \|\mathbf{v}^{k+1}-\mathbf{v}^k\|_{2,T}^2+| |\rho ^{k+1}-\rho ^k\|_{2,T}^2\leq \zeta ^k\left( \|\mathbf{v}^1-\mathbf{v}^0\| _{2,T}^2+\|\rho ^1-\rho ^0\|_{2,T}^2\right)$$ where $0<\zeta <1$. Next, we estimate \label{e3.19} \begin{aligned} |\rho ^{k+1}-\rho ^0|^2 &\leq C\sum_{j=0}^k| \rho ^{j+1}-\rho ^j|_{L^\infty }^2\\ &\leq C\sum_{j=0}^k\| \rho ^{j+1}-\rho ^j\|_{s_0}^2 \leq C\sum_{j=0}^k\|\rho^{j+1}-\rho^j\|_{2,T}^2 \\ &\leq C\sum_{j=0}^k{\zeta }^j\left( \|\rho ^1-\rho ^0\|_{2,T}^2+\|\mathbf{v}^1-\mathbf{v}^0\| _{2,T}^2\right) \end{aligned} where we used \eqref{e3.18} in the last step. We also used the triangle inequality and Cauchy's inequality and Sobolev's inequality $|h|_{L^{\infty}} \leq C\|h\|_{s_0}$. Here $s_0=[\frac N2]+1=2$ for $N=2,3$. Thus, we have $|\rho ^{k+1}-\rho ^0|\leq \delta$, provided that $\|\rho ^1-\rho ^0\|_{2,T}$ and $\| \mathbf{v}^1-\mathbf{v}^0\|_{2,T}$ are sufficiently small. We now proceed to show that $\|\rho ^1-\rho ^0\|_{2,T}$ and $\|\mathbf{v}^1-\mathbf{v}^0\|_{2,T}$ can be made sufficiently small. From equations \eqref{e3.1}, \eqref{e3.2}, we obtain the following equations for $\mathbf{v}^1-\mathbf{v}^0$ and $\rho ^1-\rho ^0$, $$\label{e3.20} \begin{gathered} \frac{D^0(\mathbf{v}^1-\mathbf{v}^0)}{Dt}+(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla (\rho ^1-\rho ^0) \\ = c\nabla \Delta (\rho ^1-\rho ^0) -\mathbf{v}^0\cdot \nabla \mathbf{v}^0 -(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0+c\nabla \Delta \rho^0, \\ \nabla \cdot (\mathbf{v}^1-\mathbf{v}^0) = 0 , \end{gathered}$$ where $D^0/{Dt}=\partial /{\partial t}+\mathbf{v}^0\cdot \nabla$, and where $(\mathbf{v}^1-\mathbf{v}^0)(\mathbf{x},0)=0$. Here, we used the fact that $\mathbf{v}^0(\mathbf{x},t)=\mathbf{v}_0(\mathbf{x})$. Now we obtain estimates for $\rho ^1-\rho ^0$ and $\mathbf{v}^1-\mathbf{v}^0$. \smallskip \textbf{Estimate for $\rho ^1-\rho ^0$:} Applying the divergence operator to \eqref{e3.20}, we obtain \label{e3.21} \begin{aligned} &c\Delta ^2(\rho ^1-\rho ^0)-\nabla \cdot ((\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla (\rho ^1-\rho ^0))\\ &=\nabla \cdot (\mathbf{v}^0\cdot \nabla ( \mathbf{v}^1-\mathbf{v}^0))+\nabla \cdot (\mathbf{v}^0\cdot \nabla \mathbf{v} ^0) \\ &\quad +\nabla \cdot ((\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0)-c \Delta^2 \rho^0 \end{aligned} Applying Lemma \ref{lem2.7} to \eqref{e3.21} with $r=2$, we obtain the estimate \label{e3.22} \begin{aligned} \|\Delta (\rho ^1-\rho ^0)\|_2^2+\|\nabla (\rho ^1-\rho ^0)\|_2^2 &\leq C_{8}( \|D\mathbf{v}^0\| _3^2\|D(\mathbf{v}^1-\mathbf{v}^0)\|_1^2+\| \mathbf{v}^0\cdot \nabla \mathbf{v}^0\|_1^2 \\ &\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0\|_1^2+\|\nabla \Delta \rho^0\|_1^2) \end{aligned} where $C_{8}=\hat{C}_{8}(L_1)$. Next, we apply Lemma \ref{lem2.4} and obtain the $L^2$ estimate \label{e3.23} \begin{aligned} \|\rho ^1-\rho ^0\|_0^2 &\leq C\|\nabla (\rho ^1-\rho ^0)\|_2^2 \\ &\leq C_{9}( \|D\mathbf{v}^0\|_3^2\|D( \mathbf{v}^1- \mathbf{v}^0)\|_1^2+\|\mathbf{v}^0\cdot \nabla \mathbf{v} ^0\|_1^2 \\ &\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0\|_1^2+\|\nabla \Delta \rho^0\|_1^2) \end{aligned} where $C_{9}=\hat{C}_{9}(L_1)$. And we used estimate \eqref{e3.22} for $\|\nabla (\rho ^1-\rho ^0)\|_2^2$. Adding the estimates \eqref{e3.22}, \eqref{e3.23} yields the following estimate for $\rho ^1-\rho ^0$: \label{e3.24} \begin{aligned} \|\rho ^1-\rho ^0\|_2^2 & \leq C\left( \|\rho ^1-\rho ^0\|_0^2+\| \nabla (\rho ^1-\rho ^0)\|_1^2\right) \\ &\leq C_{10}( \|D\mathbf{v}^0\|_3^2\| D(\mathbf{v}^1-\mathbf{v}^0)\|_1^2+\| \mathbf{v}^0\cdot \nabla \mathbf{v}^0\|_2^2\\ &\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0\|_2^2 +\|\nabla \Delta \rho^0\|_1^2) \\ &\leq C_{11}( \|D\mathbf{v}^0\|_3^2\| \mathbf{v}^1-\mathbf{v}^0\|_2^2 +\|\mathbf{v}^0\|_2^2\|D\mathbf{v}^0\|_2^2 \\ &\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\|_2^2\|\nabla \rho ^0\|_2^2+\|\nabla \rho^0\|_3^2) \end{aligned} where $C_{10}=\hat{C}_{10}(L_1)$, $C_{11}=\hat{C}_{11}(L_1)$. Here we used the Sobolev calculus inequality $\|f g \|_r\leq C\|f\|_r\|g\|_r$ for $r=2>\frac N2$. \smallskip \textbf{Estimate for $\mathbf{v}^1\mathbf{-v}^0$:} After applying Lemma \ref{lem2.3} to \eqref{e3.20}, using $r=2$, we obtain \label{e3.25} \begin{aligned} \|\mathbf{v}^1-\mathbf{v}^0\|_2^2 & \leq C_{12}\int_0^t\left( \|\mathbf{v}^0\cdot \nabla \mathbf{v}^0\|_2^2+\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0\|_2^2\right) d\tau\\ &\quad +C_{12}\int_0^t\left(\|\nabla \Delta \rho^0\|_2^2 +\|\nabla (\rho ^1-\rho ^0)\|_1^2\right) d\tau \\ &\leq C_{13}\int_0^t \left(\|\mathbf{v}^0\|_2^2\| D\mathbf{v}^0\|_2^2 +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\|_2^2\|\nabla \rho ^0\| _2^2\right) d\tau \\ &\quad +C_{13}\int_0^t\left(\|\nabla \rho^0\|_4^2+\| D\mathbf{v}^0\|_3^2\| \mathbf{v}^1-\mathbf{v}^0\|_2^2 \right) d\tau \end{aligned} where $C_{12}=\hat{C}_{12}(L_1,L_2)$, $C_{13}=\hat{C}_{13}(L_1,L_2)$, and where we substituted the estimate for $\|\nabla (\rho ^1-\rho ^0)\|_1^2$ from \eqref{e3.24}. Next, we apply Gronwall's inequality to \eqref{e3.25}, which yields \label{e3.26} \begin{aligned} \|\mathbf{v}^1-\mathbf{v}^0\|_2^2 &\leq C_{14}\int_0^t \left(\|\mathbf{v}^0\|_2^2\|D\mathbf{v}^0\|_2^2 +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\|_2^2\|\nabla \rho ^0\| _2^2+\|\nabla \rho^0\|_4^2\right) d\tau \\ &\leq C_{15}\left( \|D \mathbf{v}_0\|_{s-1}^2+\| \nabla \rho ^0\|_{4,T}^2\right) \leq 2 C_{15}\| D\mathbf{v}_0\|_{s-1}^2 \end{aligned} where we used the fact that $\mathbf{v}^0(\mathbf{x},t)=\mathbf{v}_0(\mathbf{x})$, and we used the fact that $\|\nabla \rho ^0\|_{4,T}^2\leq \|\nabla \rho ^0\|_{s+1,T}^2\leq \|D\mathbf{v}_0\|_{s-1}^2$, by hypothesis. Here $C_{14}=\hat{C}_{14}(L_1,L_2,T)$ and $C_{15}=\hat{C}_{15}(L_1,L_2,T)$. Finally, we substitute the above estimate \eqref{e3.26} into the right-hand side of the estimate \eqref{e3.24} for $\|\rho ^1-\rho ^0\|_2^2$, which yields $$\label{e3.27} \|\rho ^1-\rho ^0\|_2^2\leq C_{16}\|D\mathbf{v }_0\|_{s-1}^2$$ where $C_{16}=\hat{C}_{16}(L_1,L_2,T)$ and where we used the fact that $\|\nabla \rho ^0\|_2^2 \leq \|\nabla \rho ^0\|_3^2\leq \|\nabla \rho ^0\|_{s+1,T}^2\leq \|D\mathbf{v}_0\|_{s-1}^2$, by hypothesis. From \eqref{e3.19}, \eqref{e3.26}, and \eqref{e3.27}, we see that $|\rho ^{k+1}-\rho ^0|_{L^\infty }\leq \delta$, for $\|D \mathbf{v}_0\|_{s-1}$ sufficiently small. And so $\rho ^{k+1}\in G_1$. It thus follows from the proof by induction on $k$ that $\rho ^k\in G_1$ for all $k$. This completes the proof of Proposition \ref{prop3.2}. \end{proof} Using Propositions \ref{prop3.1} and \ref{prop3.2}, we now complete the proof of Theorem \ref{thm3.1}. From \eqref{e3.18}, it follows that $\sum_{k=1}^\infty \Big(\| \mathbf{v}^{k+1}-\mathbf{v}^k\|_{2,T}^2 +\|\rho ^{k+1}-\rho ^k\|_{2,T}^2 \Big) < \infty .$ Hence, $\|\mathbf{v}^{k+1}-\mathbf{v}^k\| _{2,T}^2\to 0$ and $\|\rho ^{k+1} -\rho ^{k}\|_{2,T}^2\to 0$ as $k\to \infty$. Therefore, we conclude that there exist $\mathbf{v}\in C([0,T],H^2(\Omega ))$ and $\rho \in C([0,T],H^{2}(\Omega ))$ so that $\|\mathbf{v}^k-\mathbf{v} \|_{2,T}\to 0$, and $\|\rho ^k-\rho \|_{2,T}\to 0$, as $k\to \infty$. Using the interpolation inequalities $\|f \|_{r'+2} \leq C\|f \|_2^\alpha \|f\|_{r+2}^{1-\alpha }$, with $\alpha =(r-r')/r$, and $\|g \|_{r'} \leq C\|g \|_2^\beta \|g \|_{r}^{1-\beta }$, with $\beta =(r-r')/(r-2)$, with $r'\frac N2+3$, Sobolev's lemma implies that $\mathbf{v}^k\to \mathbf{v}$ in $C([0,T],C^3)$ and $\rho ^k\to \rho$ in $C([0,T],C^5)$. From the linear system of equations \eqref{e3.1}, \eqref{e3.2} it follows that $\| \mathbf{v}_t^k-\mathbf{v}_t \|_{s'-1,T}\to 0$, as $k\to \infty$, so that $\mathbf{v}_t^k\to \mathbf{v}_t$ in $C([0,T],C^2)$, and $\rho$, $\mathbf{v}$, is a classical solution of the system of equations \eqref{e1.2}, \eqref{e1.3}. The additional facts that $\mathbf{v}\in L^\infty([0,T],H^s)$, and $\rho \in L^\infty([0,T],H^{s+2})$ can be deduced using boundedness in high norm and a standard compactness argument (see, for example, \cite{e1,m1}). To prove uniqueness, let $\rho^1$, $\mathbf{v}^1$, and let $\rho^2$, $\mathbf{v}^2$, be any solutions of \eqref{e1.2}, \eqref{e1.3} having the regularity of solutions from Theorem \ref{thm3.1}, and satisfying $\rho ^{1}(\mathbf{x}_0,t)=\rho ^2(\mathbf{x}_0,t)=\rho ^0(\mathbf{x}_0,t)$ at a single fixed point $\mathbf{x}_0\in \Omega$. Here, $\rho^0$ is the initial iterate from Theorem \ref{thm3.1}. And we assume that $\|D\mathbf{v}_0\|_{s-1}$ is sufficiently small. We next show that $\rho^1=\rho^2$ and $\mathbf{v}^1=\mathbf{v}^2$. Let $\rho = \lim_{k\to \infty} \rho^{k+1}$, $\mathbf{v} = \lim_{k\to \infty} \mathbf{v}^{k+1}$, where $\{\rho^{k+1}$, $\mathbf{v}^{k+1}\}$, $k\geq 0$, is a sequence of solutions to the linear equations \eqref{e3.1}, \eqref{e3.2}, so that $\rho$, $\mathbf{v}$ is a solution of \eqref{e1.2}, \eqref{e1.3} from Theorem \ref{thm3.1}. We next show that $\rho^1=\rho$ and $\mathbf{v}^1=\mathbf{v}$. It then follows that the solution to \eqref{e1.2}, \eqref{e1.3} is unique, because repeating the proof given below will also show that $\rho^2=\rho$ and $\mathbf{v}^2=\mathbf{v}$, and so we will obtain \begin{align*} &\|\rho^2-\rho^1\|_{2,T} +\|\mathbf{v}^2-\mathbf{v}^1\|_{2,T} \\ &\leq \|\rho^2-\rho\|_{2,T}+ \|\rho^1-\rho\|_{2,T} +\|\mathbf{v}^2-\mathbf{v}\|_{2,T}+\|\mathbf{v}^1-\mathbf{v}\|_{2,T} = 0 \end{align*} Therefore, we now show that $\rho^1=\rho$ and $\mathbf{v}^1=\mathbf{v}$. Subtracting the equations \eqref{e1.2}, \eqref{e1.3} for $\rho^{1}$ and $\mathbf{v}^{1}$ from the equations \eqref{e1.2}, \eqref{e1.3} for $\rho$ and $\mathbf{v}$ yields equations which we write in the form \begin{gather*} \begin{aligned} &\frac{D^1(\mathbf{v}-\mathbf{v}^{1})}{Dt}+ (\rho^1)^{-1} \hat{p}' (\rho^1)\nabla (\rho -\rho ^{1}) \\ &= c\nabla \Delta (\rho -\rho ^{1}) -(\mathbf{v}-\mathbf{v}^{1})\cdot \nabla \mathbf{v} -((\rho)^{-1} \hat{p}' (\rho)-(\rho^{1})^{-1} \hat{p}' (\rho^{1}))\nabla \rho, \end{aligned} \\ \nabla \cdot (\mathbf{v}-\mathbf{v}^{1}) = 0, \end{gather*} where $D^1/{Dt}= \partial/{\partial t}+\mathbf{v}^1 \cdot\nabla$, where $(\mathbf{v}-\mathbf{v}^{1})(\mathbf{x},0)=0$. Repeating the proof of Proposition \ref{prop3.2} yields the estimate $\|\rho -\rho ^{1}\|_{2,T}^2+\|\mathbf{v} -\mathbf{v} ^{1}\|_{2,T}^2 \leq \zeta \left(\| \rho-\rho^{1}\|_{2,T}^2 + \| \mathbf{v}-\mathbf{v}^{1}\|_{2,T}^2 \right)$ where $0<\zeta<1$. Note that in repeating the proof of the estimates \eqref{e3.14}, \eqref{e3.15} in the proof of Proposition \ref{prop3.2}, we use the fact that $\|D\mathbf{v}\|_{s-1,T}^2\leq \epsilon_0$ and $\|\nabla \rho\|_{s+1,T}^2\leq \epsilon_0$, which holds since $\rho$, $\mathbf{v}$ is a solution from Theorem \ref{thm3.1}. Finally, after moving the right-hand side of the above inequality to the left-hand side, we obtain $\|\rho -\rho ^{1}\|_{2,T}^2+\|\mathbf{v}-\mathbf{v} ^{1}\|_{2,T}^2=0$ and so the solution is unique. \end{proof} \subsection*{A Final Remark} We sketch a proof that $\partial \rho/\partial t$ is small. Then, since we already know that $\nabla \rho$ is small, it follows that the conservation of mass equation is approximately satisfied. Therefore, the model equations \eqref{e1.2}, \eqref{e1.3} might be useful as an approximation in the case of almost constant density, and nearly incompressible fluid flow with a small velocity gradient. Proceeding formally, we differentiate equation \eqref{e1.3} with respect to $t$, obtaining $\frac{\partial ^2 \mathbf{v}}{\partial t ^2}+\mathbf{v}\cdot \nabla \mathbf{v}_t+\mathbf{v}_t\cdot \nabla \mathbf{v} + \rho^{-1} \hat{p}' (\rho)\nabla \rho_t +\frac{\partial}{\partial \rho}(\rho^{-1} \hat{p}' (\rho))\rho_t\nabla \rho = c\nabla \Delta \rho_t$ Applying the divergence operator to this equation yields $c\Delta^2 \rho_t -\nabla \cdot ((\rho)^{-1} \hat{p}' (\rho)\nabla \rho_t)=\nabla \cdot (\frac{\partial}{\partial \rho}(\rho^{-1} \hat{p}' (\rho))\rho_t\nabla \rho )+ \nabla \cdot (\mathbf{v}\cdot \nabla \mathbf{v}_t)+\nabla \cdot (\mathbf{v}_t\cdot \nabla \mathbf{v})\quad$ where we used the fact that $\nabla \cdot \mathbf{v}=0$, and where $c$ is a positive constant. Applying Lemma \ref{lem2.7} we obtain from the above equation the following estimate for $\nabla \rho_t$, \label{e3.28} \begin{aligned} \|\nabla \rho_t \|_{3}^2 &\leq C(\|\Delta \rho_t \|_2^2+\|\nabla \rho_t \|_2^2) \\ &\leq C[1+\sum_{j=1}^{2}\|D((\rho)^{-1} \hat{p}' (\rho))\|_{2}^{2j}](\|D \mathbf{v}\| _{3}^2\|D\mathbf{v}_t\|_{1}^2+\|\mathbf{v}_t \cdot \nabla\mathbf{v}\|_{1}^2 \\ &\quad + \|\frac{\partial}{\partial \rho}(\rho^{-1} \hat{p}' (\rho))\rho_t\nabla \rho\|_{1}^2) \\ &\leq C_{17}(\|D\mathbf{v}\|_{3}^2\|\mathbf{v}_t\| _{2}^2+\|\rho_t\|_2^2\|\nabla \rho\|_{2}^2) \end{aligned} where $C_{17}=\hat{C}_{17}(L_1,T)$. Next, by Lemma \ref{lem2.4} we have the $L^2$ estimate \label{e3.29} \begin{aligned} \|\rho_t\|_{0}^2 &\leq C_0\|\rho_{t}^{0}\|_0^2+C_0\|\nabla\rho_{t}^{0}\|_2^2 +C_0\|\nabla \rho_t\|_2^2\\ &\leq C_{18}(\|\rho_{t}^{0}\|_0^2+\|\nabla\rho_{t}^{0}\|_2^2 +\|D\mathbf{v}\|_{3}^2\|\mathbf{v}_t\| _{2}^2+\|\rho_t\|_2^2\|\nabla \rho\| _{2}^2)\end{aligned} where $C_{18}=\hat{C}_{18}(L_1,T)$ and where we used the estimate for $\|\Delta \rho_t \|_2^2+\|\nabla \rho_t\|_2^2$ from \eqref{e3.28}. Here, $\rho^0$ is the initial iterate as defined in Theorem \ref{thm3.1}. Note that by definition of $\rho^0$, we have $\rho_t(\mathbf{x}_0,t)=\rho_t^0(\mathbf{x}_0,t)$ at the single fixed point $\mathbf{x}_0 \in \Omega$. Also, we now specify that $\|\rho_{t}^{0}\|_{0,T}\leq \|D\mathbf{v}_0\|_{s-1}$ and $\|\nabla\rho_{t}^{0}\|_{2,T} \leq \|D\mathbf{v}_0\|_{s-1}$, as an additional hypothesis for $\rho^0$. Recall that $\epsilon_0=C_6\|D\mathbf{v}_0\|_{s-1}^2$ where $C_6=\hat{C}_6(L_1,T)$, from Proposition \ref{prop3.1}. Then adding the estimates \eqref{e3.28}, \eqref{e3.29} yields the following estimate for $\rho_t$: \begin{align*} \|\rho_t \|_{3}^2 &\leq C(\|\rho_{t}\|_0^2+\|\nabla \rho_t\| _{2}^2) \leq C_{19} (\|\rho_{t}^{0}\|_0^2+\|\nabla\rho_{t}^{0}\|_2^2 +\|D\mathbf{v}\|_{3}^2\|\mathbf{v}_t\|_{2}^2+\|\rho_t\|_2^2\| \nabla \rho\|_{2}^2) \\ &\leq C_{20} (\epsilon_0+\epsilon_0\|\mathbf{v}_t\|_{2}^2 +\epsilon_0 \|\rho_t\|_3^2) \end{align*} where $C_{19}=\hat{C}_{19}(L_1,T)$, $C_{20}=\hat{C}_{20}(L_1,T)$ and where we used the facts that $\|D\mathbf{v}\| _{3}^2 \leq \epsilon_0$, and $\|\nabla \rho\|_{2}^2 \leq \epsilon_0$ from Proposition \ref{prop3.1}. After moving the term $\epsilon_0 \|\rho_t\|_3^2$ to the left-hand side, and substituting the estimate for $\mathbf{v}_t$ from Proposition \ref{prop3.1}, we then obtain the estimate $\|\rho_t \|_{3}^2 \leq \epsilon_0 C_{21}$ where $C_{21}=\hat{C}_{21}(L_1,L_3,T)$. Therefore, $\rho_t$ is small, since $\epsilon_0$ will be small if $\|D\mathbf{v}_0\|_{s-1}$ is sufficiently small. \section{Existence for the linear case} In this section, we sketch the proof of the existence of a solution to the linear equations \eqref{e3.1}, \eqref{e3.2}. Before going into the existence proof, which appears in Lemma \ref{lemA.5}, we first collect some preliminary technical facts. We recall that every vector field $\mathbf{u}\in L^2(\mathbb{T}^N)$ admits a unique orthogonal decomposition in terms of a solenoidal vector field $\mathbf{w}$ and a potential $\nabla \phi$, so that $\mathbf{u}=\mathbf{w}+\nabla \phi$, where $\mathbf{w}=P\mathbf{u}$ and $\nabla \phi =Q\mathbf{u.}$ Moreover, if $\mathbf{u}\in H^s(\mathbb{T}^N)$, with $s\geq 1$, then $\mathbf{w}$ and $\phi$ satisfy $\nabla \cdot \mathbf{w=}0$, $\Delta \phi =\nabla \cdot \mathbf{u.}$ The next lemma, due to Embid \cite{e1}, provides us with an estimate that will be useful for the existence proof. \begin{lemma}\label{lemA.1} If $\mathbf{u}$, $\mathbf{v}$ $\in H^r(\Omega )$, $r>\frac N2+3$, $\Omega =\mathbb{T}^N$, then $Q(\mathbf{v\cdot }\nabla P\mathbf{u})\in H^r(\Omega )$ and $\|Q(\mathbf{v\cdot }\nabla P\mathbf{u})\|_r\leq C\|\mathbf{v}\| _r\|\mathbf{u}\|_r$. \end{lemma} For a proof of this lemma, see Embid \cite{e1}. To prove existence of a solution to \eqref{e3.1}, \eqref{e3.2}, we need the following lemmas. \begin{lemma} \label{lemA.2} Given $a\in H^r(\Omega )$, $f\in H^r(\Omega )$, $r>\frac N2+3$, $\Omega =\mathbb{T}^N$, then $P(a\nabla f)\in H^r(\Omega )$, and $\|P(a\nabla f)\|_r\leq C\|Da \| _{r-1}\|\nabla f\|_{r-1}$, where $P$ is the projection onto the solenoidal vector field. \end{lemma} \begin{proof} First, from the orthogonality property of the projection $P$, we obtain an $L^2$ estimate as follows: $\|P(a\nabla f)\|_0=\|P(\bar{f} \nabla a)\|_0 \leq \|\bar{f} \nabla a\|_0 \leq C|Da| _{L^\infty }\|\bar{f}\|_0\leq C|Da| _{L^\infty }\|\nabla f\|_0.$ where $\bar{f}=f-\frac {1}{|\Omega|} \int_{\Omega}fdx$. Here, we used Poincar\'{e}'s inequality to estimate $\|\bar{f}\|_0\leq C\|\nabla f\|_0$. Next, from the orthogonality property of the projection $P$, and using the commutator estimate from Lemma \ref{lem2.2}, as well as using the triangle inequality, we obtain for $|\alpha|\geq 1$: \begin{align*} \|P(a\nabla f)_\alpha\|_0 &\leq \|P(a\nabla f_\alpha)\|_0 +\|P\big[ (a\nabla f)_\alpha -a\nabla f_\alpha \big]\|_0 \\ &= \|P(\bar{f}_\alpha \nabla a)\|_0+\|P[(a\nabla f)_\alpha -a\nabla f_\alpha ]\|_0\\ &\leq C|Da|_{L^\infty }\|\bar{f}\|_k +C\|Da\|_{k_1}\|\nabla f\|_{k-1} \\ &\leq C\|Da\|_{k_1}\|\nabla f\|_{k-1}, \end{align*} where $|\alpha|=k$, where $k_1=\max (k-1,s_0)$, and where $s_0=[ \frac N2]+1$. And we used the facts that $\|\bar{f}\|_k\leq C(\|\bar{f}\| _0+\|\nabla f\|_{k-1})$, and $\|\bar{f}\| _0\leq C\|\nabla f\|_0$, by Poincar\'{e}'s inequality. Finally, by adding the above estimates over all $0\leq | \alpha |\leq r$, we obtain the estimate $\|P(a\nabla f)\|_r \leq C\|Da\|_{r-1}\|\nabla f\|_{r-1}$. \end{proof} \begin{lemma}\label{lemA.3} Given $\mathbf{v}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^r)$, and $\mathbf{f}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$, with $r>\frac N2+3$, for $\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$, $0\leq t\leq T$, there is a unique, classical solution $\mathbf{u}\in C([0,T],C^3)\cap L^\infty ([0,T],H^r)$ of ${D\mathbf{u}}/{Dt} =\mathbf{f},$ with $\mathbf{u}(\mathbf{x},0) =\mathbf{u}_0(\mathbf{x})\in H^r$, where $D/{Dt}=\partial/ {\partial t}+\mathbf{v\cdot }\nabla$. Moreover, $\mathbf{u}$ satisfies the estimate $\| \mathbf{u}\|_r^2\leq e^{\alpha (t)}\left(\| \mathbf{u}_0\|_r^2+\int_0^tC\|\mathbf{f}\| _r^2d\tau \right)$, where $\alpha (t)=\int_0^tC(1+\| D\mathbf{v}\|_{r-1})d\tau$ and $C$ depends on $r$. \end{lemma} The proof of the above lemma is standard; see, for example, \cite{e1}. \begin{lemma}\label{lemA.4} Given $a$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$ and $\mathbf{f}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r-1})$, where $r>\frac N2+3$, $a(\mathbf{x,}t)\geq c_1$, with $c_1>0$ for $\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$, $0\leq t\leq T$, there is a classical solution $\rho \in C([0,T],H^0)\cap L^\infty ([0,T],H^{r+2})$ of $$\label{A.1} c\Delta^2 \rho-\nabla\cdot (a\nabla \rho)=\nabla \cdot \mathbf{f}$$ Here, $c$ is a positive constant. And $\rho$ satisfies the estimate $\|\nabla\rho\|_{r+1}^2 \leq C\left( \|\Delta \rho \|_r^2+\|\nabla\rho\|_r^2\right) \leq C\Big[1+\sum_{j=1}^{r}\|Da\| _{r-1}^{2j}\Big]\|\mathbf{f}\|_{r-1}^2 \leq C\|\mathbf{f}\|_{r-1}^2$ \end{lemma} \begin{proof} The operator in \eqref{A.1} is linear with $a(\mathbf{x,}t)\geq c_1$, where $c_1>0$ for $\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$, and $c$ is a positive constant, and the compatibility condition $\int_{\Omega}\nabla \cdot \mathbf{f}dx=0$ is satisfied. The existence of a solution $\rho$ follows from the standard theory for elliptic equations, specifically, the Lax-Milgram Lemma (see, for example, \cite{e2}). The a priori estimate follows from Lemma \ref{lem2.7}. \end{proof} Now we prove the existence of a solution to \eqref{e3.1}, \eqref{e3.2}. \begin{lemma}\label{lemA.5} Given $\mathbf{v}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^r)$ and $a$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$, and given $r>\frac N2+3$, $a(\mathbf{x},t)\geq c_1$, with $c_1>0$ for $\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$, $0\leq t\leq T$, there exists a classical solution $\rho$, $\mathbf{w}$ of \begin{gather} \frac{D\mathbf{w}}{Dt} =-a\nabla \rho+c\nabla \Delta \rho, \label{A.2}\\ \nabla \cdot \mathbf{w} = 0, \label{A.3} \\ \mathbf{w}(\mathbf{x},0) = \mathbf{w}_0(\mathbf{x}),\quad \nabla \cdot \mathbf{w}_0 = 0, \label{A.4} \end{gather} with $\mathbf{w}\in C([0,T],C^3)\cap L^\infty ([0,T],H^r)$, and $\rho \in C([0,T],C^5)\cap L^\infty ([0,T],H^{r+2})$. Here, $c$ is a positive constant, and $D/{Dt}=\partial/ {\partial t}+\mathbf{v\cdot }\nabla$, and $\nabla \cdot \mathbf{v}=0$. \end{lemma} \begin{proof} First, we reduce the equations to an equivalent system by employing the projections $P$ and $Q=I-P$, where $P$ is the orthogonal projection of $L^2$ onto the solenoidal vector field and $Q$ is the orthogonal projection of $L^2$ onto the gradient field. Applying the operator $P$ to \eqref{A.2}, and using the fact that $P\mathbf{w=w}$, we obtain the equation $$\label{A.5} \frac{D\mathbf{w}}{Dt}=J:= Q(\mathbf{v}\cdot \nabla P\mathbf{w})-P(a\nabla\rho).$$ Applying the operator $Q$ to \eqref{A.2}, we obtain the equation $$\label{A.6} Q(\mathbf{v}\cdot \nabla P\mathbf{w}+ a\nabla \rho -c\nabla \Delta \rho)=0.$$ This equation is equivalent to the equation $$\label{A.7} Q(Q(\mathbf{v}\cdot \nabla P\mathbf{w})+a\nabla \rho -c\nabla \Delta \rho)=0.$$ From the definition of $Q$, we observe that equation \eqref{A.7} is equivalent to $$\label{A.8} c\Delta^2 \rho -\nabla \cdot (a\nabla \rho) =\nabla \cdot Q(\mathbf{v}\cdot \nabla P \mathbf{w}).$$ With the given initial data \eqref{A.4}, the two systems of equations \eqref{A.2}, \eqref{A.3}, and \eqref{A.5}, \eqref{A.8} are equivalent. (The proof is standard; see, for example, \cite{e1}). Next, we construct the solution of the system \eqref{A.5}, \eqref{A.8} using the method of successive approximation as follows: Set $\mathbf{w}^0=$ $\mathbf{w}_0(\mathbf{x})$ and set $\rho^0$ to be the initial iterate from Theorem \ref{thm3.1}. For $k=0,1,2,\dots$ define $\mathbf{w}^{k+1}$, $\rho ^{k+1}$ as the solution of the equations \begin{gather} \frac{D\mathbf{w}^{k+1}}{Dt} =J^k, \label{A.9} \\ c\Delta \rho^{k+1}-\nabla \cdot (a\nabla \rho^{k+1}) = \nabla \cdot Q(\mathbf{v}\cdot \nabla (P\mathbf{w}^{k+1})), \label{A.10} \end{gather} where $J^k=Q(\mathbf{v}\cdot \nabla P\mathbf{w}^{k}) -P(a\nabla \rho ^{k})$. And $\mathbf{w}^{k+1}(\mathbf{x},0)=\mathbf{w}_0\in H^r$, with $\nabla \cdot \mathbf{w}_0=0$. The first step is to solve \eqref{A.9} for $\mathbf{w}^{k+1}$. By Lemma \ref{lemA.1} and by the induction hypothesis we have $Q(\mathbf{v}\cdot \nabla P\mathbf{w}^k)\in C([0,T],H^0)\cap L^\infty ([0,T],H^r)$. Furthermore, from Lemma \ref{lemA.2} and the induction hypothesis we have $P(a\nabla \rho ^{k})\in C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$. Therefore, the existence of a solution $\mathbf{w}^{k+1}\in C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$ to \eqref{A.9} follows from Lemma \ref{lemA.3}. The next step is, given the solution $\mathbf{w}^{k+1}$ just obtained, to solve \eqref{A.10} for $\rho^{k+1}$. From Lemma \ref{lemA.1} we have $Q(\mathbf{v}\cdot \nabla (P\mathbf{w}^{k+1}))\in C([0,T],H^0)\cap L^\infty ([0,T],H^r)$, and therefore it follows from Lemma \ref{lemA.4} that there is a solution $\rho^{k+1} \in C([0,T],H^0)\cap L^\infty ([0,T],H^{r+2})$ to \eqref{A.10}. And since $\rho^{k+1}$ is unique up to an arbitrary function of $t$, we specify that $\rho^{k+1}(\mathbf{x}_0,t)=\rho^k(\mathbf{x}_0,t)$ for all $k\geq 0$, at the single fixed point $\mathbf{x}_0 \in \Omega$, so $\rho^{k+1}(\mathbf{x}_0,t)=\rho^0(\mathbf{x}_0,t)$. Next, we derive estimates for $\|\rho^{k+1}-\rho^k \|_{r+2,T}$ and $\|\mathbf{w}^{k+1}\mathbf{-w}^k\|_{r,T}$. We see that $\rho^{k+1}-\rho ^k$ and $\mathbf{w}^{k+1}\mathbf{-w}^k$ solve the system of equations \begin{gather} \frac{D(\mathbf{w}^{k+1}-\mathbf{w}^k)}{Dt} =J^k-J^{k-1}, \label{A.11} \\ c\Delta^2(\rho^{k+1}-\rho^k) -\nabla \cdot (a\nabla(\rho ^{k+1}-\rho ^k)) = \nabla \cdot Q(\mathbf{v}\cdot \nabla(P(\mathbf{w}^{k+1}-\mathbf{w}^{k}))), \label{A.12} \end{gather} where $J^k-J^{k-1}=Q(\mathbf{v}\cdot \nabla P(\mathbf{w}^k-\mathbf{w} ^{k-1}))$ $-P(a\nabla(\rho^k-\rho^{k-1}))$. Initially we have $(\mathbf{w}^{k+1}-\mathbf{w}^k)(0)=0$. First, we derive the estimate for $\|\mathbf{w}^{k+1}-\mathbf{w} ^k\|_{r,T}$. From \eqref{A.11} and from Lemmas \ref{lemA.1}, \ref{lemA.2}, and \ref{lemA.3}, we obtain the estimate \label{A.13} \begin{aligned} \|\mathbf{w}^{k+1}-\mathbf{w}^k\|_r^2 &\leq Ce^{\alpha(t)}\int_0^t \|Q(\mathbf{v\cdot }\nabla P(\mathbf{w}^k -\mathbf{w}^{k-1}))\|_r^2+\|P(a\nabla (\rho ^{k}-\rho ^{k-1}))\|_{r}^2 d\tau \\ &\leq C_1\int_0^t \|\mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2 +\|\nabla(\rho ^{k}-\rho ^{k-1})\|_{r-1}^2 d\tau \end{aligned} Here $\alpha (t)=C\int_0^t(1+\|D\mathbf{v}\| _{r-1})d\tau$, $C$ depends on $r$, and $C_1$ depends on $r$, $\|\mathbf{v}\|_{r,T}$ and $\|a\| _{r,T}$. Next, from \eqref{A.12}, and from Lemmas \ref{lemA.1} and \ref{lemA.4}, we obtain the estimate \label{A.14} \begin{aligned} \|\nabla( \rho ^{k+1}-\rho ^k)\|_{r+1}^2 &\leq C[1+\sum_{j=1}^{r}\|Da\|_{r-1}^{2j}]\| Q(\mathbf{v}\cdot \nabla (P(\mathbf{w}^{k+1}-\mathbf{w}^{k}))) \|_{r-1}^2 \\ &\leq C_2 \|\mathbf{w}^{k+1}-\mathbf{w}^{k}\|_{r}^2 \\ &\leq C_3\int_0^t \left(\| \mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2+\|\nabla( \rho ^{k}-\rho ^{k-1})\|_{r-1}^2 \right)d\tau \end{aligned} where $C_2$ and $C_3$ depend on $r$, $\|a\|_{r,T}$, and $\|\mathbf{v}\|_{r,T}$. Here, we used the estimate \eqref{A.13} in the last step. From Lemma \ref{lem2.4} and \eqref{A.14}, we have the estimate \label{A.15} \begin{aligned} \|\rho ^{k+1}-\rho ^k\|_{0}^2 &\leq C\|\nabla(\rho^{k+1} -\rho^{k})\|_2^2 \\ &\leq C_4 \|\mathbf{w}^{k+1}-\mathbf{w}^{k}\|_{r}^2 \\ &\leq C_5 \int_0^t \left(\|\mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2 +\|\nabla( \rho^{k}-\rho ^{k-1})\|_{r-1}^2 \right)d\tau \end{aligned} where $C_4$ and $C_5$ depend on $r$, $\|a\|_{r,T}$, and $\|\mathbf{v}\|_{r,T}$, and where $\rho^{k+1}(\mathbf{x}_0,t)=\rho^k(\mathbf{x}_0,t)$ for all $k\geq 0$, at the single fixed point $\mathbf{x}_0 \in \Omega$. Adding the estimates \eqref{A.13}, \eqref{A.14}, \eqref{A.15}, we obtain $$\label{A.16} \|\mathbf{w}^{k+1}-\mathbf{w}^k\|_r^2+\|\rho ^{k+1}-\rho ^k\|_{r+2}^2 \leq C_6 \int_0^t \left(\| \mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2+\|\rho ^{k}-\rho ^{k-1}\|_{r+2}^2\right)d\tau$$ where $C_6$ depends on $r$, $\|\mathbf{v}\|_{r,T}$, $\|a\|_{r,T}$. Repeated application of \eqref{A.16} yields $\|\mathbf{w}^{k+1}-\mathbf{w}^k\|_{r,T}^2+\| \rho ^{k+1}-\rho ^k\|_{r+2,T}^2 \leq \frac{(C_6T)^k}{k!} \left(\|\mathbf{w}^1-\mathbf{w}^0\|_{r,T}^2+ \| \rho ^{1}-\rho ^{0}\|_{r+2,T}^2\right)$ from which it follows that $\sum_{k=1}^\infty \left(\| \mathbf{w}^{k+1}-\mathbf{w}^k\|_{r,T}^2+\|\rho ^{k+1}-\rho ^k\|_{r+2,T}^2 \right)<\infty .$ Hence, $\|\mathbf{w}^{k+1}-\mathbf{w}^k\| _{r,T}^2\to 0$ as $k\to \infty$ and $\|\mathbf{\rho }^{k+1} -\mathbf{\rho }^{k}\|_{r+2,T}^2\to 0$ as $k\to \infty$. Therefore, we deduce that there exist $\mathbf{w}$ $\in C([0,T],H^0)\cap L^\infty ([0,T],H^r)$ such that $\mathbf{w}^k\to \mathbf{w}$ as $k\to \infty$ strongly in $C([0,T],H^0)\cap$ $L^\infty ([0,T],H^r)$, and there exists $\rho \in C([0,T],H^0)\cap L^\infty ([0,T],H^{r+2})$ such that $\rho ^k\to \rho$ as $k\to \infty$ strongly in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r+2})$. The fact that $\mathbf{w}$ and $\rho$ is a solution to the system of equations \eqref{A.5}, \eqref{A.8} follows by a standard argument. And therefore $\mathbf{w}$, $\rho$ is a solution to the equivalent system \eqref{A.2}, \eqref{A.3}. \end{proof} \begin{thebibliography}{00} \bibitem{a1} D. M. Anderson, G. B. McFadden and A. A. Wheeler, \textit{Diffuse-interface methods in fluid mechanics}, Ann. Rev. Fluid Mech. 30 (1998), 139--65. \bibitem{b1} D. Bresch, B. Desjardins, and C.K. Lin, \textit{On some compressible fluid models: Korteweg, lubrication, and shallow water systems}, Comm. in Partial Differential Equations 28 (2003), nos. 3 and 4, 843--68. \bibitem{d1} D. L. Denny, \textit{Existence of solutions fluids with capillary effects}, Communications on Pure and Applied Analysis, 2, no. 4, (2003), 1-21., \bibitem{d2} J. E. Dunn and J. Serrin, \textit{On the thermomechanics of interstitial working}, Arch. Rational Mech. Anal. 88 (1985), 95--133. \bibitem{e1} P. Embid, \textit{On the Reactive and Non-diffusive Equations for Zero Mach Number Combustion}, Comm. in Partial Differential Equations (nos. 8 and 9) 14 (1989), 1249--1281. \bibitem{e2} L. Evans, \textit{Partial Differential Equations}, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, Rhode Island, 1998. \bibitem{h1} H. Hattori and D. Li, \textit{Global solutions of a high dimensional system for Korteweg materials}, J. Math. Anal. Appl. 198 (1996) 84--97. \bibitem{h2} H. Hattori and D. Li, \textit{The existence of global solutions to a fluid dynamic model for materials for Korteweg type}, J. Partial Diff. Eqs. 9 (1996) 323--342. \bibitem{k1} S. Klainerman and A. Majda, \textit{Singular Limits of Quasilinear Hyperbolic Systems with Large Parameters and the Incompressible Limit of Compressible Fluids}, Comm. Pure Appl. Math. 34 (1981), 481--524. \bibitem{m1} A. Majda, \textit{Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables}, Springer-Verlag: New York, 1984. \bibitem{m2} J. Moser, \textit{A Rapidly Convergent Iteration Method and Non-linear Differential Equations}, Ann. Scuola Norm. Sup., Pisa 20 (1966), 265--315. \end{thebibliography} \end{document}