\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. {\em Electronic Journal of Differential Equations}, Conf. 19 (2010), pp. 45--64.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{45} \title[\hfilneg EJDE-2010/Conf/19/\hfil Compressible fluid flow] {Solution to a system of equations modelling compressible fluid flow with capillary stress effects} \author[D. L. Denny\hfil EJDE/Conf/19 \hfilneg] {Diane L. Denny} \address{Diane L. Denny \newline Department of Mathematics and Statistics, Texas A\&M University - Corpus Christi \\ Corpus Christi, TX 78412, USA} \email{diane.denny@tamucc.edu} \thanks{Published September 25, 2010.} \subjclass[2000]{35A05} \keywords{Existence; capillary; compressible fluid} \begin{abstract} We study the initial-value problem for a system of nonlinear equations that models the flow of a compressible fluid with capillary stress effects. The system includes hyperbolic equations for the density and for the velocity, and an algebraic equation (the equation of state) for the pressure. We prove the existence of a unique classical solution to an initial-value problem for this system of equations under periodic boundary conditions. The key to the proof is an a priori estimate for the density and velocity in a high Sobolev norm. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} We begin by considering a system of equations which arises from a model of the multi-dimensional flow of a compressible fluid with capillary stresses. When viscosity is neglected, the model consists of the following equations: \begin{gather*} \frac{D\rho }{Dt} =-\rho \nabla \cdot \mathbf{v} \\ \frac{D\mathbf{v}}{Dt} =-\rho ^{-1}\nabla p+c\nabla \Delta \rho \end{gather*} where $\rho$ is the density, $p$ is the pressure, and $\mathbf{v}$ is the velocity. Here $c$ is a coefficient of capillarity which is a small, positive constant. The material derivative $D/{Dt}=\partial/{\partial t}+\mathbf{v}\cdot\nabla$. The term $c\nabla \Delta \rho$ is due to capillary stresses, from the theory of Korteweg-type materials described by Dunn and Serrin \cite{DS}. The fluid's thermodynamic state is determined by the density $\rho$, and the pressure $p$ is then determined from the density by an equation of state $p=\hat{p}(\rho)$. A derivation of the model's equations appears in \cite{DP1}. Anderson, McFadden and Wheeler \cite{AMW} have reviewed related theories, as well as applications to diffuse-interface modelling. Other researchers have proven the existence of solutions to other versions of this model which include viscosity and an evolution equation for temperature (see, e.g., \cite{BDL, HL1, HL2, HL3}). To our knowledge, this system of equations for inviscid fluid flow with capillary stresses has not been previously studied. With the change of variables $\mathbf{u}=\rho \mathbf{v},$ the system of equations becomes \begin{gather} \frac{\partial \rho }{\partial t} =- \nabla \cdot \mathbf{u} \label{e1.1} \\ \begin{aligned} \frac{\partial\mathbf{u}}{\partial t} &=-\rho^{-1}\mathbf{u}\cdot\nabla \mathbf{u}+\rho^{-2}(\mathbf{u}\cdot\nabla \rho)\mathbf{u} \\ &\quad -\rho^{-1}(\nabla \cdot \mathbf{u})\mathbf{u}-\nabla p+c\rho\nabla \Delta \rho \end{aligned}\label{e1.2} \end{gather} Let $\bar{\rho}=\rho-|\Omega|^{-1}\int_{\Omega} \rho d \mathbf{x}$. We assume that $\bar{\rho}$ is small. Since the capillary coefficient $c$ is very small, we assume that $c\bar{\rho}$ is neglibly small, and we will approximate the capillary stress term as follows: $c\rho\nabla \Delta \rho = c\Big(\bar{\rho}+|\Omega|^{-1}\int_{\Omega} \rho d \mathbf{x} \Big)\nabla \Delta \rho \approx c \Big(|\Omega|^{-1}\int_{\Omega} \rho d \mathbf{x} \Big)\nabla \Delta \rho$ Then using the equation of state for the pressure, we make the following approximation to equation \eqref{e1.2}: \begin{eqnarray} \frac{\partial\mathbf{u}}{\partial t} &= -\rho^{-1}\mathbf{u}\cdot\nabla \mathbf{u}+\rho^{-2}(\mathbf{u}\cdot\nabla \rho)\mathbf{u} -\rho^{-1}(\nabla \cdot \mathbf{u})\mathbf{u}- p' (\rho)\nabla \rho \nonumber\\ &\quad +c\left(|\Omega|^{-1}\int_{\Omega} \rho d \mathbf{x}\right)\nabla \Delta \rho \label{e1.3} \end{eqnarray} The purpose of this paper is to prove the existence of a unique classical solution $\mathbf{u}$, $\rho$ to the initial-value problem for equations \eqref{e1.1}, \eqref{e1.3}, for $0\leq t \leq T$, using periodic boundary conditions. Hence, we choose for our domain the N-dimensional torus $\mathbb{T}^N$, where $N=2$ or $N=3$. We will show that a unique solution exists, provided that $T\|D \mathbf{u}_0\|_s$ and $T\| \nabla \rho_0\|_{s+1}$ are sufficiently small, where $\mathbf{u}_0$, $\rho_0$ is the given initial data. \section{Existence theorem} In this section, we prove the existence of a unique classical solution to the initial-value problem for equations \eqref{e1.1}, \eqref{e1.3} with periodic boundary conditions. We will be using 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 $% \| f\| _s^2=\sum_{| \alpha | \leq s} \int_\Omega | D^\alpha f| ^2d \mathbf{x}$. We use the standard multi-index notation. We will be using the standard function spaces $L^\infty ([0,T],H^s(\Omega))$ and $C([0,T],H^s(\Omega))$. $L^\infty ([0,T],H^s(\Omega))$ is the space of bounded measurable functions from $[0,T]$ into $H^s(\Omega)$, with the norm $\| f\| _{s,T}^2=\operatorname{ess\, sup}_{0\leq t\leq T} \| f(t)\| _s^2$. The set $C([0,T],H^s(\Omega))$ is the space of continuous functions from $[0,T]$ into $H^s(\Omega )$. We will also be using the notation $| f|_{L^{\infty},T}=\operatorname{ess\,sup}_{0\leq t\leq T}$ $| f(t)|_{L^{\infty}(\Omega)}$. \begin{theorem} \label{T3.1} Let $\rho_0(\mathbf{x})=\rho(\mathbf{x},0)\in H^{s+2}(\Omega )$, $\mathbf{u}_0(\mathbf{x})=\mathbf{u}(\mathbf{x},0)\in H^{s+1}(\Omega )$ be the given initial data, with $s>$ $\frac N2+1$, and $\Omega =\mathbb{T}^N$, with $N=2$ or $N=3$. Let max $\{| \rho _0|_{L^\infty},| \mathbf{u}_0|_{L^\infty}\}\leq L_0$, for some positive constant $L_0$. Let $p=\hat{p}(\rho)$ be a given equation of state for the pressure $p$ as a function of $\rho$. We assume that $p$ is a sufficiently smooth function of $\rho$ for any $\rho \in G$, where $G\subset \mathbf{R}$ is an open set. We assume that in $G$, $\rho$ is positive and $p'(\rho)$ is positive. We fix convex, 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 data satisfy $\rho_0(\mathbf{x})\in G_0$, for all $\mathbf{x}\in\Omega$. Then the initial-value problem for \eqref{e1.1}, \eqref{e1.3} with $\Omega =$ $\mathbb{T}^N$ has a unique, classical solution $\rho$, $\mathbf{u}$ for $0\leq t\leq T$, where $\rho \in \bar{G}_1$, and \begin{gather*} \rho \in C([0,T],C^3(\Omega))\cap L^\infty ([0,T],H^{s+2}(\Omega)) \\ \mathbf{u}\in C([0,T],C^2(\Omega))\cap L^\infty ([0,T],H^{s+1}(\Omega)) \end{gather*} provided $T\|D \mathbf{u}_0\|_s$ and $T\| \nabla \rho_0\|_{s+1}$ are sufficiently small. \end{theorem} \begin{proof} The proof of the 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 is sought. Convergence of the sequence is proven in two steps: first, we prove the uniform boundedness of the approximating sequence $\{\rho^k\}$, $\{\mathbf{u}^k \}$, in a high Sobolev norm, and then we prove contraction of the sequence in a low Sobolev norm. Standard compactness arguments complete the proof. \end{proof} We will construct the solution of the initial-value problem for \eqref{e1.1}, \eqref{e1.3} with $\Omega =\mathbb{T}^N$ through the following iteration scheme. Set $\rho ^0(\mathbf{x},t)=\rho _0(\mathbf{x})$, and $\mathbf{u}^0(\mathbf{x},t)=\mathbf{u}_0(\mathbf{x})$. For $k=0,1,2,\dots$. construct $\rho ^{k+1}$, $\mathbf{u}^{k+1}$ from the previous iterates $\rho ^{k}$, $\mathbf{u}^{k}$ by solving \begin{gather} \frac{\partial\rho ^{k+1}}{\partial t} = -\nabla \cdot \mathbf{u}^{k+1} \label{e3.1} \\ \begin{aligned} \frac{\partial \mathbf{u}^{k+1}}{\partial t} &= -(\rho^k)^{-1}\mathbf{u}^k\cdot \nabla \mathbf{u}^{k+1}+(\rho^k)^{-2}\mathbf{u}^k\cdot \nabla \rho^{k+1}\mathbf{u}^k -(\rho^k)^{-1}(\nabla \cdot\mathbf{u}^{k+1}) \mathbf{u}^k\\ &\quad - p'(\rho^k) \nabla \rho^{k+1}+c\Big(\frac {1}{|\Omega|}\int_{\Omega} \rho^k d \mathbf{x}\Big)\nabla \Delta \rho ^{k+1} \end{aligned} \label{e3.2} \end{gather} with initial data $\rho ^{k+1}(\mathbf{x,}0)=\rho _0(\mathbf{x})$, $\mathbf{u}^{k+1}(\mathbf{x,}0)= \mathbf{u}_0(\mathbf{x})$. Existence of a solution to equations \eqref{e3.1}, \eqref{e3.2} for fixed $k$ is proven in Appendix A. The a priori estimates used in the proof are proven in Appendix B. We proceed now to prove convergence of the iterates as $k\to \infty$ to a unique, classical solution of \eqref{e1.1}, \eqref{e1.3}. Since $\rho ^k(\mathbf{x},0)=\rho _0\in G_0$, where $\bar{G}% _0\subset G_1$ and $\bar{G}_1\subset G$, we fix $\delta =\hat{\delta}% (G_1)$ so that if $| \rho -\rho _0|_{L^\infty,T} \leq\delta$, then $\rho\in \bar{G}_1$. And we fix $c_1=\hat{c}_1(G_1)>0$ and $c_2=\hat{c}_2(G_1)>0$, where $c_1<1$, so that $c_1<\rho 1$ which depends on $L_0$ and $R$. Then applying Lemma \ref{L2.2} from Appendix B to equations \eqref{e3.1}--\eqref{e3.2}, where we let $\mathbf{F}=0$ and $Q_k \mathbf{g}=0$ in equation \eqref{e2.2} of Lemma \ref{L2.2}, yields the estimate \begin{aligned} &\|D \mathbf{u}^{k+1}\|_s^2+\| \nabla \rho^{k+1}\|_s^2 +\| \Delta \rho^{k+1}\|_s^2 \\ &\leq C_4(1+C_4K_4Te^{C_4K_4T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2) \end{aligned} \label{e3.7} where $C_4=\hat{C}_4(s,c,c_1,c_2,c_3)$, where $s>$ $\frac N2+1$ with $N=2$ or $N=3$, so that $s\geq 3$, and where from Lemma \ref{L2.2} \begin{align*} K_4&=\max \Big\{1, \; \| (\rho^k)^{-1}\| _{s+1,T}^2\| \mathbf{u}^k\| _{s+1,T}^2, \; \|p'(\rho^k)\|_{s+1,T}^2, \; \|(\rho^k)^{-2}\| _{s+1,T}^2\| \mathbf{u}^k\| _{s+1,T}^4, \\ &\quad \|(\rho^k)^{-1}_{t}\| _{2, T }^2\| \mathbf{u}^k\|_{2,T }^2, \; \|(\rho^k)^{-1}\| _{2, T }^2\| (\mathbf{u}^k)_{t}\| _{2, T}^2, \; \| (\rho^k)_t\| _{2, T }, \; \| (p'(\rho^k))_t\| _{2, T } \Big\} \end{align*} We estimate $K_4 \leq C_6$, where the constant $C_6=\hat{C} _6(c_1, L_1,L_2)$, by the induction hypothesis. Then after using this estimate for $K_4$ in equation \eqref{e3.7}, we obtain \begin{aligned} &\|D \mathbf{u}^{k+1}\|_s^2+\| \nabla \rho^{k+1}\|_s^2 +\| \Delta \rho^{k+1}\|_s^2\\ &\leq C_4(1+C_4C_6Te^{C_4C_6T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2) \\ &= (C_4+C_7Te^{C_7T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2) \end{aligned} where $C_7=\hat{C} _7(s,c, c_1, c_2, c_3, L_1, L_2)$. Recall that $C_4$ does not depend on $L_1$ or $L_2$. Therefore, it follows that $\|D \mathbf{u}^{k+1}\|_{s,T}^2+\| \nabla \rho^{k+1}\|_{s,T}^2 +\| \Delta \rho^{k+1}\|_{s,T}^2\leq L_1^2$ provided that we choose $L_1$ large enough so that $$\frac{L_1^2}{2}\geq C_4(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2), \label{e3.9}$$ and provided that $T\|D \mathbf{u}_0\|_s$ and $T\| \nabla \rho_0\|_{s+1}$ are sufficiently small so that $$C_7Te^{C_7T}(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2) \leq \frac{L_1^2}{2}.\label{e3.11}$$ Thus, either the time interval $0\leq t\leq T$ is chosen to be sufficiently small, or the norms of the initial gradients, $\|D \mathbf{u}_0\|_s$ and $\| \nabla \rho_0\|_{s+1}$, are sufficiently small, or both are small. This completes the proof of part (a). Next, from \eqref{e3.1} for $\rho^{k+1}$, we have \begin{aligned} | \rho ^{k+1}-\rho _0| &\leq \int_0^t| \rho_t^{k+1}| _{L^\infty }d\tau \leq C\int_0^T\| \nabla \cdot \mathbf{u}^{k+1}\| _{s}dt \\ &\leq C T\|D\mathbf{u}^{k+1}\| _{s,T}\\ & \leq C T\Big((C_4+C_7Te^{C_7T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2)\Big)^{1/2} \end{aligned}\label{e3.12} Similarly, from equation \eqref{e3.2}, we obtain \begin{aligned} | \mathbf{u} ^{k+1}-\mathbf{u} _0| &\leq \int_0^t| \mathbf{u}_t^{k+1}| _{L^\infty }d\tau \\ &\leq C\int_0^T \|(\rho^k)^{-1}\|_{s-1}\|\mathbf{u}^k\|_{s-1}\|D \mathbf{u}^{k+1}\|_{s-1}d\tau \\ &\quad +C\int_0^T\|(\rho^k)^{-2}\|_{s-1}\|\mathbf{u}^k\|_{s-1}^2\|\nabla \rho^{k+1}\|_{s-1}d\tau \\ &\quad +C\int_0^T\|(\rho^k)^{-1}\|_{s-1}\|\mathbf{u}^k\|_{s-1} \|\nabla \cdot\mathbf{u}^{k+1}\|_{s-1}d\tau \\ &\quad +C\int_0^T\|p'(\rho^k) \|_{s-1}\|\nabla \rho^{k+1}\|_{s-1}d\tau \\ &\quad +C\int_0^T\|c\Big(\frac {1}{|\Omega|}\int_{\Omega} \rho^k d \mathbf{x}\Big)\nabla \Delta \rho ^{k+1}\|_{s-1}d\tau \\ &\leq C_8T(\|D \mathbf{u}^{k+1}\|_{s,T}+\| \nabla \rho^{k+1}\|_{s,T}+ \|\Delta \rho^{k+1}\|_{s,T}) \\ &\leq 3C_8T\Big((C_4+C_7Te^{C_7T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2)\Big)^{1/2} \end{aligned} \label{e3.13} where $C_8=\hat{C}_{8}(s,c,c_1,c_2,L_1)$. It follows from \eqref{e3.12}, \eqref{e3.13} that \begin{gather*} | \rho ^{k+1}-\rho _0|_{L^\infty,T} \leq \delta , \\ |\mathbf{u} ^{k+1}-\mathbf{u} _0 |_{L^{\infty}, T} \leq R \end{gather*} provided that $T\|D \mathbf{u}_0\|_s$ and $T\| \nabla \rho_0\|_{s+1}$ are small enough to satisfy $$C T\Big((C_4+C_7Te^{C_7T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2)\Big)^{1/2} \leq \delta \label{e3.14}$$ and provided that $T\|D \mathbf{u}_0\|_s$ and $T\| \nabla \rho_0\|_{s+1}$ are small enough to satisfy $$3 C_{8} T\Big((C_4+C_7Te^{C_7T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2)\Big)^{1/2}\leq R \label{e3.15}$$ This completes the proof of part (b). Using the fact that max $\{| \rho _0|_{L^\infty}, | \mathbf{u}_0|_{L^\infty}\}\leq L_0$, and the result just obtained for part (b), it follows that $| \rho^{k+1}|_{L^\infty,T} \leq | \rho_0|_{L^\infty}+\delta \leq L_0+ \delta$ and $| \mathbf{u} ^{k+1}|_{L^\infty,T} \leq | \mathbf{u} _0|_{L^\infty}+R \leq L_0+R$. Therefore, we have $\|\rho ^{k+1}\|_{0,T}\leq |\Omega|^{1/2}| \rho^{k+1}|_{L^\infty,T} \leq |\Omega|^{1/2}(L_0+\delta) \leq L_1$ and $\| \mathbf{u}^{k+1}\|_{0,T} \leq |\Omega|^{1/2} | \mathbf{u} ^{k+1}| _{L^\infty,T} \leq |\Omega|^{1/2}( L_0+R) \leq L_1$ provided that we choose $L_1$ large enough so that $$L_1\geq |\Omega|^{1/2} (L_0+ \delta) \label{e3.16}$$ and we choose $L_1$ large enough so that $$L_1\geq |\Omega|^{1/2} (L_0+ R) \label{e3.17}$$ This completes the proof of part (c). Since $\|\nabla \rho ^k\|_{s+1,T}^2\leq C\|\Delta \rho ^k\|_{s,T}^2$ when $\Omega =\mathbb{T}^N$ (a proof appears in \cite{DD1}), it follows from parts (a) and (c) that $\rho^{k+1} \in L^\infty ([0,T],H^{s+2})$. Finally, using equations \eqref{e3.1}, \eqref{e3.2}, and using the results just obtained in parts (a) and (c), we can directly estimate $\| \rho _t^{k+1}\| _{s,T} \leq C_{9},\quad \| \mathbf{u} _t^{k+1}\| _{s-1,T} \leq C_{10}$ where $C_{9}=\hat{C}_{9}(s,L_1)$ and $C_{10}=\hat{C} _{10}(s,c, c_1,L_1)$. Therefore, $\| \rho _t^{k+1}\|_{s,T}\leq L_2$ and $\| \mathbf{u} _t^{k+1}\| _{s-1,T} \leq L_2$ provided we choose $L_2$ large enough so that $$L_2\geq C_{9}, \quad L_2\geq C_{10} \label{e3.18}$$ This completes the proof of part (d). Summarizing, if we fix $L_1$, $L_2$, a priori and independent of $k$, so that \eqref{e3.9}, \eqref{e3.11}, \eqref{e3.14}, \eqref{e3.15}, \eqref{e3.16}, \eqref{e3.17}, \eqref{e3.18} are satisfied, then $\rho ^k$ and $\mathbf{u}^k$ satisfy (a)--(d) for all $k\geq 0$. This completes the proof. \end{proof} Next, we give the proof of contraction in low norm. \begin{proposition} \label{P3.2} Assume that the hypotheses of Theorem \ref{T3.1} hold. Then it follows that $\sum_{k=1}^{\infty}\big(\| \rho ^{k+1}-\rho ^k \| _{3,T}^2+\| \mathbf{u}^{k+1}-\mathbf{u} ^k\|_{2,T}^2\big)<\infty$ \end{proposition} \begin{proof} Subtracting \eqref{e3.1}, \eqref{e3.2} for $\rho ^k$, $\mathbf{u}^k$ from \eqref{e3.1}, \eqref{e3.2} for $\rho ^{k+1}$, $\mathbf{u}^{k+1}$ yields \begin{gather} \frac{\partial(\rho ^{k+1}-\rho ^k)}{\partial t} = -\nabla \cdot (\mathbf{u} ^{k+1}-\mathbf{u}^k), \label{e3.25} \\ \begin{aligned} \frac{\partial(\mathbf{u}^{k+1}-\mathbf{u}^k)}{\partial t} &= -(\rho^k)^{-1}\mathbf{u}^k\cdot \nabla (\mathbf{u}^{k+1}-\mathbf{u}^k)+(\rho^k)^{-2}\mathbf{u}^k\cdot \nabla (\rho^{k+1}-\rho^k)\mathbf{u}^k \\ &\quad -(\rho^k)^{-1}(\nabla \cdot(\mathbf{u}^{k+1}-\mathbf{u}^k)) \mathbf{u}^k-p'(\rho^k) \nabla (\rho^{k+1}-\rho^k) \\ &\quad +c\Big(|\Omega|^{-1}\int_{\Omega} \rho^k d \mathbf{x}\Big)\nabla \Delta(\rho ^{k+1}-\rho ^k) +\mathbf{F} \end{aligned}\label{e3.27} \end{gather} where $(\rho ^{k+1}-\rho ^k)(\mathbf{x},0) =0$, and $(\mathbf{u}^{k+1}-\mathbf{u}^k)(\mathbf{x},0) =0$, and where \begin{align*} \mathbf{F} &= -((\rho^k)^{-1}\mathbf{u}^k-(\rho^{k-1})^{-1}\mathbf{u}^{k-1})\cdot \nabla \mathbf{u}^{k} \\ &\quad +(((\rho^k)^{-2}\mathbf{u}^k-(\rho^{k-1})^{-2}\mathbf{u}^{k-1})\cdot \nabla \rho^{k})\mathbf{u}^k+(\rho^{k-1})^{-2}(\mathbf{u}^{k-1}\cdot \nabla \rho^{k})(\mathbf{u}^k-\mathbf{u}^{k-1}) \\ &\quad -(\nabla \cdot\mathbf{u}^{k})((\rho^k)^{-1} \mathbf{u}^k-(\rho^{k-1})^{-1} \mathbf{u}^{k-1}) -(p'(\rho^k) -p'(\rho^{k-1}) )\nabla \rho^{k} \\ &\quad +c\Big(|\Omega|^{-1}\int_{\Omega} (\rho^k-\rho^{k-1}) d \mathbf{x}\Big)\nabla \Delta \rho^k \end{align*} From Lemma \ref{L2.2} in Appendix B, using $r=1$, where we let $Q_k \mathbf{g}=0$ in equation \eqref{e2.2} of Lemma \ref{L2.2}, we obtain the following inequality $$\|D (\mathbf{u}^{k+1}-\mathbf{u}^k)\|_1^2+\| \nabla (\rho^{k+1}-\rho^k)\|_1^2 +\| \Delta (\rho^{k+1}-\rho^k)\|_1^2 \leq C_{11}\int_0^t\| \mathbf{F}\|_2^2 d\tau\quad \label{e3.28}$$ where $C_{11}=\hat{C}_{11}(c, c_1,c_2,c_3,L_1,L_2,T)$, and where we have used the results from Proposition \ref{P3.1}. From Lemma \ref{L2.2} in Appendix B, where we let $Q_k \mathbf{g}=0$ in equation \eqref{e2.2} of Lemma \ref{L2.2}, and using the results from Proposition \ref{P3.1}, we obtain the $L^2$ estimate \begin{aligned} &\| \mathbf{u}^{k+1}-\mathbf{u}^k\|_0^2+\| \rho^{k+1}-\rho^k\|_0^2+\| \nabla (\rho^{k+1}-\rho^k)\|_0^2 \\ &\leq C_{12}\int_0^t(\|D (\mathbf{u}^{k+1}-\mathbf{u}^k)\|_0^2+\| \mathbf{F}\|_0^2 )d\tau \end{aligned} \label{e3.29} where $C_{12}=\hat{C}_{12}(c, c_1,c_2,c_3,L_1,L_2,T)$. After adding \eqref{e3.28}, \eqref{e3.29}, and putting additional terms on the right-hand side, we obtain \begin{aligned} &\| \mathbf{u}^{k+1}-\mathbf{u}^k\|_0^2+\| \rho^{k+1}-\rho^k\|_0^2+\| \nabla (\rho^{k+1}-\rho^k)\|_0^2 \\ & +\|D (\mathbf{u}^{k+1}-\mathbf{u}^k)\|_1^2+\| \nabla (\rho^{k+1}-\rho^k)\|_1^2 +\| \Delta (\rho^{k+1}-\rho^k)\|_1^2 \\ &\leq C_{13}\int_0^t(\| \mathbf{u}^{k+1}-\mathbf{u}^k\|_0^2+\| \rho^{k+1}-\rho^k\|_0^2+\| \nabla (\rho^{k+1}-\rho^k)\|_0^2)d\tau \\ &\quad + C_{13}\int_0^t(\|D (\mathbf{u}^{k+1}-\mathbf{u}^k)\|_1^2+\| \nabla (\rho^{k+1}-\rho^k)\|_1^2)d\tau \\ &\quad +C_{13}\int_0^t( \| \Delta (\rho^{k+1}-\rho^k)\|_1^2+\| \mathbf{F}\|_2^2) d\tau \end{aligned} \label{e3.30} where $C_{13}=\hat{C}_{13}(c, c_1,c_2,c_3,L_1,L_2,T)$. From the definition of $\mathbf{F}$, and using Proposition \ref{P3.1}, we obtain the estimate $$\| \mathbf{F}\|_2^2 \leq C_{14}(\|\mathbf{u}^k-\mathbf{u}^{k-1}\|_2^2+ \| \rho ^k-\rho ^{k-1}\|_2^2) \label{e3.31}$$ where $C_{14}=\hat{C}_{14}(c,c_1,L_1)$. Here, we used the fact that $s > \frac N2+1$, so that $s \geq 3$, and we used the Sobolev inequality $|f|_{L^\infty} \leq C\|f\|_{s_0}$ (see, e.g., \cite{DD1}, \cite{E1}), where $s_0=[ \frac N2] +1=2$, when we estimated the term $\|c\big(|\Omega|^{-1}\int_{\Omega} (\rho^k-\rho^{k-1}) d \mathbf{x}\big)\nabla \Delta \rho^k\|_2^2\leq c|\rho^k-\rho^{k-1}|_{L^\infty}^2\|\nabla \Delta \rho^k\|_2^2 \leq C L_1^2 \|\rho^k-\rho^{k-1}\|_2^2$ in the definition of $\mathbf{F}$. Applying Gronwall's inequality to \eqref{e3.30}, and using \eqref{e3.31}, yields \begin{aligned} &\| \mathbf{u}^{k+1}-\mathbf{u}^k\|_0^2+\| \rho^{k+1}-\rho^k\|_0^2+\| \nabla (\rho^{k+1}-\rho^k)\|_0^2 \\ &+\|D (\mathbf{u}^{k+1}-\mathbf{u}^k)\|_1^2+\| \nabla (\rho^{k+1}-\rho^k)\|_1^2 +\| \Delta (\rho^{k+1}-\rho^k)\|_1^2 \\ &\leq C_{15}\int_0^t\| \mathbf{F}\|_2^2 d\tau \\ &\leq C_{16}\int_0^t(\| \rho ^k-\rho ^{k-1}\| _{2}^2 +\| \mathbf{u}^k-\mathbf{u} ^{k-1}\|_{2}^2) d\tau \end{aligned}\label{e3.32} where $C_{15}=\hat{C}_{15}(c, c_1,c_2,c_3,L_1,L_2,T)$, $C_{16}=\hat{C}_{16}(c, c_1,c_2,c_3,L_1,L_2,T)$. It follows that $$\| \rho ^{k+1}-\rho ^k\| _{3}^2+\| \mathbf{u}^{k+1}-\mathbf{u}^k \|_{2}^2 \leq C_{17}\int_0^t(\| \rho ^k-\rho ^{k-1}\| _{3}^2 +\| \mathbf{u}^k-\mathbf{u} ^{k-1}\|_{2}^2) d\tau \label{e3.33}$$ where $C_{17}=\hat{C}_{17}(c, c_1,c_2,c_3,L_1,L_2, T)$. Here we used the fact that $\| \nabla (\rho^{k+1}-\rho^k)\|_2^2 \leq C\| \Delta (\rho^{k+1}-\rho^k)\|_1^2$ when $|\Omega|=\mathbb{T}^N$ (a proof appears in \cite{DD1}). Repeatedly applying \eqref{e3.33} yields $\| \rho ^{k+1}-\rho ^k\| _{3,T}^2+\| \mathbf{u}^{k+1}-\mathbf{u}^k \|_{2,T}^2 \leq\frac{ (C_{17}T)^k}{k!}(\| \rho ^1-\rho ^{0}\| _{3,T}^2+\| \mathbf{u}^1-\mathbf{u} ^{0}\|_{2,T}^2)$ It follows that $\sum_{k=1}^{\infty}\left(\| \rho ^{k+1}-\rho ^k \| _{3,T}^2+\| \mathbf{u}^{k+1}-\mathbf{u} ^k\|_{2,T}^2\right)<\infty$ This completes the proof. \end{proof} Using Propositions \ref{P3.1} and \ref{P3.2}, we now complete the proof of Theorem \ref{T3.1} by using a standard argument (see, for example, \cite{E1}, \cite{M1}). From Proposition \ref{P3.2}, we conclude that there exist $\rho \in C([0,T],H^3(\Omega ))$, and $\mathbf{u}\in C([0,T],H^2(\Omega ))$ so that $\| \rho ^k-\rho \| _{3,T}\to 0$, and $\| \mathbf{u}^k-\mathbf{u}\| _{2,T}\to 0$ as $k\to \infty$. Using the standard interpolation inequalities (see, e.g., \cite{E1}) \begin{gather*} \| \rho^{k+1}-\rho^k \| _{s'+2} \leq C\| \rho^{k+1}-\rho^k \| _3^\beta \| \rho^{k+1}-\rho^k \| _{s+2}^{1-\beta } \\ \| \mathbf{u}^{k+1}-\mathbf{u}^k \| _{s'+1} \leq C\| \mathbf{u}^{k+1}-\mathbf{u}^k\| _2^\beta \| \mathbf{u}^{k+1}-\mathbf{u}^k\| _{s+1}^{1-\beta } \end{gather*} with $\beta =\frac{s-s'}{s-1}$ , and Propositions \ref{P3.1} and \ref{P3.2}, we can conclude that $\| \rho ^k-\rho \|_{s'+2,T}\to 0$, and $\| \mathbf{u}^k-\mathbf{u}\|_{s'+1,T}\to 0$ as $k\to \infty$ for any $s'\frac N 2+1$, Sobolev's lemma implies that $\rho ^k\to \rho$ in $C([0,T],C^3(\Omega))$, and $\mathbf{u}^k\to \mathbf{u}$ in $C([0,T],C^2(\Omega))$. From the linear system of equations \eqref{e3.1}, \eqref{e3.2} it follows that $\| \rho _t^k-\rho _t\|_{s',T}\to 0$, and $\| \mathbf{u}_t^k-\mathbf{u}_t\| _{s'-1,T}\to 0$ as $k\to \infty$, so that $\rho _t^k\to \rho _t \in C([0,T],C^1(\Omega))$, and $\mathbf{u}_t^k\to \mathbf{u}_t$ in $C([0,T],C(\Omega))$, and $\rho$, $\mathbf{u}$ is a classical solution of the system of equations \eqref{e1.1}, \eqref{e1.3}. The additional facts that $\rho \in L^\infty ([0,T],H^{s+2}(\Omega))$, $\mathbf{u}\in L^\infty ([0,T],H^{s+1}(\Omega))$, can be deduced from the uniform boundedness of $\{\rho^k\}$ in $L^\infty ([0,T],H^{s+2}(\Omega))$ and of $\{\mathbf{u}^k\}$ in $L^\infty ([0,T],H^{s+1}(\Omega))$ from Proposition \ref{P3.1}, and from the weak-* compactness of bounded sets in $L^\infty ([0,T],H^{r}(\Omega))$, i.e., by Alaoglu's theorem (see, for example, \cite{E1}, \cite{M1}). The uniqueness of the solution follows by a standard proof, using estimates similar to the proof of Proposition \ref{P3.2}. \begin{appendix} \section{Existence for the linear problem} We now present a proof of the existence of a classical solution $\rho$, $\mathbf{u}$ to the linear equations \eqref{e3.1}, \eqref{e3.2}: \begin{gather} \frac{\partial\rho }{\partial t} = -\nabla \cdot \mathbf{u} \label{A.1} \\ \begin{aligned} \frac{\partial\mathbf{u}}{\partial t} &= -a_1^{-1}\mathbf{v}\cdot\nabla \mathbf{u}-a_1^{-1}(\nabla \cdot \mathbf{u})\mathbf{v} +a_1^{-2}(\mathbf{v}\cdot\nabla \rho)\mathbf{v} -a_2\nabla \rho \\ &\quad +c\Big(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}\Big)\nabla \Delta \rho \end{aligned} \label{A.2} \end{gather} \begin{lemma} \label{LA.1} Given \begin{gather*} \mathbf{v} \in C([0,T],H^0(\Omega))\cap L^\infty ([0,T],H^{s+1}(\Omega)), \\ a_1 \in C([0,T],H^0(\Omega))\cap L^\infty ([0,T],H^{s+2}(\Omega)), \\ a_2 \in C([0,T],H^0(\Omega))\cap L^\infty([0,T],H^{s+2}(\Omega)), \\ \mathbf{v}_t \in L^\infty ([0,T],H^{s-1}(\Omega)), \\ (a_1)_t, \; (a_2)_t \in L^\infty ([0,T],H^{s}(\Omega)), \end{gather*} where $s>\frac N2+1$, $\Omega =\mathbb{T}^N$, with $N=2$ or $N=3$, and where $0< c_1<$ $a_1(\mathbf{x,}t)1$ and $0\leq t\leq T$, there is a classical solution $\rho$, $\mathbf{u}$ of the initial value problem for \eqref{A.1}, \eqref{A.2}, with initial data $\rho (\mathbf{x},0)=\rho _0(\mathbf{x})\in H^{s+2}(\Omega)$, $\mathbf{u}(\mathbf{x},0)=\mathbf{u}_0(\mathbf{x}) \in H^{s+1}(\Omega)$, and \begin{gather*} \rho \in C([0,T],C^3(\Omega))\cap L^\infty ([0,T],H^{s+2}(\Omega)), \\ \mathbf{u} \in C([0,T],C^2(\Omega))\cap L^\infty ([0,T],H^{s+1}(\Omega)) . \end{gather*} \end{lemma} \begin{proof} Since we are solving the initial-value problem under periodic boundary conditions, we will use Galerkin's method, with the standard orthonormal basis in $L^2$ of trigonometric functions $\{w _i\}_{i=1}^\infty$, to construct the solution. Here $w_i$ has the form $cos(2\pi \mathbf{n}_i\cdot \mathbf{x})$ or $sin(2\pi \mathbf{n}_i\cdot \mathbf{x})$ with $\mathbf{n}_i \in \mathbb{Z}_{+}^N$. The proof by Galerkin's method is a standard one, and is included here for the sake of completeness. We will write the system of equations \eqref{A.1}, \eqref{A.2} equivalently as follows: \begin{gather} \frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{u}, \label{A.4} \\ \begin{aligned} \frac{\partial u_i}{\partial t} &= -a_1^{-1}\mathbf{v}\cdot\nabla u_i-a_1^{-1}(\nabla \cdot \mathbf{u})v_i +a_1^{-2}(\mathbf{v}\cdot\nabla \rho)v_i \\ &\quad -a_2\frac{\partial \rho}{\partial x_i}+c\Big(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}\Big)\frac {\partial}{\partial x_i} (\Delta \rho), \end{aligned}\label{A.6} \end{gather} where $i=1,\dots,N$. Here $u_i$ is the $ith$ component of the vector $\mathbf{u}$ and $v_i$ is the $ith$ component of the vector $\mathbf{v}$. Let $P_k$ denote the orthogonal projection of $L^2$ onto the finite dimensional subspace $V_k=$ span$\{w_1,\ldots ,w_k\}$. The finite-dimensional approximation $\rho ^k\in V_k$ and $u_i^k\in V_k$, where $u_i^k$ is the $ith$ component of $\mathbf{u}^k$, is the solution of the equations \begin{gather} \frac{\partial \rho ^k}{\partial t} = - \nabla \cdot \mathbf{u}^k, \label{A.7} \\ \begin{aligned} \frac{\partial u_i^k}{\partial t} &= -P_k(a_1^{-1}\mathbf{v}\cdot \nabla u_i^k) -P_k(a_1^{-1} ( \nabla \cdot \mathbf{u}^k) v_i)+P_k(a_1^{-2}(\mathbf{v}\cdot \nabla \rho^k)v_i) \\ &\quad -P_k\Big(a_2\frac {\partial \rho^k}{\partial x_i}\Big)+ P_k\Big(c\Big(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}\Big) \frac {\partial}{\partial x_i}(\Delta \rho ^k) \Big), \end{aligned}\label{A.8} \end{gather} with $\rho ^k(\mathbf{x,}0) =P_k\rho(\mathbf{x},0)$, and $u_i^k(\mathbf{x,}0)=P_k u_i(\mathbf{x},0)$, for $i=1,\dots,N$. Because $\rho ^k\in V_k$ and $u_i^k\in V_k$, we can write \begin{gather} \rho^k= \sum_{j=1}^k\alpha _j(t)w_j,\label{A.9}\\ u_i^k= \sum_{j=1}^k \gamma_{i,j}(t)w_j. \label{A.10} \end{gather} After substituting \eqref{A.9}, \eqref{A.10} into \eqref{A.7} and \eqref{A.8} we take the $L^2$ inner product of \eqref{A.7} and \eqref{A.8} with $w_l$ for $l=1,\ldots ,k$, which transforms \eqref{A.7} and \eqref{A.8} into the following equivalent linear system of ordinary differential equations for the coefficients $\alpha _l(t)$ and $\gamma_{i,l}(t)$, where $i=1,\dots,N$, and $l=1,\dots,k$: $\frac{d\alpha_{l}}{dt} = -\sum_{j=1}^k (\sum_{m=1}^N\gamma _{m,j}(t)\frac{\partial w_j}{\partial x_m} ,w_l) ,$ \begin{align*} \frac{d \gamma _{i,l}}{dt} &= -\sum_{j=1}^k \Big((a_1^{-1}\mathbf{v}\cdot \nabla w_j, w_l)\gamma _{i,j}(t)-(a_1^{-1} (\sum_{m=1}^N\gamma _{m,j}(t)\frac{\partial w_j}{\partial x_m}) v_i,w_l)\Big) \\ &\quad +\sum_{j=1}^k\Big((a_1^{-2}(\mathbf{v}\cdot \nabla w_j)v_i, w_l)\alpha_{j}(t)-(a_2 \frac {\partial w_j}{\partial x_i},w_l)\alpha_j(t)\Big) \\ &\quad +\sum_{j=1}^k\Big(\Big(c\Big(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}\Big)\frac {\partial}{\partial x_i} (\Delta w_j),w_l\Big))\alpha _j(t)\Big). \end{align*} Also $\alpha _l(0) =(\rho(\mathbf{x},0),w_l)$, and $\gamma_{i,l}(0)=(u_i(\mathbf{x},0),w_l)$. The coefficients in this system of equations are continuous, and it has a unique solution $\{\alpha _l(t)\}_{l=1}^k$ $\in C^1([0,T])$ and $\{\gamma _{i,l}(t)\}_{l=1}^k$ $\in C^1([0,T])$, for $i=1,\dots,N$. It follows that $\rho ^k\in C^1([0,T],H^r(\Omega))$ and $u_i^k\in C^1([0,T],H^r(\Omega))$ for any $r\geq 0$. Next, we obtain estimates for $\rho^k$, $\mathbf{u}^k$ in high Sobolev norm. Let $Q_k =I -P_k$, where $I$ is the identity operator. Then we write \eqref{A.7}, \eqref{A.8} equivalently as follows: \begin{gather} \frac{\partial\rho^k }{\partial t} = -\nabla \cdot \mathbf{u}^k \label{A.51} \\ \begin{aligned} \frac{\partial\mathbf{u}^k}{\partial t} &=-a_1^{-1}\mathbf{v}\cdot\nabla \mathbf{u}^k-a_1^{-1}(\nabla \cdot \mathbf{u}^k)\mathbf{v} +a_1^{-2}(\mathbf{v}\cdot\nabla \rho^k)\mathbf{v} -a_2\nabla \rho^k \\ &\quad +c\Big(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}\Big)\nabla \Delta \rho^k - Q_k \mathbf{g} \end{aligned}\label{A.52} \end{gather} where $Q_k\mathbf{g}= -Q_k(a_1^{-1}\mathbf{v}\cdot\nabla \mathbf{u^k})-Q_k(a_1^{-1}(\nabla \cdot \mathbf{u}^k)\mathbf{v}) +Q_k(a_1^{-2}(\mathbf{v}\cdot\nabla \rho^k)\mathbf{v}) -Q_k(a_2\nabla \rho^k) %\label{e2.29}$ Note that by the orthogonality of the projections $P_k$ and $Q_k$, we have $(Q_k \mathbf{g},\mathbf{u}^k)=0$, $(\nabla \cdot(Q_k \mathbf{g})_{\alpha},\nabla \cdot \mathbf{u}^k_{\alpha})=0$, and $(\nabla \times (Q_k\mathbf{g})_{\alpha},\nabla \times\mathbf{u}^k_{\alpha})=0$ for $|\alpha| \geq 0$. Also, note that $Q_k(c\left(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}\right)\nabla \Delta \rho^k )=0$. Then applying Lemma \ref{L2.2} in Appendix B to equations \eqref{A.51}, \eqref{A.52} yields the following estimates $$\|D \mathbf{u}^k\|_s^2+\| \nabla \rho^k\|_s^2 +\| \Delta \rho^k\|_s^2 \leq C_4(1+C_4K_4Te^{C_4K_4T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2) \label{A.53}$$ and \begin{aligned} &\| \mathbf{u}^k\|_0^2+\|\rho^k\|_0^2+\| \nabla \rho^k\|_0^2 \\ &\leq C_5(1+C_5K_4Te^{C_5K_4T})(\|\mathbf{u}_0\|_0^2 +\| \rho_0\|_0^2+\| \nabla \rho_0\|_{0}^2) \\ &\quad +C_5(1+C_5 K_4Te^{C_5K_4T})\int_0^t \|D \mathbf{u}^k\|_0^2 d\tau \\ &\leq C_5(1+C_5K_4Te^{C_5K_4T})(\|\mathbf{u}_0\|_0^2 +\| \rho_0\|_0^2+\| \nabla \rho_0\|_{0}^2) \\ &\quad +C_5(1+C_5 K_4Te^{C_5K_4T})TC_4(1+C_4K_4Te^{C_4K_4T})(\|D \mathbf{u}_0\|_s^2+\| \nabla \rho_0\|_{s+1}^2) \end{aligned} \label{A.54} where the constants $C_4$, $C_5$, $K_4$ are defined in Lemma \ref{L2.2}. Here, we used the fact that $\|P_k \rho_0\|_r \leq \|\rho_0\|_r$ and $\|P_k \mathbf{u}_0\|_r \leq \|\mathbf{u}_0\|_r$. And we used estimate \eqref{A.53} in the right-hand side of estimate \eqref{A.54}. From \eqref{A.53}, \eqref{A.54} it follows that $\{\rho ^k \}$ is bounded in $L^\infty([0,T],H^{s+2}(\Omega))$ and $\{\mathbf{u}^k\}$ is bounded in $L^\infty([0,T],H^{s+1}(\Omega))$. Here we used the fact that $\|\nabla \rho ^k\|_{s+1,T}^2\leq C\|\Delta \rho ^k\|_{s,T}^2$ when $\Omega =\mathbb{T}^N$ (a proof appears in \cite{DD1}). From equations \eqref{A.51}, \eqref{A.52}, it follows that $\| \rho^k_t \| _{0}$ and $\| \mathbf{u}^k_t\| _{0}$ are bounded for all $k\geq 1$. Here we used the fact that $\|Q_k \mathbf{g}\|_0 \leq \|\mathbf{g}\|_0$. It follows that $\{\rho^k\}$ and $\{\mathbf{u}^k\}$ are bounded and equicontinuous in $C([0,T],H^0(\Omega))$. Using the Arzela-Ascoli theorem together with the weak-* compactness of bounded sets in $L^\infty([0,T],H^{r}(\Omega))$, it follows that there exist subsequences $\rho^{k_j}$ of $\rho^k$ and $\mathbf {u}^{k_j}$ of $\mathbf{u}^k$, and there exist functions $\rho \in C([0,T],H^0(\Omega))\cap L^\infty ([0,T],H^{s+2}(\Omega))$, $\mathbf{u}\in C([0,T],H^0(\Omega))\cap L^\infty ([0,T],H^{s+1}(\Omega))$, such that as $j \to \infty$, \begin{gather*} \rho^{k_j} \to \rho \quad \text{strongly in } C([0,T],H^0(\Omega)), \\ \rho^{k_j} \to \rho \quad \text{weak-* in } L^\infty ([0,T],H^{s+2}(\Omega)),\\ \mathbf{u}^{k_j}\to \mathbf{u} \quad \text{strongly in } C([0,T],H^0(\Omega)),\\ \mathbf{u}^{k_j}\to \mathbf{u} \quad \text{weak-* in } L^\infty ([0,T],H^{s+1}(\Omega)) \end{gather*} Using the standard interpolation inequalities (see, e.g., \cite{E1}), \begin{gather*} \| \mathbf{u}^{k_{j+1}}-\mathbf{u}^{k_j} \| _{s'+1} \leq C\| \mathbf{u}^{k_{j+1}}-\mathbf{u}^{k_j} \|_0^{\theta_1} \| \mathbf{u}^{k_{j+1}} -\mathbf{u}^{k_j} \| _{s+1}^{1-\theta_1 } \\ \| \rho^{k_{j+1}} -\rho^{k_j} \| _{s'+2} \leq C\| \rho^{k_{j+1}} -\rho^{k_j} \| _0^{\theta_2} \| \rho^{k_{j+1}} -\rho^{k_j} \| _{s+2}^{1-\theta_2 } \end{gather*} with $\theta_1 =\frac{s-s'}{s+1}$, $\theta_2 =\frac{s-s'}{s+2}$, it follows that $\rho^{k_j}\to \rho$ in $C([0,T],H^{s'+2}(\Omega))$ and $\mathbf{u}^{k_j} \to \mathbf{u}$ in $C([0,T],H^{s'+1}(\Omega))$ for any $s'1$. Here, $0\leq t \leq T$, and the domain $\Omega =\mathbb{T}^N$. Let $\rho_0(\mathbf{x})=\rho(\mathbf{x},0)$, $\mathbf{u}_0(\mathbf{x})=\mathbf{u}(\mathbf{x},0)$ be the given initial data, which is assumed to be sufficiently smooth. Then $\rho$, $\mathbf{u}$ satisfy the following two inequalities \begin{align*} \|D \mathbf{u}\|_r^2+\| \nabla \rho\|_r^2 +\| \Delta \rho\|_r^2 &\leq C_4(1+C_4K_4Te^{C_4K_4T})(\|D \mathbf{u}_0\|_r^2+\| \nabla \rho_0\|_{r+1}^2) \\ &\quad +C_4(1+C_4K_4Te^{C_4K_4T})\int_0^t\| \mathbf{F}\|_{r+1}^2 d\tau \end{align*} and \begin{align*} \| \mathbf{u}\|_0^2+\| \rho\|_0^2+\|\nabla \rho\|_0^2 &\leq C_5(1+C_5K_4Te^{C_5K_4T})(\|\mathbf{u}_0\|_0^2 +\| \rho_0\|_0^2+\| \nabla \rho_0\|_{0}^2) \\ &\quad +C_5(1+C_5K_4Te^{C_5K_4T})\int_0^t(\|D \mathbf{u}\|_0^2+ \| \mathbf{F}\|_0^2) d\tau\,, \end{align*} where $C_4=\hat{C}_4(r,c, c_1,c_2,c_3)$, $C_5=\hat{C}_5(c, c_1,c_2)$, and $r\geq 1$, and where \begin{align*} K_4&= \max \Big\{1,\; \| a_1^{-1}\| _{q+1,T}^2\|\mathbf{v}\| _{q+1,T}^2, \; \| a_2\|_{q+1,T}^2, \quad \|a_1^{-2}\| _{q+1,T}^2\| \mathbf{v}\| _{q+1,T}^4, \\ &\quad \|( a_1^{-1})_{t}\| _{2, T }^2\| \mathbf{v}\| _{2,T}^2, \; \| a_1^{-1}\| _{2, T}^2\| \mathbf{v}_{t}\| _{2, T}^2, \; \|(a_1)_t\| _{2, T }, \; \| (a_2)_t\| _{2, T } \Big\} \end{align*} where $q=\max\{r,s_0\}$, where $r\geq 1$, and where $s_0=[\frac N2]+1=2$ for $N=2$ or $N=3$. \end{lemma} \begin{proof} First, we will obtain an $L^2$ estimate. Then we will obtain estimates for $\nabla \cdot \mathbf{u}$ and for $\nabla \times \mathbf{u}$, which will be combined to obtain an estimate for $D\mathbf{u}$. Using the fact that $(Q_k \mathbf{g},\mathbf{u})=0$, we obtain an $L^2$ estimate as follows: \begin{align} \frac 12\frac d{dt}\| \mathbf{u}\| _0^2 &= (\mathbf{u}_t,\mathbf{u)} \nonumber \\ &= -(a_1^{-1}\mathbf{v}\cdot \nabla \mathbf{u,u})-(a_1^{-1}(\nabla \cdot \mathbf{u})\mathbf{v}, \mathbf{u})+(a_1^{-2}(\mathbf{v}\cdot\nabla \rho)\mathbf{v}, \mathbf{u}) \nonumber \\ &\quad -(a_2\nabla \rho, \mathbf{u})+c((|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x})\nabla \Delta \rho ,\mathbf{u})+(\mathbf{F},\mathbf{u}) -(Q_k\mathbf{g},\mathbf{u}) \nonumber \\ &= \frac 12(\mathbf{u}\nabla \cdot (a_1^{-1}\mathbf{v}),\mathbf{u}) -(a_1^{-1}(\nabla \cdot \mathbf{u})\mathbf{v}, \mathbf{u})+(a_1^{-2}(\mathbf{v}\cdot\nabla \rho)\mathbf{v}, \mathbf{u}) \nonumber \\ &\quad +(\rho \nabla a_2, \mathbf{u})+(a_2\rho, \nabla \cdot\mathbf{u}) -c((|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x})\Delta \rho ,\nabla \cdot \mathbf{u})+(\mathbf{F},\mathbf{u}) \nonumber \\ &\leq C(| a_1^{-1}| _{L^\infty }| \nabla \cdot \mathbf{v}| _{L^\infty }+| D(a_1^{-1})| _{L^\infty }| \mathbf{v}| _{L^\infty })\| \mathbf{u}\| _0^2 \nonumber \\ &\quad +C| a_1^{-1}|_{L^\infty }| \mathbf{v}| _{L^\infty }\| \nabla \cdot\mathbf{u}\| _0\| \mathbf{u}\| _0 +C| a_1^{-2}| _{L^\infty }| \mathbf{v}| _{L^\infty }^2\| \nabla \rho\| _0\| \mathbf{u}\| _0 \nonumber \\ &\quad +C| Da_2| _{L^\infty }\| \rho\| _0\| \mathbf{u}\| _0-(a_2\rho, \rho_t) +c((|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x})\Delta \rho ,\rho_t) \nonumber \\ &\quad +C\|\mathbf{F}\|_0\|\mathbf{u}\| _0 \nonumber \\ &\leq C(1+| a_1^{-1}| _{L^\infty }| D \mathbf{v}| _{L^\infty }+| D (a_1^{-1})| _{L^\infty }| \mathbf{v}| _{L^\infty })\| \mathbf{u}\| _0^2 \nonumber \\ &\quad +C(| a_1^{-1}| _{L^\infty }^2| \mathbf{v}| _{L^\infty }^2+| a_1^{-2}| _{L^\infty }^2| \mathbf{v}| _{L^\infty }^4+|D a_2| _{L^\infty}^2) \| \mathbf{u}\| _0^2 + C\|\nabla \cdot\mathbf{u}\| _0^2 \nonumber \\ &\quad -\frac 12 \frac {d}{dt}(a_2\rho, \rho)+\frac 12((a_2)_t\rho, \rho)+C\| \rho\| _0^2+C\| \nabla \rho\| _0^2+C\|\mathbf{F}\|_0^2 \nonumber \\ &\quad -\frac{c}{2}\frac{d}{dt}((|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x})\nabla \rho ,\nabla\rho)+\frac{c}{2}((|\Omega|^{-1}\int_{\Omega} (a_1)_t d \mathbf{x})\nabla \rho ,\nabla\rho) \label{e2.4} \end{align} where $C$ is a generic constant, and where we used equation \eqref{e2.1} to substitute for $\nabla \cdot \mathbf{u}$. Here, we have used Holder's inequality $(f,g)\leq \| f\|_0\| g\| _0$. Also, we used Cauchy's inequality $fg\leq \frac 12(f^2 +g^2)$. Integrating \eqref{e2.4} with respect to time, and using the fact that $0\frac N2$ (see, e.g., \cite{E1}). We integrate equation \eqref{e2.13} with respect to time, and use estimate \eqref{e2.16} on the right-hand side, and then add over $0\leq |\alpha|\leq r$, where $r\geq 1$, which yields the estimate \begin{aligned} &\|\nabla \cdot \mathbf{u}\|_r^2+\| \nabla \rho\|_r^2+\| \Delta \rho\|_r^2\\ &\leq C_2 (\|\nabla \cdot \mathbf{u}_0\|_r^2+\| \nabla \rho_0\|_r^2+\| \Delta \rho_0\|_r^2)+C_2\int_0^t\| \mathbf{F}\|_{r+1}^2d\tau \\ &\quad +C_2 K_2 \int_0^t (\|D \mathbf{u}\|_r^2+\| \nabla \rho\|_r^2+\| \Delta \rho\|_r^2) d\tau \end{aligned} \label{e2.17} where $C_2=\hat{C}_2(r, c,c_1,c_2)$, and where we define $K_2$, which is an upper bound for the coefficients in \eqref{e2.13}, \eqref{e2.16}, as follows: \begin{aligned} K_2&= \max \Big\{1, \; \| a_1^{-1}\| _{q,T}^2\| D\mathbf{v}\| _{q,T}^2, \; \| D (a_1^{-1})\|_{q,T}^2\|\mathbf{v}\| _{q,T}^2, \\ &\quad \|D(a_1^{-2})\| _{q,T}^2\| \mathbf{v}\|_{q,T}^4, \; \|a_1^{-2}\| _{q,T}^2\| D\mathbf{v}\| _{q,T}^2\| \mathbf{v}\| _{q,T}^2, \\ &\quad | a_1^{-2}| _{L^\infty,T}^2|\mathbf{v}|_{L^\infty,T }^4, \; \| D a_2\| _{q,T}^2, \quad | (a_1)_t| _{L^\infty,T },\; | (a_2)_{t}| _{L^\infty,T } \Big\} \end{aligned} \label{e2.18} where $q=\max\{r,s_0\}$, where $r\geq 1$, and where $s_0=[\frac N2]+1=2$ for $N=2$ or $N=3$. Here we have used the fact that $01$, we define $\beta=c_1^2/(2c_3^2)$ (so that we have $\beta <1$), and we have already defined $\epsilon=1/4$. We obtain the following estimate for one of the terms from \eqref{e2.28}: $\frac{\beta}{4\epsilon}|a_1^{-1}| _{L^\infty }^2| \mathbf{v}| _{L^\infty }^2\|\nabla \rho\|_r^2 =\frac{c_1^2}{2c_3^2}|a_1^{-1}| _{L^\infty }^2| \mathbf{v}| _{L^\infty }^2\|\nabla \rho\|_r^2 \leq \frac 12\|\nabla \rho\|_r^2$ Similarly, we obtain the estimate $\frac{\beta}{4\epsilon}|a_1(\mathbf{x},0)^{-1}| _{L^\infty }^2| \mathbf{v}_0| _{L^\infty }^2\|\nabla \rho_0\|_r^2\leq \frac 12 \|\nabla \rho_0\|_r^2$ Using these estimates in the right-hand side of \eqref{e2.28} and then moving the term $\frac 12 \|\nabla \rho\|_r^2$ to the left-hand side, and applying Gronwall's inequality yields the desired estimate \begin{aligned} \|D \mathbf{u}\|_r^2+\| \nabla \rho\|_r^2 +\| \Delta \rho\|_r^2 &\leq C_4(1+C_4K_4Te^{C_4K_4T})(\|D \mathbf{u}_0\|_r^2+\| \nabla \rho_0\|_{r+1}^2) \\ &\quad +C_4(1+C_4K_4Te^{C_4K_4T})\int_0^t\| \mathbf{F}\|_{r+1}^2 d\tau \end{aligned} \label{e2.30} where $C_4=\hat{C}_4(r,c, c_1,c_2,c_3)$. From \eqref{e2.8}, we obtain the $L^2$ estimate \begin{align*} \| \mathbf{u}\|_0^2+\| \rho\|_0^2+\| \nabla \rho\|_0^2 &\leq C_5(1+C_5K_4Te^{C_5K_4T})(\|\mathbf{u}_0\|_0^2 +\| \rho_0\|_0^2+\| \nabla \rho_0\|_{0}^2) \\ &\quad +C_5(1+C_5 K_4Te^{C_5K_4T})\int_0^t(\|D \mathbf{u}\|_0^2+\| \mathbf{F}\|_0^2 )d\tau \end{align*} where $C_5=\hat{C}_5(c, c_1,c_2)$, and where we used the fact that $K_1\leq C K_4$, where $K_1$ was defined in \eqref{e2.7}. The preceding two estimates are the desired result. \end{proof} \end{appendix} \begin{thebibliography} {99} \bibitem{AMW} D. M. Anderson, G. B. McFadden and A. A. Wheeler; \emph{Diffuse-interface methods in fluid mechanics}, Ann. Rev. Fluid Mech., 30 (1998), 139--65. \bibitem{BDL} D. Bresch, B. Desjardins, C.K. Lin; \emph{On some compressible fluid models: Korteweg, lubrication, and shallow water systems}, Comm. in Partial Differential Equations, 28, nos. 3 and 4, (2003), 843--68. \bibitem{DD1} D. L. 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