\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 $c_1<\rho <c_2$ and $c_1<p'(\rho)<c_2$ for
$\rho\in \bar{G}_1$.

Next, we proceed with the proof of uniform boundedness of the
approximating sequence in a high Sobolev norm.

\begin{proposition} \label{P3.1}
 Assume that the hypotheses of Theorem \ref{T3.1} hold.
 Let $\delta$, $R$ be given positive constants.
Then there are constants $L_1$, $L_2$, such that for $
k=0,1,2,3..$. the following estimates hold
\begin{itemize}
\item[(a)] $\|\nabla \rho ^k\|_{s,T} \leq  L_1$,
$\|\Delta \rho ^k\|_{s,T} \leq  L_1$,
$\|D\mathbf{u}^k\|_{s,T} \leq L_1$;

\item[(b)] $| \rho ^k-\rho _0|_{L^\infty,T} \leq \delta$,
$| \mathbf{u} ^k-\mathbf{u} _0|_{L^\infty,T} \leq R$;

\item[(c)] $\|\rho ^k\|_{0,T} \leq  L_1$,
$\|\mathbf{u}^k\|_{0,T} \leq L_1$;

\item[(d)] $\| \frac{\partial \rho ^k}{\partial t}\|_{s,T}
\leq  L_2$, $\| \frac{\partial \mathbf{u}^k}{\partial
t}\|_{s-1,T} \leq L_2$ %  \label{e3.5}
\end{itemize}
provided $T\|D \mathbf{u}_0\|_s$ and $T\| \nabla
\rho_0\|_{s+1}$ are sufficiently small.
\end{proposition}

\begin{proof}
The proof is by induction on $k$, of which we show only the
inductive step. We will derive estimates for $\rho ^{k+1}$ and
$\mathbf{u}^{k+1}$, and then use these estimates to prescribe
$L_1$ and $L_2$ a priori, independent of $k$, so that if $\rho^k$
and $\mathbf{u}^k$ satisfy (a)--(d), then $\rho ^{k+1}$ and
$\mathbf{u}^{k+1}$ also satisfy (a)--(d).

 From the statement of the theorem, we have max $\{| \rho
_0|_{L^\infty},| \mathbf{u}_0|_{L^\infty}\}\leq
L_0$, for some positive constant $L_0$. By the induction
hypothesis, we have $| \mathbf{u} ^k-\mathbf{u}
_0|_{L^\infty,T} \leq R$. It follows that $| \mathbf{u}
^k|_{L^\infty,T} \leq | \mathbf{u}
_0|_{L^\infty}+R \leq L_0+R <c_3$, for some constant $c_3>1$
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{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
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
\begin{equation}
\frac{L_1^2}{2}\geq C_4(\|D \mathbf{u}_0\|_s^2+\|
\nabla \rho_0\|_{s+1}^2), \label{e3.9}
\end{equation}
and provided that $T\|D \mathbf{u}_0\|_s$ and $T\|
\nabla \rho_0\|_{s+1}$ are sufficiently small so that
\begin{equation}
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}
\end{equation}
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{equation}
\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}
\end{equation}
Similarly, from equation \eqref{e3.2}, we obtain
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
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}
\end{equation}
and provided that $T\|D \mathbf{u}_0\|_s$ and
$T\| \nabla \rho_0\|_{s+1}$ are small enough to satisfy
\begin{equation}
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}
\end{equation}
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
\begin{equation}
L_1\geq |\Omega|^{1/2} (L_0+ \delta) \label{e3.16}
\end{equation}
and we choose $L_1$ large enough so that
\begin{equation}
L_1\geq |\Omega|^{1/2} (L_0+ R) \label{e3.17}
\end{equation}
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
\begin{equation}
L_2\geq C_{9},  \quad
L_2\geq C_{10} \label{e3.18}
\end{equation}
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
\begin{equation}
\|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}
\end{equation}
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{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
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
\begin{equation}
\| \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}
\end{equation}
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{equation}
\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}
\end{equation}
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
\begin{equation}
\| \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}
\end{equation}
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 $s'<s$.  For $s'>\frac N 2+1$,
 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 $0< c_1<$ $a_1(\mathbf{x,}t)<c_2$, $0< c_1<$
$a_2(\mathbf{x,}t)<c_2$, and
$|\mathbf{v}(\mathbf{x},t)|< c_3$ for some constants
$c_1$, $c_2$, $c_3$, with $c_1<1$, $c_3>1$ 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
\begin{equation}
\|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}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
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 $s'<s$.

 From applying the Lebesgue dominated convergence theorem to
equations \eqref{A.51}, \eqref{A.52} and using a standard argument
(see, for example, Embid \cite{E1} and Majda \cite{M1}),
it follows that $\rho$, $\mathbf{u}$ is a classical solution of
\eqref{A.1}, \eqref{A.2}.
\end{proof}

\section{A priori estimates}

To obtain a priori estimates, 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. For convenience, we will  be
denoting derivatives by $f_\alpha =D^\alpha f$. And we will be
letting $Df$ denote the gradient of $f$. In addition, we will be
denoting the $L^2$ inner product by $(f,g)=\int_\Omega f\cdot g$
$d \mathbf{x}$. We will also be using the notation
$| f|_{L^{\infty},T}=\operatorname{ess\, sup}_{0\leq t\leq T}
|f(t)|_{L^{\infty}(\Omega)}$. The following lemmas will yield
the a priori estimates needed for the proof of Theorem \ref{T3.1}.


\begin{lemma}[Low-Norm Commutator Estimate] \label{L2.1}
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}

The proof of the above lemma is based on standard Sobolev calculus
inequalities and appears in   \cite{DD1}.
The next lemma provides the key a priori estimate for the
existence proof.

\begin{lemma} \label{L2.2}
Let $a_1$, $a_2$, $\mathbf{v}$, $\mathbf{F}$  be sufficiently
smooth given functions in the system of equations
\begin{gather}
\frac{\partial\rho }{\partial t}
= -\nabla \cdot \mathbf{u}  \label{e2.1} \\
\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+\mathbf{F}-Q_k \mathbf{g}
\end{aligned}\label{e2.2}
\end{gather}
where $Q_k$ is the orthogonal projection operator from Lemma
\ref{LA.1} in Appendix A and
\begin{equation}
Q_k\mathbf{g}= -Q_k(a_1^{-1}\mathbf{v}\cdot\nabla
\mathbf{u})-Q_k(a_1^{-1}(\nabla \cdot \mathbf{u})\mathbf{v})
+Q_k(a_1^{-2}(\mathbf{v}\cdot\nabla \rho)\mathbf{v})
-Q_k(a_2\nabla \rho)\quad \label{e2.39}
\end{equation}
and where $(Q_k \mathbf{g},\mathbf{u})=0$,
 $(\nabla \cdot(Q_k \mathbf{g})_{\alpha},\nabla \cdot
\mathbf{u}_{\alpha})=0$,
$(\nabla \times (Q_k\mathbf{g})_{\alpha},\nabla
\times\mathbf{u}_{\alpha})=0$ for $|\alpha| \geq 0$.
And $0< c_1<$
$a_1(\mathbf{x,}t)<c_2$, $0< c_1<$ $a_2(\mathbf{x,}t)<c_2$, and
$|\mathbf{v}(\mathbf{x},t)|< c_3$ for some constants
$c_1$, $c_2$, $c_3$, where $c_1<1$, $c_3>1$. 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 $0<c_1<a_1(\mathbf{x},t)<c_2$ and
$0<c_1<a_2(\mathbf{x,}t)<c_2$ yields
\begin{equation}
\begin{aligned}
&\| \mathbf{u}\| _0^2+\| \rho\| _0^2+\|\nabla \rho\| _0^2\\
&\leq  C_1\Big(\| \mathbf{u}_0\|_0^2+\| \rho_0\| _0^2
+\| \nabla \rho_0\| _0^2\Big)\\
&\quad + C_1 K_1 \int_0^t (\| \mathbf{u}\| _0^2+\|
\rho\| _0^2+\| \nabla \rho\| _0^2)d\tau
+C_1\int_0^t (\|D\mathbf{u}\|
_0^2+\|\mathbf{F}\|_0^2) d\tau
\end{aligned} \label{e2.6}
\end{equation}
where $C_1=\hat{C}_1(c,c_1,c_2)$, and where we define $K_1$, which
is an upper bound for the coefficients in \eqref{e2.4}, as
follows:
\begin{equation}
\begin{aligned}
K_1&=  \max \Big\{1, \; |a_1^{-1}|_{L^\infty,
T}^2|D \mathbf{v}| _{L^\infty, T }^2, \; | D
(a_1^{-1})| _{L^\infty, T }^2| \mathbf{v}|
_{L^\infty, T }^2, \; | a_1^{-1}| _{L^\infty, T
}^2| \mathbf{v}|
_{L^\infty, T }^2, \\
&\quad | a_1^{-2}| _{L^\infty, T }^2| \mathbf{v}|
_{L^\infty, T }^4, \; | D a_2| _{L^\infty, T }^2,
\; |(a_2)_t|_{L^\infty, T},
\; |(a_1)_t|_{L^\infty, T}\}
\end{aligned} \label{e2.7}
\end{equation}
where in $K_1$ we have used Cauchy's inequality
$fg\leq \frac 12(f^2 +g^2)$, with $g=1$ for some of the terms.
Applying Gronwall's inequality to \eqref{e2.6} yields
\begin{equation}
\begin{aligned}
\| \mathbf{u}\| _0^2+\| \rho\| _0^2+\|
\nabla \rho\| _0^2 &\leq  C_1(1+ C_1K_1Te^{C_1 K_1
T})(\| \mathbf{u}_0\|
_0^2+\| \rho_0\| _0^2+\| \nabla \rho_0\| _0^2) \\
&\quad +C_1(1+ C_1K_1Te^{C_1 K_1 T})\int_0^t (\|D
\mathbf{u}\| _0^2
+\|\mathbf{F}\|_0^2)d\tau
\end{aligned} \label{e2.8}
\end{equation}

Next, we will obtain estimates for $\nabla \cdot \mathbf{u}$ and
for $\nabla \times \mathbf{u}$. Recall that we use the notation
$f_{\alpha}=D^{\alpha}f$. We will let $C$ denote a generic
constant which may change from one instance to the next, but which
will depend only on $r$, where $|\alpha|\leq r$.

After applying the operator $D^\alpha $ to \eqref{e2.1}, \eqref{e2.2},
we obtain
\begin{gather}
\frac{\partial\rho_{\alpha} }{\partial t} =-\nabla \cdot
\mathbf{u}_{\alpha}
\label{e2.9}\\
\begin{aligned}
\frac{\partial \mathbf{u}_{\alpha}}{\partial t}
&= -a_1^{-1}\mathbf{v}\cdot\nabla \mathbf{u}_{\alpha}
-a_1^{-1}(\nabla \cdot
\mathbf{u}_{\alpha})\mathbf{v}+a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\mathbf{v} \\
&\quad -a_2\nabla\rho_{\alpha} +c\Big(\frac {1}{|\Omega|}\int_{\Omega}
a_1 d \mathbf{x}\Big)\nabla \Delta
\rho_{\alpha}-(Q_k\mathbf{g})_\alpha+\mathbf{G}_{\alpha}
\end{aligned}\label{e2.10}
\end{gather}
where we define $\mathbf{G}_{\alpha}$ as follows:
\begin{equation}
\begin{aligned}
\mathbf{G}_{\alpha}
&= \mathbf{F}_{\alpha}-[(a_1^{-1}\mathbf{v}\cdot \nabla
\mathbf{u})_\alpha -a_1^{-1}\mathbf{v}\cdot \nabla
\mathbf{u}_\alpha ] -[(a_1^{-1}(\nabla \cdot
\mathbf{u})\mathbf{v})_{\alpha}-a_1^{-1}(\nabla \cdot
\mathbf{u}_{\alpha})\mathbf{v}] \\
&\quad +[(a_1^{-2}(\mathbf{v}\cdot\nabla
\rho)\mathbf{v})_{\alpha}-a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\mathbf{v}]-[(a_2\nabla \rho)_{\alpha}-a_2\nabla
\rho_{\alpha}]
\end{aligned} \label{e2.11}
\end{equation}
Next, we will obtain an estimate for $\nabla \cdot \mathbf{u}$. We
apply the divergence operator to equation \eqref{e2.10}, and
obtain
\begin{equation}
\begin{aligned}
\frac{\partial \nabla \cdot\mathbf{u}_{\alpha}}{\partial t}
&= -2a_1^{-1}\mathbf{v}\cdot\nabla(\nabla \cdot
\mathbf{u}_{\alpha})-\nabla(a_1^{-1})\cdot(\mathbf{v}\cdot\nabla
\mathbf{u}_{\alpha}) -a_1^{-1}(\nabla\mathbf{v}^{T}:\nabla
\mathbf{u}_{\alpha})  \\
&\quad -(\nabla
\cdot\mathbf{u}_{\alpha})\mathbf{v}\cdot\nabla(a_1^{-1})
-a_1^{-1}(\nabla \cdot \mathbf{u}_{\alpha})\nabla \cdot\mathbf{v}
+(\mathbf{v}\cdot\nabla \rho_{\alpha})\mathbf{v}\cdot\nabla
(a_1^{-2}) \\
&\quad +a_1^{-2}\nabla(\mathbf{v}\cdot \nabla
\rho_{\alpha})\cdot\mathbf{v}+a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\nabla \cdot\mathbf{v} -\nabla \cdot(a_2\nabla
\rho_{\alpha})  \\
&\quad +c\left(\frac {1}{|\Omega|}\int_{\Omega} a_1 d \mathbf{x}\right)
\Delta^2 \rho_{\alpha}-\nabla \cdot (Q_k \mathbf{g})_\alpha+\nabla
\cdot \mathbf{G}_{\alpha}
\end{aligned} \label{e2.12}
\end{equation}
 From equation \eqref{e2.12}, and using the fact that $(\nabla
\cdot(Q_k \mathbf{g})_{\alpha},\nabla \cdot
\mathbf{u}_{\alpha})=0$ we obtain the estimate
\begin{align}
&\frac 12 \frac {d}{dt}\|\nabla \cdot
\mathbf{u}_{\alpha}\|_0^2 \nonumber \\
&= (\frac{\partial \nabla
\cdot\mathbf{u}_{\alpha}}{\partial t},\nabla \cdot
\mathbf{u}_{\alpha})  \nonumber \\
&= -2(a_1^{-1}\mathbf{v}\cdot\nabla(\nabla \cdot
\mathbf{u}_{\alpha}),\nabla \cdot
\mathbf{u}_{\alpha})-(\nabla(a_1^{-1})\cdot(\mathbf{v}\cdot\nabla
\mathbf{u}_{\alpha}),\nabla \cdot
\mathbf{u}_{\alpha}) \nonumber  \\
&\quad -(a_1^{-1}(\nabla\mathbf{v}^{T}:\nabla
\mathbf{u}_{\alpha}),\nabla \cdot \mathbf{u}_{\alpha})-((\nabla
\cdot\mathbf{u}_{\alpha})\mathbf{v}\cdot\nabla(a_1^{-1}),\nabla
\cdot \mathbf{u}_{\alpha})  \nonumber  \\
&\quad -(a_1^{-1}(\nabla \cdot \mathbf{u}_{\alpha})\nabla
\cdot\mathbf{v},\nabla \cdot \mathbf{u}_{\alpha})
+((\mathbf{v}\cdot\nabla \rho_{\alpha})\mathbf{v}\cdot\nabla
(a_1^{-2}),\nabla \cdot \mathbf{u}_{\alpha})  \nonumber \\
&\quad +(a_1^{-2}\nabla (\mathbf{v}\cdot\nabla
\rho_{\alpha})\cdot\mathbf{v},\nabla \cdot \mathbf{u}_{\alpha})
+(a_1^{-2}(\mathbf{v}\cdot\nabla \rho_{\alpha})\nabla
\cdot\mathbf{v},\nabla \cdot \mathbf{u}_{\alpha}) \nonumber \\
&\quad -(\nabla \cdot(a_2\nabla \rho_{\alpha}),\nabla \cdot
\mathbf{u}_{\alpha}) +(c(|\Omega|^{-1}\int_{\Omega} a_1 d
\mathbf{x})
\Delta^2 \rho_{\alpha},\nabla \cdot\mathbf{u}_{\alpha}) \nonumber  \\
&\quad -(\nabla \cdot(Q_k \mathbf{g})_{\alpha},\nabla \cdot
\mathbf{u}_{\alpha})+(\nabla \cdot \mathbf{G}_{\alpha},\nabla
\cdot \mathbf{u}_{\alpha}) \nonumber  \\
&= (\nabla \cdot(a_1^{-1}\mathbf{v})\nabla \cdot
\mathbf{u}_{\alpha},\nabla \cdot
\mathbf{u}_{\alpha})-(\nabla(a_1^{-1})\cdot(\mathbf{v}\cdot\nabla
\mathbf{u}_{\alpha}),\nabla \cdot
\mathbf{u}_{\alpha})  \nonumber \\
&\quad -(a_1^{-1}(\nabla\mathbf{v}^{T}:\nabla
\mathbf{u}_{\alpha}),\nabla \cdot \mathbf{u}_{\alpha}) -((\nabla
\cdot\mathbf{u}_{\alpha})\mathbf{v}\cdot\nabla(a_1^{-1}),\nabla
\cdot \mathbf{u}_{\alpha})  \nonumber \\
&\quad -(a_1^{-1}(\nabla \cdot \mathbf{u}_{\alpha})\nabla
\cdot\mathbf{v},\nabla \cdot \mathbf{u}_{\alpha})
+((\mathbf{v}\cdot\nabla \rho_{\alpha})\mathbf{v}\cdot\nabla
(a_1^{-2}),\nabla \cdot \mathbf{u}_{\alpha})  \nonumber  \\
&\quad +(a_1^{-2}\nabla(\mathbf{v}\cdot\nabla
\rho_{\alpha})\cdot\mathbf{v},\nabla \cdot \mathbf{u}_{\alpha})
+(a_1^{-2}(\mathbf{v}\cdot\nabla \rho_{\alpha})\nabla
\cdot\mathbf{v},\nabla \cdot
\mathbf{u}_{\alpha}) \nonumber  \\
&\quad +(\nabla\cdot(a_2\nabla \rho_{\alpha}),\rho_{t,\alpha})
-(c(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}) \Delta^2
\rho_{\alpha},\rho_{t,\alpha})  \nonumber \\
&\quad +(\nabla \cdot \mathbf{G}_{\alpha},\nabla \cdot
\mathbf{u}_{\alpha})   \label{e2.13} \\
&\leq  C(|D( a_1^{-1})| _{L^\infty }|
\mathbf{v}| _{L^\infty }+| a_1^{-1}| _{L^\infty
}| \nabla \cdot\mathbf{v}| _{L^\infty })\|\nabla
\cdot \mathbf{u}_{\alpha}\|_0^2  \nonumber \\
&\quad +C(|D( a_1^{-1})| _{L^\infty }|
\mathbf{v}| _{L^\infty }+| a_1^{-1}| _{L^\infty
}| D\mathbf{v}| _{L^\infty })\|\nabla \cdot
\mathbf{u}_{\alpha}\|_0\|D\mathbf{u}_{\alpha}\|_0 \nonumber  \\
&\quad +C(| \mathbf{v}| _{L^\infty }^2|
D(a_1^{-2})| _{L^\infty }+| a_1^{-2}| _{L^\infty
}| D\mathbf{v}| _{L^\infty }| \mathbf{v}|
_{L^\infty })\| \nabla \rho_{\alpha}\|_0\|\nabla
\cdot\mathbf{u}_{\alpha}\|_0   \nonumber \\
&\quad +C| a_1^{-2}| _{L^\infty }|\mathbf{v}|
_{L^\infty }^2\| \nabla \rho_{\alpha}\|_1\|\nabla
\cdot\mathbf{u}_{\alpha}\|_0 -(a_2\nabla
\rho_{\alpha},\nabla\rho_{t,\alpha})  \nonumber \\
&\quad -(c(|\Omega|^{-1}\int_{\Omega} a_1 d \mathbf{x}) \Delta
\rho_{\alpha},\Delta\rho_{t,\alpha})+\|\nabla \cdot
\mathbf{G}_{\alpha}\|_0\|\nabla \cdot
\mathbf{u}_{\alpha}\|_0  \nonumber  \\
&\leq  C(1+|D( a_1^{-1})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2+| a_1^{-1}| _{L^\infty
}^2| D\mathbf{v}| _{L^\infty }^2)\|\nabla \cdot
\mathbf{u}_{\alpha}\|_0^2   \nonumber \\
&\quad +C(| \mathbf{v}| _{L^\infty }^4|
D(a_1^{-2})| _{L^\infty }^2+ | a_1^{-2}|
_{L^\infty }^2| D\mathbf{v}| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2  )\|\nabla
\cdot\mathbf{u}_{\alpha}\|_0^2  \nonumber \\
&\quad +C\| \nabla \rho_{\alpha}\|_0^2 +C|
a_1^{-2}| _{L^\infty }^2|\mathbf{v}| _{L^\infty
}^4\|\nabla \cdot\mathbf{u}_{\alpha}\|_0^2 +C\|
\Delta \rho_{\alpha}\|_0^2 +C\|D\mathbf{u}_{\alpha}\|_0^2  \nonumber \\
&\quad -\frac 12 \frac {d}{dt}(c(|\Omega|^{-1}\int_{\Omega} a_1 d
\mathbf{x}) \Delta \rho_{\alpha},\Delta\rho_{\alpha}) +\frac
12(c(|\Omega|^{-1}\int_{\Omega} (a_1)_t d \mathbf{x}) \Delta
\rho_{\alpha},\Delta\rho_{\alpha})  \nonumber \\
&\quad -\frac {1}{2} \frac {d}{dt}(a_2\nabla
\rho_{\alpha},\nabla\rho_{\alpha})+\frac 12 ((a_2)_t\nabla
\rho_{\alpha},\nabla\rho_{\alpha})+C\|D
\mathbf{G}_{\alpha}\|_0^2
\end{align} 
where we used Cauchy's inequality $fg\leq \frac 12(f^2 +g^2)$, and
where for some of the terms, we let $g=1$. We also used the fact
that $\| \nabla \rho_{\alpha}\| _{1}^2\leq C\|
\Delta \rho_{\alpha}\| _{0}^2$ when $\Omega=\mathbb{T}^N$ (a
proof appears in   \cite{DD1}). And we used equation \eqref{e2.9}
to substitute for $\nabla \cdot \mathbf{u}_\alpha$.

Next, we estimate the term
$\|D \mathbf{G}_{\alpha}\|_0^2$ in \eqref{e2.13}. We apply the
$D^{\gamma}$ differentiation operator, where the multi-index
$|\gamma|=1$, to equation \eqref{e2.11} for $\mathbf{G}_{\alpha}$,
which yields
\begin{align*}
D^{\gamma}(\mathbf{G}_{\alpha})
&= \mathbf{F}_{\alpha+\gamma}-[(a_1^{-1}\mathbf{v}\cdot \nabla
\mathbf{u})_{\alpha+\gamma} -a_1^{-1}\mathbf{v}\cdot \nabla
\mathbf{u}_{\alpha+\gamma} ]   \\
&\quad +(a_1^{-1})_{\gamma}\mathbf{v}\cdot \nabla \mathbf{u}_{\alpha}
+a_1^{-1}\mathbf{v}_{\gamma}\cdot \nabla
\mathbf{u}_{\alpha}  \\
&\quad -[(a_1^{-1}(\nabla \cdot
\mathbf{u})\mathbf{v})_{\alpha+\gamma}-a_1^{-1}(\nabla \cdot
\mathbf{u}_{\alpha+\gamma})\mathbf{v}]  \\
&\quad +(a_1^{-1})_{\gamma}(\nabla \cdot
\mathbf{u}_{\alpha})\mathbf{v}+a_1^{-1}(\nabla \cdot
\mathbf{u}_{\alpha})\mathbf{v}_{\gamma}  \\
&\quad +[(a_1^{-2}(\mathbf{v}\cdot\nabla
\rho)\mathbf{v})_{\alpha+\gamma}-a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha+\gamma})\mathbf{v}]  \\
&\quad -(a_1^{-2})_{\gamma}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\mathbf{v}-a_1^{-2}(\mathbf{v}_{\gamma}\cdot\nabla
\rho_{\alpha})\mathbf{v}-a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\mathbf{v}_{\gamma}  \\
&\quad -[(a_2\nabla \rho)_{\alpha+\gamma}-a_2\nabla
\rho_{\alpha+\gamma}] +(a_2)_{\gamma}\nabla \rho_{\alpha}
\end{align*}
For $|\gamma|=1$ and $|\alpha|=k-1$, where
$0\leq k-1\leq r$, and by applying Lemma \ref{L2.1} to the
terms of the form $\|(fg)_{\alpha+\gamma}-fg_{\alpha+\gamma}\|_0^2$,
we obtain the estimate
\begin{equation}
\begin{aligned}
\| D^{\gamma}(\mathbf{G}_{\alpha})\|_0^2
&\leq C\| \mathbf{F}_{\alpha+\gamma}\|_{0}^2+C\|
(a_1^{-1}\mathbf{v}\cdot \nabla \mathbf{u})_{\alpha+\gamma}
-a_1^{-1}\mathbf{v}\cdot \nabla \mathbf{u}_{\alpha+\gamma}
\|_0^2 \\
&\quad +C\|(a_1^{-1})_{\gamma}\mathbf{v}\cdot \nabla
\mathbf{u}_{\alpha} \|_0^2+C\|
a_1^{-1}\mathbf{v}_{\gamma}\cdot \nabla \mathbf{u}_{\alpha}
\|_0^2 \\
&\quad +C\|(a_1^{-1}(\nabla \cdot
\mathbf{u})\mathbf{v})_{\alpha+\gamma}-a_1^{-1}(\nabla \cdot
\mathbf{u}_{\alpha+\gamma})\mathbf{v}
\|_0^2 \\
&\quad +C\|(a_1^{-1})_{\gamma}(\nabla \cdot
\mathbf{u}_{\alpha})\mathbf{v}
\|_0^2+C\|a_1^{-1}(\nabla \cdot
\mathbf{u}_{\alpha})\mathbf{v}_{\gamma}
\|_0^2 \\
&\quad +C\|(a_1^{-2}(\mathbf{v}\cdot\nabla
\rho)\mathbf{v})_{\alpha+\gamma}-a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha+\gamma})\mathbf{v}
\|_0^2 \\
&\quad +C\|(a_1^{-2})_{\gamma}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\mathbf{v} \|_0^2
+C\|a_1^{-2}(\mathbf{v}_{\gamma}\cdot\nabla
\rho_{\alpha})\mathbf{v}
\|_0^2 \\
&\quad +C\|a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\mathbf{v}_{\gamma} \|_0^2 +C\|(a_2\nabla
\rho)_{\alpha+\gamma}-a_2\nabla \rho_{\alpha+\gamma}
\|_0^2 \\
&\quad +C\|(a_2)_{\gamma}\nabla \rho_{\alpha} \|_0^2
 \\
&\leq  C\| \mathbf{F}\|_{k}^2 +C(\|
a_1^{-1}\| _{k_1}^2\| D\mathbf{v}\|
_{k_1}^2+\| D (a_1^{-1})\| _{k_1}^2\|
\mathbf{v}\| _{k_1}^2)\| D \mathbf{u}\|
_{k-1}^2 \\
&\quad +C(| D(a_1^{-1})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty}^2+| a_1^{-1}| _{L^\infty
}^2| D\mathbf{v}| _{L^\infty}^2)\| D
\mathbf{u}_{\alpha}\| _{0}^2 \\
&\quad +C(\| a_1^{-1}\| _{k_1}^2\| D\mathbf{v}\|
_{k_1}^2+\| D (a_1^{-1})\| _{k_1}^2\|
\mathbf{v}\| _{k_1}^2)\| \nabla \cdot\mathbf{u}\|
_{k-1}^2 \\
&\quad +C(| D(a_1^{-1})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty}^2+| a_1^{-1}| _{L^\infty
}^2| D\mathbf{v}| _{L^\infty}^2)\| \nabla \cdot
\mathbf{u}_{\alpha}\| _{0}^2 \\
&\quad +C(\|D(a_1^{-2})\| _{k_1}^2\| \mathbf{v}\|
_{k_1}^4+\|a_1^{-2}\| _{k_1}^2\|
D\mathbf{v}\| _{k_1}^2\| \mathbf{v}\|
_{k_1}^2)\| \nabla \rho\| _{k-1}^2 \\
&\quad +C(| D(a_1^{-2})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty}^4+| a_1^{-2}| _{L^\infty
}^2| D\mathbf{v}| _{L^\infty}^2| \mathbf{v}|
_{L^\infty }^2)\| \nabla \rho_{\alpha}\| _{0}^2 \\
&\quad +C\| D a_2\| _{k_1}^2\| \nabla \rho\|
_{k-1}^2 +C|D a_2| _{L^\infty }^2\| \nabla
\rho_{\alpha}\| _{0}^2  \\
&\leq  C\| \mathbf{F}\|_{k}^2 +C(\|
a_1^{-1}\| _{k_1}^2\| D\mathbf{v}\|
_{k_1}^2+\| D (a_1^{-1})\| _{k_1}^2\|
\mathbf{v}\| _{k_1}^2)\| D \mathbf{u}\|_{k-1}^2 \\
&\quad +C(\|D(a_1^{-2})\| _{k_1}^2\|
\mathbf{v}\| _{k_1}^4+\|a_1^{-2}\|
_{k_1}^2\| D\mathbf{v}\| _{k_1}^2\|
\mathbf{v}\| _{k_1}^2)\| \nabla \rho\| _{k-1}^2   \\
&\quad +\| D a_2\| _{k_1}^2\| \nabla \rho\|
_{k-1}^2
\end{aligned} \label{e2.16}
\end{equation}
where $k_1=$ max$\{k-1,s_0\}$ and
$s_0=[\frac N2]+1=2$ for $N=2$ or $N=3$.
Here, we used the Sobolev inequality
$|f|_{L^\infty} \leq C\|f\|_{s_0}$. We also
used the Sobolev calculus inequality $\| fg\| _{s}\leq
C\| f\| _{s}$ $\| g\| _{s}$ for $s>\frac N2$
(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{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
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 $0<c_1<a_1(\mathbf{x},t)<c_2$ and
$0<c_1<a_2(\mathbf{x},t)<c_2$. We also used the Sobolev inequality
$|f|_{L^\infty} \leq C\|f\|_{s_0}$.

Next, we obtain an estimate for $\nabla \times \mathbf{u}$.
Applying the curl operator to equation \eqref{e2.10} yields
\begin{equation}
\begin{aligned}
\frac{\partial \nabla \times \mathbf{u}_{\alpha}}{\partial t}
&= -a_1^{-1}\mathbf{v}\cdot\nabla (\nabla \times \mathbf{u}_{\alpha})
-(\nabla \cdot \mathbf{u}_{\alpha})\nabla
\times(a_1^{-1}\mathbf{v}) \\
&\quad -\nabla(\nabla \cdot
\mathbf{u}_{\alpha})\times(a_1^{-1}\mathbf{v}) +\nabla
\times(a_1^{-2}(\mathbf{v}\cdot\nabla
\rho_{\alpha})\mathbf{v}) \\
&\quad -\nabla a_2\times\nabla \rho_{\alpha}-\nabla \times (Q_k
\mathbf{g})_{\alpha} +\nabla \times
\mathbf{G}_{\alpha}+\mathbf{H}_{\alpha}
\end{aligned}\label{e2.19}
\end{equation}
where
\begin{equation}
\mathbf{H}_{\alpha}
=-[\nabla \times(a_1^{-1}\mathbf{v}\cdot\nabla
\mathbf{u}_{\alpha})-a_1^{-1}\mathbf{v}\cdot\nabla (\nabla \times
\mathbf{u}_{\alpha})]
\end{equation}
and where we estimate $\| \mathbf{H}_{\alpha}\|_0^2$ as
follows:
\begin{equation}
\begin{aligned}
\| \mathbf{H}_{\alpha}\|_0^2
&=  \| \nabla \times(a_1^{-1}\mathbf{v}\cdot\nabla
\mathbf{u}_{\alpha})-a_1^{-1}\mathbf{v}\cdot\nabla (\nabla \times
\mathbf{u}_{\alpha})\|_0^2 \\
&\leq  C(| a_1^{-1}| _{L^\infty }^2|
D\mathbf{v}| _{L^\infty }^2+|D( a_1^{-1})|
_{L^\infty }^2| \mathbf{v}| _{L^\infty
}^2)\|D\mathbf{u}_{\alpha} \|_0^2
\end{aligned} \label{e2.20}
\end{equation}
 From \eqref{e2.19}, and using the fact that
$(\nabla \times (Q_k\mathbf{g})_{\alpha},\nabla
\times\mathbf{u}_{\alpha})=0$, we obtain the estimate
\begin{equation}
\begin{aligned}
&\frac 12\frac {d}{dt} \|\nabla \times
\mathbf{u}_{\alpha}\|_0^2\\
&= (\frac{\partial (\nabla \times
\mathbf{u}_{\alpha})}{\partial t},\nabla \times
\mathbf{u}_{\alpha}) \\
&=  -(a_1^{-1}\mathbf{v}\cdot\nabla (\nabla \times
 \mathbf{u}_{\alpha}),\nabla \times
 \mathbf{u}_{\alpha})
-((\nabla \cdot \mathbf{u}_{\alpha})\nabla
\times(a_1^{-1}\mathbf{v}),\nabla
\times \mathbf{u}_{\alpha}) \\
&\quad -(\nabla(\nabla\cdot
\mathbf{u}_{\alpha})\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{\alpha}) \\
&\quad +(\nabla \times(a_1^{-2}(\mathbf{v}\cdot\nabla \rho_{\alpha})
\mathbf{v}),\nabla \times \mathbf{u}_{\alpha})
 -(\nabla a_2\times\nabla \rho_{\alpha},\nabla \times
\mathbf{u}_{\alpha})  \\
&\quad -(\nabla \times (Q_k\mathbf{g})_{\alpha},\nabla
\times\mathbf{u}_{\alpha})+(\nabla \times
\mathbf{G}_{\alpha},\nabla
\times\mathbf{u}_{\alpha})+(\mathbf{H}_{\alpha},\nabla \times
\mathbf{u}_{\alpha}) \\
&\leq  C(| a_1^{-1}| _{L^\infty }| \nabla \cdot
\mathbf{v}| _{L^\infty }+| D(a_1^{-1})|
_{L^\infty }|\mathbf{v}| _{L^\infty })\|\nabla
\times
\mathbf{u}_{\alpha}\|_0^2 \\
&\quad +C(| D(a_1^{-1})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2+| a_1^{-1}| _{L^\infty
}^2|\nabla\times \mathbf{v}| _{L^\infty
}^2)\|\nabla
\times \mathbf{u}_{\alpha}\|_0^2 \\
&\quad +C\| \nabla \cdot\mathbf{u}_{\alpha}\|_0^2
-(\nabla(\nabla\cdot
\mathbf{u}_{\alpha})\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{\alpha}) \\
&\quad +C(| a_1^{-2}| _{L^\infty }^2| \mathbf{v}|
_{L^\infty }^2| D\mathbf{v}| _{L^\infty }^2+|
D(a_1^{-2})| _{L^\infty }^2| \mathbf{v}|
_{L^\infty }^4)\|\nabla \times
\mathbf{u}_{\alpha}\|_0^2 \\
&\quad +| a_1^{-2}| _{L^\infty }^2| \mathbf{v}|
_{L^\infty }^4\|\nabla \times
\mathbf{u}_{\alpha}\|_0^2+C\|\nabla
\rho_{\alpha}\|_0^2+C\|\nabla
\rho_{\alpha}\|_1^2 \\
&\quad +C|D a_2| _{L^\infty }^2\|\nabla
\rho_{\alpha}\|_0^2+C\|\nabla \times
\mathbf{u}_{\alpha}\|_0^2
 +C\|D\mathbf{G}_{\alpha}\|_0^2
+C\|\mathbf{H}_{\alpha}\|_0^2
\end{aligned}\label{e2.21}
\end{equation}
where we used the fact that $-(a_1^{-1}\mathbf{v}\cdot\nabla
(\nabla \times \mathbf{u}_{\alpha}),\nabla \times
\mathbf{u}_{\alpha})=\frac 12((\nabla
\cdot(a_1^{-1}\mathbf{v}))(\nabla \times
\mathbf{u}_{\alpha}),\nabla \times \mathbf{u}_{\alpha})$.

Next, we estimate the term $-(\nabla(\nabla\cdot
\mathbf{u}_{\alpha})\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{\alpha})$ from \eqref{e2.21} above.
When $|\alpha|=0$, we obtain the estimate
\begin{equation}
-(\nabla(\nabla\cdot
\mathbf{u}_{\alpha})\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{\alpha})\leq C| a_1^{-1}| _{L^\infty
}^2| \mathbf{v}| _{L^\infty }^2\|\nabla \times
\mathbf{u}\|_0^2+C\|\nabla \cdot
\mathbf{u}\|_1^2 \label{e2.22}
\end{equation}
When $|\alpha|\geq 1$, we substitute equation \eqref{e2.9} for
$\nabla \cdot \mathbf{u}_{\alpha}$, to obtain the estimate
\begin{equation}
\begin{aligned}
&-(\nabla(\nabla\cdot
\mathbf{u}_{\alpha})\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{\alpha})\\
&= (\nabla\rho_{t,{\alpha}}\times(a_1^{-1}\mathbf{v}),\nabla
\times \mathbf{u}_{\alpha}) \\
&= \frac {d}{dt}(\nabla
 \rho_{\alpha}\times(a_1^{-1}\mathbf{v}),\nabla \times
 \mathbf{u}_{\alpha})
 -(\nabla \rho_{\alpha}\times((a_1^{-1})_t\mathbf{v}), \nabla
\times\mathbf{u}_{\alpha})  \\
&\quad -(\nabla \rho_{\alpha}\times(a_1^{-1}\mathbf{v}_t), \nabla
\times\mathbf{u}_{\alpha})
 -(\nabla \rho_{\alpha}\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{t,\alpha}) \\
&\leq \frac {d}{dt}(\nabla
 \rho_{\alpha}\times(a_1^{-1}\mathbf{v}),\nabla \times
 \mathbf{u}_{\alpha})
+C |( a_1^{-1})_{t}| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2\|\nabla\times
\mathbf{u}_{\alpha}\|_0^2
 \\
&\quad +C| a_1^{-1}| _{L^\infty }^2|
\mathbf{v}_{t}| _{L^\infty }^2\|\nabla\times
\mathbf{u}_{\alpha}\|_0^2+C\|\nabla
\rho_{\alpha}\|_0^2 \\
&\quad -(\nabla \rho_{\alpha}\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{t,\alpha})
\end{aligned}\label{e2.23}
\end{equation}
 and then we integrate by parts once to estimate the term
 $-(\nabla \rho_{\alpha}\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{t,\alpha})$ from \eqref{e2.23} above as follows:
\begin{equation}
\begin{aligned}
&-(\nabla \rho_{\alpha}\times(a_1^{-1}\mathbf{v}),\nabla \times
\mathbf{u}_{t,\alpha})\\
&=  (\nabla \rho_{\alpha}\times((a_1^{-1})_{\gamma}\mathbf{v}),\nabla
\times\mathbf{u}_{t,\alpha-\gamma})
 +(\nabla \rho_{\alpha}\times(a_1^{-1}\mathbf{v}_{\gamma}),\nabla
\times \mathbf{u}_{t,\alpha-\gamma})  \\
&\quad +(\nabla \rho_{\alpha+\gamma}\times(a_1^{-1}\mathbf{v}),\nabla
\times
\mathbf{u}_{t,\alpha-\gamma})  \\
&\leq C(| D(a_1^{-1})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2+|a_1^{-1}| _{L^\infty
}^2| D\mathbf{v}| _{L^\infty }^2)\|\nabla
\rho_{\alpha}\|_0^2\\
&\quad +C|a_1^{-1}| _{L^\infty }^2| \mathbf{v}|
_{L^\infty }^2\|\nabla
\rho_{\alpha+\gamma}\|_0^2+C\|\nabla \times
\mathbf{u}_{t,\alpha-\gamma}\|_0^2 \quad \quad\label{e2.24}
\end{aligned}
\end{equation}
where $|\gamma|=1$.  From  \eqref{e2.19}, we obtain the
estimate
\begin{equation}
\begin{aligned}
\|\nabla \times \mathbf{u}_{t,\alpha-\gamma}\|_0^2
&\leq
C|a_1^{-1}|_{L^\infty}^2|\mathbf{v}|_{L^\infty}^2
\|\nabla \times \mathbf{u}_{\alpha-\gamma}\|_1^2 \\
&\quad +C(|D(a_1^{-1})|_{L^\infty}^2|\mathbf{v}|_{L^\infty}^2
+C|a_1^{-1}|_{L^\infty}^2|\nabla
\times\mathbf{v}|_{L^\infty}^2)\|\nabla \cdot
\mathbf{u}_{\alpha-\gamma}\|_0^2 \\
&\quad +C|a_1^{-1}|_{L^\infty}^2|\mathbf{v}|_{L^\infty}^2\|\nabla
\cdot\mathbf{u}_{\alpha-\gamma}\|_1^2
+C|a_1^{-2}|_{L^\infty}^2|\mathbf{v}|_{L^\infty}^4
\|\nabla \rho_{\alpha-\gamma}\|_1^2 \\
&\quad +C(|D(a_1^{-2})|_{L^\infty}^2|\mathbf{v}|_{L^\infty}^4+
|a_1^{-2}|_{L^\infty}^2|D\mathbf{v}|_{L^\infty}^2
|\mathbf{v}|_{L^\infty}^2)\| \nabla
\rho_{\alpha-\gamma}\|_0^2 \\
&\quad +C|D a_2|_{L^\infty}^2\|\nabla
\rho_{\alpha-\gamma}\|_0^2 +C\|\nabla \times (Q_k
\mathbf{g})_{\alpha-\gamma}\|_0^2+C\|D\mathbf{G}_{\alpha-\gamma}\|_0^2
 \\
&\quad +C\|\mathbf{H}_{\alpha-\gamma}\|_0^2
\end{aligned}\label{e2.25}
\end{equation}
Note that if $|\alpha|=1$, then we choose $\gamma =\alpha$.

When we estimate the term $C\|\nabla \times (Q_k
\mathbf{g})\|_{r-1}^2\leq C\|Q_k \mathbf{g}\|_{r}^2$, which comes
from adding inequality \eqref{e2.25} over $1\leq |\alpha| \leq r$,
where $|\gamma|=1$, we will use the fact that $\|Q_k
\mathbf{f}\|_r^2$ $\leq \| \mathbf{f}\|_r^2 $, for any function $f
\in H^r(\Omega)$, which follows by the definition of the
projection operator $Q_k$ in Lemma \ref{LA.1}. And we will use the
definition \eqref{e2.39} of $Q_k \mathbf{g}$.

Integrating equation \eqref{e2.21} with respect to time, and using
the estimates \eqref{e2.16}, \eqref{e2.20},
\eqref{e2.22}-\eqref{e2.25} on the right-hand side, and using the
definition \eqref{e2.39} of $Q_k \mathbf{g}$, and adding over
$0\leq |\alpha|\leq r$, where $r\geq 1$, yields
\begin{align}
&\frac 12\|\nabla \times \mathbf{u}\|_r^2 \nonumber\\
& \leq \frac 12\|\nabla \times \mathbf{u}_0\|_r^2
+C\int_0^t (| a_1^{-1}| _{L^\infty }| \nabla
\cdot \mathbf{v}| _{L^\infty}+|D( a_1^{-1})|
_{L^\infty }|\mathbf{v}| _{L^\infty})\|\nabla
\times \mathbf{u}\|_r^2 d\tau  \nonumber\\
&\quad +C\int_0^t(| D(a_1^{-1})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2+| a_1^{-1}| _{L^\infty
}^2|\nabla\times \mathbf{v}| _{L^\infty
}^2)\|\nabla
\times \mathbf{u}\|_r^2 d\tau  \nonumber\\
&\quad +C\int_0^t(\|\nabla \rho\|_{r+1}^2+\| \nabla
\cdot\mathbf{u}\|_r^2+\|\nabla
\rho\|_r^2)d\tau  \nonumber \\
&\quad + \sum_{0\leq |\alpha|\leq r}|a_1^{-1}| _{L^\infty
}| \mathbf{v}| _{L^\infty }\|\nabla
\rho_{\alpha}\|_0\|\nabla \times
\mathbf{u}_{\alpha}\|_0  \nonumber \\
&\quad + \sum_{0\leq |\alpha|\leq
r}|a_1(\mathbf{x},0)^{-1}|_{L^\infty}|
\mathbf{v}_0| _{L^\infty }\|\nabla
 (\rho_0)_{\alpha}\|_0\|\nabla \times
(\mathbf{u}_0)_{\alpha}\|_0 \nonumber\\
&\quad +C\int_0^t (|( a_1^{-1})_{t}| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2+| a_1^{-1}| _{L^\infty
}^2| \mathbf{v}_{t}| _{L^\infty }^2)
\|\nabla\times \mathbf{u}\|_r^2d\tau \nonumber\\
&\quad +C\int_0^t(| a_1^{-2}| _{L^\infty }^2|
D\mathbf{v}| _{L^\infty }^2| \mathbf{v}|
_{L^\infty }^2+|D( a_1^{-2})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^4)\|\nabla \times
\mathbf{u}\|_r^2 d\tau \nonumber \\
&\quad +C\int_0^t(| a_1^{-2}| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^4+| a_1^{-1}| _{L^\infty
}^2| \mathbf{v}| _{L^\infty
}^2)\|\nabla\times \mathbf{u}\|_r^2d\tau \nonumber \\
&\quad +C\int_0^t(|D a_2| _{L^\infty
}^2+|D(a_1^{-1})| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2+|a_1^{-1}| _{L^\infty
}^2| D\mathbf{v}| _{L^\infty }^2)\|\nabla
\rho\|_r^2 d\tau  \nonumber \\
&\quad +C\int_0^t(|a_1^{-1}| _{L^\infty }^2|
\mathbf{v}| _{L^\infty }^2\|\nabla
\rho\|_{r+1}^2+\|\nabla \times
\mathbf{u}_{t}\|_{r-1}^2+\|\nabla
\times \mathbf{u}\|_r^2)d\tau  \nonumber\\
&\quad +C\sum_{0\leq |\alpha|\leq r}\int_0^t(\|D
\mathbf{G}_{\alpha}\|_0^2
+\|\mathbf{H}_{\alpha}\|_0^2)d\tau \nonumber \\
&\leq  (\frac 12 +\epsilon)\|\nabla \times
\mathbf{u}_0\|_r^2+\frac {1}{4\epsilon}
|a_1(\mathbf{x},0)^{-1}|_{L^\infty}^2|
\mathbf{v}_0| _{L^\infty }^2\|\nabla
 \rho_0\|_r^2 \nonumber\\
&\quad +\epsilon\|\nabla \times
\mathbf{u}\|_r^2+\frac{1}{4\epsilon}|a_1^{-1}|
_{L^\infty }^2| \mathbf{v}| _{L^\infty }^2\|\nabla
\rho\|_r^2 \nonumber \\
&\quad +C K_3\int_0^t(\|D \mathbf{u}\|_r^2+\|\nabla
\rho\|_{r}^2 +\|\Delta \rho\|_r^2)d\tau+C
\int_0^t\| \mathbf{F}\|_{r+1}^2 d\tau
\label{e2.26}
\end{align}
where we used Cauchy's inequality with $\epsilon$, namely $fg\leq
\frac{1}{4\epsilon} f^2 +\epsilon g^2$, where we choose
$\epsilon=1/4$, and where we define $K_3$, which is an upper bound
for the coefficients, as follows
\begin{equation}
\begin{aligned}
K_3&= \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, \;
\| D a_2\| _{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^{-1}\| _{q,T}^2\| \mathbf{v}\|_{q,T}^2, \;
\|  a_2\| _{q,T}^2, \;
\|a_1^{-2}\| _{q,T}^2\| \mathbf{v}\|_{q,T}^4,\;
| a_1^{-1}| _{L^\infty, T }^2|\mathbf{v}|_{L^\infty, T }^2,  \\
&| a_1^{-2}| _{L^\infty, T }^2| \mathbf{v}|_{L^\infty, T }^4, \;
|( a_1^{-1})_{t}| _{L^\infty, T }^2| \mathbf{v}| _{L^\infty, T }^2, \;
| a_1^{-1}| _{L^\infty, T }^2| \mathbf{v}_{t}|
_{L^\infty, T }^2 \}
\end{aligned} \label{e2.27}
\end{equation}
where $q=\max\{r,s_0\}$, where $r\geq 1$, and where
$s_0=[\frac N2]+1=2$ for $N=2$ or $N=3$.

After multiplying estimate \eqref{e2.26} by $\beta$, where
$0<\beta<1$ is a constant, and then adding the resulting
inequality to the estimate \eqref{e2.17} for $\nabla \cdot
\mathbf{u}$, and using the fact that $\epsilon=1/4$, and using the
fact that $\| D \mathbf{u}\| _{r}^2 = \|\nabla
\cdot \mathbf{u}\|_r^2+\|\nabla \times
\mathbf{u}\|_r^2$, which follows from the identity
$\Delta\mathbf{u}_{\alpha}=\nabla(\nabla
\cdot\mathbf{u}_{\alpha})-\nabla \times(\nabla
\times\mathbf{u}_{\alpha})$, we obtain
\begin{equation}
\begin{aligned}
&\frac {\beta}{4}\| D \mathbf{u}\| _{r}^2+\| \nabla
\rho\|_r^2+ \| \Delta \rho\|_r^2 \\
&=  \beta(\frac 12-\epsilon) (\|\nabla \times \mathbf{u}\|_r^2+
\|\nabla \cdot \mathbf{u}\|_r^2)+\| \nabla
\rho\|_r^2 +\|\Delta \rho\|_r^2 \\
&\leq C_3(\|\nabla \times
\mathbf{u}_0\|_r^2+\|\nabla \cdot
\mathbf{u}_0\|_r^2+\| \nabla \rho_0\|_r^2+
\| \Delta \rho_0\|_r^2)\\
&\quad +\frac{\beta}{4\epsilon}|a_1(\mathbf{x},0)^{-1}|
_{L^\infty }^2| \mathbf{v}_0| _{L^\infty
}^2\|\nabla \rho_0\|_r^2
+\frac{\beta}{4\epsilon}|a_1^{-1}| _{L^\infty
}^2| \mathbf{v}| _{L^\infty }^2\|\nabla \rho\|_r^2 \\
&\quad +C_3 K_4\int_0^t (\| D \mathbf{u}\| _{r}^2+\|
\nabla \rho\|_r^2+\| \Delta \rho\|_r^2)d\tau
+C_3\int_0^t\| \mathbf{F}\|_{r+1}^2 d\tau
\end{aligned}\label{e2.28}
\end{equation}
where $C_3=\hat{C}_3(r,c,c_1,c_2)$, and where we define
\begin{equation}
\begin{aligned}
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 }, \quad \| (a_2)_t\| _{2, T }
\Big\}
\end{aligned} \label{e2.29}
\end{equation}
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 used
the Sobolev inequality $|f|_{L^\infty} \leq
C\|f\|_{s_0}$. Note that $K_2 \leq K_4$ and $K_3\leq
K_4$.

Next, using the fact that $0<c_1<a_1(\mathbf{x},t)<c_2$ where
$c_1<1$, and using the fact that $|
\mathbf{v}(\mathbf{x},t)|< c_3$, where $c_3
>1$, 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{equation}
\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}
\end{equation}
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}

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\end{document}