\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 39, pp. 1--23.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/39\hfil Existence and uniqueness of global solutions]
{Existence and uniqueness of global solutions \\ to a model for
the flow of an incompressible, barotropic fluid with capillary
effects}

\author[D. L. Denny\hfil EJDE-2007/39\hfilneg]
{Diane L. Denny}

\address{Diane Denny \newline
Department of Mathematics and Statistics\\
Texas A\&M University - Corpus Christi\\
Corpus Christi, TX 78412, USA}
\email{diane.denny@tamucc.edu}

\thanks{Submitted August 10, 2006. Published March 6, 2007.}
\subjclass[2000]{35A05}
\keywords{Existence; uniqueness; capillary; incompressible;
\hfill\break\indent inviscid fluid}

\begin{abstract}
 We study the initial-value problem for a system of nonlinear
 equations that models the flow of an inviscid, incompressible,
 barotropic fluid with capillary stress effects. We prove the
 global-in-time existence of a unique, classical solution to this
 system of equations, with a small initial velocity gradient. The
 key to the proof lies in using an $L^2$ estimate for the density
 $\rho$, and using the smallness of the initial velocity gradient,
 to obtain uniqueness for the density.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

In this paper, we consider equations which arise from a model of
the multi-dimensional flow of an incompressible, barotropic fluid
with capillary stresses. When viscosity is neglected, these
equations reduce to the following system, written in terms of the
density $\rho$, the pressure $p$, and velocity $\mathbf{v}$:
\begin{gather}
\frac{D\mathbf{v}}{Dt} + \rho^{-1} \nabla p = c\nabla \Delta \rho,
\label{e1.1} \\
\nabla\cdot \mathbf{v} = 0. \label{e1.2}
\end{gather}
Here $c$ is a coefficient of capillarity which is a small,
positive constant, and the material derivative $D/{Dt}=
\partial/{\partial t}+\mathbf{v}\cdot\nabla$.
The term $c\nabla \Delta \rho$ arises from capillary stresses, as
described in the theory of Korteweg-type materials developed by
Dunn and Serrin \cite{d2}. The fluid's thermodynamic state is
determined by the density $\rho$. The pressure $p$ is determined
from the density by an equation of state $p=\hat{p}(\rho)$. In
related work, the existence of a solution to a similar system of
equations for the case of viscous fluid flow, which also includes
a hyperbolic equation for density and  a parabolic equation for
temperature, has been proven by Hattori and Li \cite{h1,h2}, and
by Bresch, Desjardins, and Lin \cite{b1}. Anderson, McFadden and
Wheeler \cite{a1} have given a review of related theories and
applications to diffuse-interface modelling.

In this model, it will be shown that $\partial \rho/\partial t$,
$\nabla \rho$, and $\nabla \mathbf{v}$ are small, for suitable
initial data. Although the conservation of mass equation is only
approximately satisfied, the model equations \eqref{e1.1}, \eqref{e1.2} might be
useful as an approximation in the case of almost constant density,
and nearly incompressible fluid flow with a small velocity
gradient. We write equation \eqref{e1.1} equivalently as
\begin{equation}
\frac{D\mathbf{v}}{Dt} + \rho^{-1} \hat{p}' (\rho)\nabla \rho =
c\nabla \Delta \rho \label{e1.3}
\end{equation}

The purpose of this paper is to prove the existence of a unique,
global-in-time, classical solution $\mathbf{v}$, $\rho$ to
equations \eqref{e1.2}, \eqref{e1.3} with suitable initial velocity data
$\mathbf{v}_0 \in H^s$, under periodic boundary conditions. That
is, we choose for our domain the N-dimensional torus
$\mathbb{T}^N$, where $N=2$ or $N=3$. The proof of the existence
theorem is based on the method of successive approximations, in
which an iteration scheme, based on solving a linearized version
of the equations, is designed and convergence of the sequence of
approximating solutions to a unique solution satisfying the
nonlinear equations is sought. The framework of the proof follows
one used, for example, by A. Majda for proving the existence of a
solution to a system of conservation laws \cite{m1}.  Embid \cite{e1} also
uses the same general framework to prove the existence of a
solution to equations for zero Mach number combustion. Under this
framework, the convergence proof is presented in two steps. In the
first step, we prove uniform boundedness of the approximating
sequence of solutions in a high Sobolev norm. The second step is
to prove contraction of the sequence in a low Sobolev norm.
Standard compactness arguments will be used to finish the proof.
The key to the proof lies in using an $L^2$ estimate for the
density $\rho$, and using the smallness of the initial velocity
gradient, to obtain uniqueness for the density.

\section{A priori estimates}

The main tools utilized in the existence proof are a priori estimates.
We will work with the Sobolev space $H^s(\Omega )$ (where $s\geq 0$ is an
integer) of real-valued functions in $L^2(\Omega )$ whose distribution
derivatives up to order $s$ are in $L^2(\Omega )$, with norm given by
$\|f\|_s^2=\sum_{|\alpha |\leq
s}\int_\Omega |D^\alpha f|^2dx$ and inner product
$(f,g)_s=\sum_{|\alpha |\leq s}\int_\Omega (D^\alpha
f)\cdot (D^\alpha g)dx$. Here, we adopt the standard multi-index
notation. For convenience, we will denote derivatives by $f_\alpha
=D^\alpha f$. We will let $Df$ denote the gradient of $f$. Also,
we will denote the $L^2$ inner product by $(f,g)=\int_\Omega
f\cdot g$ $dx$. We will use standard function spaces. $L^\infty
([0,T],H^s)$ is the space of bounded measurable functions from
$[0,T]$ into $H^s(\Omega)$, with the norm $\|f\|
_{s,T}^2=\mathop{\rm ess\,sup}_{0\leq t\leq T}\|f(t)\|_s^2$.
$C([0,T],H^s)$ is the space of continuous functions
from $[0,T]$ into $H^s(\Omega )$. The following technical lemmas
will be needed for the proof of the existence of a classical
solution to the initial-value problem for the system
\eqref{e1.2}, \eqref{e1.3}.

\begin{lemma}[Standard Calculus Inequalities] \label{lem2.1} \quad
\begin{itemize}
\item[(a)] If $f\in H^{s_1}(\Omega)$, $g\in H^{s_2}(\Omega)$ and
$s_3=\min\{s_1,s_2, s_1+s_2-s_{0}\}\geq 0$, where
$s_0=[\frac N2] +1$, then $fg\in H^{s_3}(\Omega)$, and
$\|fg\|_{s_3}\leq C\|f\|_{s_1}\|g\|_{s_2}$. We note that $s_0=2$
for $N=2$ or $N=3$. \item[(b)] If $f\in H^s(\Omega)$, $g\in
H^{s-1}(\Omega)\cap L^\infty(\Omega) $, $Df\in L^\infty(\Omega)$,
and $| \alpha | \leq s$, then
$\| D^\alpha (fg)-fD^{\alpha}g\|_0\leq C(|Df|_{L^\infty}
\|g\|_{s-1}+|g|_{L^\infty}\|Df\|_{s-1})$.
\end{itemize}
\end{lemma}

In $(a)$ the constant $C$ depends on $s_1$, $s_2$, and $\Omega$,
while in $(b)$ the constant $C$ depends on $s$ and $\Omega$. These
inequalities are well known. Proofs may be found, for example, in
\cite{k1,m2}.

\begin{lemma}[Low-Norm Commutator Estimate]\label{lem2.2}
If $Df\in H^{r_1}(\Omega )$, $g\in H^{r-1}(\Omega )$, where
$r_1=\max\{r-1,s_0\}$, $s_0=[ \frac N2]+1$, then for
any $r\geq 1$, $f,g$ satisfy the estimate
$\|D^\alpha (fg)-fD^\alpha g\|_0\leq C\|Df\|_{r_1}\|g\|_{r-1}$,
where $r=|\alpha |$, and the constant $C$ depends on $r$,
$\Omega $.
\end{lemma}

\begin{proof}
 The proof is based on the Sobolev calculus
inequalities from Lemma \ref{lem2.1}. We
consider separately the cases $r-1<s_0$ and $r-1\geq s_0$, where
$r\geq 1$.  If $r-1<s_0$, we expand the term $D^\alpha (fg)$ using
the Leibniz rule and then apply inequality (a) from Lemma \ref{lem2.1} to
obtain the desired estimate. If $r-1\geq s_0$, we apply the
inequality (b) from Lemma \ref{lem2.1} and the Sobolev
inequality $|h|_{L^{\infty} }\leq C\|h\|_{s_0}$ for
$s_0=[ \frac N2]+1$, to obtain the estimate for this
case. Combining these two results then completes the proof.
\end{proof}

\begin{lemma}\label{lem2.3}
If $\mathbf{u}$, $\mathbf{v}$, $a$, $\mathbf{f}$, and $\rho$ are
sufficiently smooth in
\begin{gather*}
{D\mathbf{u}}/{Dt} = -a\nabla \rho +c\nabla \Delta \rho
+\mathbf{f},\\
\nabla \cdot \mathbf{u}=0,
\end{gather*}
where $\mathbf{u(x},0) =\mathbf{u}_0(\mathbf{x})$, $\nabla \cdot
\mathbf{u}_0=0$, $\Omega=\mathbb{T}^N$, where $N=2,3$, and where
$c$ is a positive constant, $D/{Dt}=
\partial/{\partial t}+\mathbf{v\cdot }\nabla $,
and $\nabla \cdot \mathbf{v}=0$, then for any $r\geq 1$,
$D\mathbf{u}$ and $\mathbf{u}$ satisfy the estimates
\[
\|D\mathbf{u}\|_{r-1}^2 \leq
Ce^{2t}(1+te^{2t}e^{\beta (t)}\|D\mathbf{v}\|
_{r_1,T}^2)(\|D\mathbf{u}_0\|_{r-1}^2+ \int_0^t(\|
\mathbf{f}\|_{r}^2 +\|D a\|_{r_1}^2\|
\nabla\rho \|_{r-1}^2 )d\tau)
\]
and
\begin{align*}
\|\mathbf{u}\|_r^2
& \leq e^{2t}\|\mathbf{u}_0\|_0^2 +Ce^{2t}(1+te^{2t}e^{\beta
(t)}\|
D\mathbf{v}\|_{r_1,T}^2)(\|D\mathbf{u}_0\|_{r-1}^2  \\
&\quad +\int_0^t (\|\mathbf{f}\|_{r}^2 +\|D a\|
_{r_1}^2\|\nabla\rho \|_{r-1}^2 )d\tau)
\end{align*}
where $\beta (t)=te^{2t}\|D\mathbf{v}\|_{r_1,T}^2 $.
Here $r_1=\max\{r-1,s_0\}$, and $s_0=[ \frac N2]+1=2$
for $N=2,3$. The constant $C$ depends on $r$, $\Omega$.
\end{lemma}

\begin{proof}
First, we obtain an $L^2$ estimate for $\mathbf{u}$.
Let $\bar{\rho}=\rho- \frac {1}{|\Omega|} \int_{\Omega}
\rho d\mathbf{x}$. We then compute
\begin{equation} \label{e2.1}
\begin{aligned}
\frac 12\frac d{dt}\|\mathbf{u}\|_0^2
&=(\mathbf{u}_t,\mathbf{u})\\
&=-(\mathbf{v}\cdot \nabla \mathbf{u,u})-(a\nabla \rho
 ,\mathbf{u})+c(\nabla \Delta \rho,\mathbf{u})+(\mathbf{f},\mathbf{u})   \\
&=\frac 12(\mathbf{u}\nabla \cdot \mathbf{v},\mathbf{u})+(a
 \bar{\rho},\nabla
 \cdot \mathbf{u})+(\bar{\rho} \nabla a,\mathbf{u})
 -c(\Delta \rho,\nabla \cdot \mathbf{u})+(\mathbf{f},\mathbf{u})   \\
&\leq  |Da|_{L^\infty }\|\mathbf{u}
\|_0\|\bar{\rho} \|_0 +\|
\mathbf{f}\|_0\|\mathbf{u} \|_0  \\
&\leq  \|\mathbf{u} \|_0^2+ \frac 12 |Da|_{L^\infty }^2\|
\bar{\rho} \|_0^2 +\frac 12\|\mathbf{f}\|_0^2
\end{aligned}
\end{equation}
where we used the facts that $\nabla \cdot \mathbf{u}=0$, and
$\nabla \cdot \mathbf{v}=0$. And we used Cauchy's inequality.
After applying Gronwall's inequality, we obtain the estimate
\begin{equation} \label{e2.2}
\|\mathbf{u}\|_{0}^2 \leq
e^{2t}\|\mathbf{u}_0\|_0^2+ e^{2t}\int_0^tC(|
Da|_{L^\infty }^2\|\bar{ \rho} \|
_{0}^2+\|\mathbf{f}\|_{0}^2)d\tau,
\end{equation}

Next, we obtain an estimate for $\|D\mathbf{u}\|
_{r-1}^2$. After applying the operator $D^{\gamma+\alpha}$ to the
equation for $\mathbf{u}$, where $0\leq |\alpha|\leq r-1$ and
$|\gamma|=1$, we obtain
\begin{equation} \label{e2.3}
\frac{D\mathbf{u}_{\gamma+\alpha}}{Dt}= -a\nabla \rho
_{\gamma+\alpha} +c\nabla \Delta \rho_{\gamma+\alpha}
+\mathbf{F}_{\gamma+\alpha}
\end{equation}
where $\mathbf{F}_{\gamma+\alpha} =\mathbf{f}_{\gamma+\alpha}
-[(\mathbf{v}\cdot \nabla \mathbf{u})_{\gamma+\alpha}
-\mathbf{v}\cdot \nabla \mathbf{u}_{\gamma+\alpha} ]-[(a\nabla
\rho)_{\gamma+\alpha} -a\nabla \rho _{\gamma+\alpha} ]$. For
\eqref{e2.3}, estimate \eqref{e2.1} becomes
\begin{equation} \label{e2.4}
\frac 12\frac d{dt}\|\mathbf{u}_{\gamma+\alpha} \|
_0^2 \leq \|\mathbf{u}_{\gamma+\alpha} \|_0^2+\frac 12 |Da|
_{L^\infty }^2\|\bar{\rho}_{\gamma+\alpha} \|_0^2
+\frac 12 \|\mathbf{F}_{\gamma+\alpha}\|_0^2
\end{equation}
Next, we estimate $\|\mathbf{F}_{\gamma+\alpha}  \|_0^2
$. Using the commutator estimate from Lemma \ref{lem2.2}, we obtain
\begin{equation} \label{e2.5}
\begin{aligned}
\|\mathbf{F}_{\gamma+\alpha} \|_0^2
&\leq C\| \mathbf{f}_{\gamma+\alpha}\|_0^2 +C\|(\mathbf{v}\cdot
\nabla \mathbf{u})_{\gamma+\alpha} -\mathbf{v}\cdot \nabla
\mathbf{u}_{\gamma+\alpha} \|_0^2 \\
&\quad +C\|(a\nabla\rho)_{\gamma+\alpha} -a\nabla
  \rho_{\gamma+\alpha}\|_0^2    \\
&\leq C\|\mathbf{f}\|_{k}^2 +C\|D\mathbf{v}\|_{k_1}^2
  \|D \mathbf{u}\|_{k-1}^2
+C\|D a\|_{k_1}^2\|\nabla\rho \|_{k-1}^2
\end{aligned}
\end{equation}
where $k=|\gamma+ \alpha |$, $k_1=\max\{k-1,s_0\}$,
and $s_0=[ \frac N2]+1$. Here, we used the triangle
inequality and Cauchy's inequality. Substituting estimate \eqref{e2.5}
into \eqref{e2.4} yields
\begin{align*}
\frac 12\frac d{dt}\|\mathbf{u}_{\gamma+\alpha} \|_0^2
&\leq \|\mathbf{u}_{\gamma+\alpha} \|_0^2+\frac 12 |Da|
_{L^\infty }^2\|\bar{\rho}_{\gamma+\alpha} \|_0^2
+C\|\mathbf{f}\|_k^2 \\
&\quad +C\|D\mathbf{v}\|_{k_1}^2\|D \mathbf{u}\|_{k-1}^2
+C\|D a\|_{k_1}^2\|\nabla\rho \|_{k-1}^2
\end{align*}
Applying Gronwall's inequality yields the following:
\begin{equation} \label{e2.6}
\begin{aligned}
\|\mathbf{u}_{\gamma+\alpha}\|_{0}^2
&\leq e^{2t}\|(\mathbf{u}_0)_{\gamma+\alpha}\|_0^2+
Ce^{2t}\int_0^t\Big(|Da|_{L^\infty }^2\|
\bar{\rho}_{\gamma+\alpha} \|_0^2+\|
\mathbf{f}\|_{k}^2 \\
&\quad +\|D\mathbf{v}\|_{k_1}^2\|D\mathbf{u}\|_{k-1}^2
 +\|D a\|_{k_1}^2\|\nabla\rho \|_{k-1}^2 \Big)d\tau  \\
&\leq Ce^{2t}\|D\mathbf{u}_0\|_{k-1}^2+ Ce^{2t}\int_0^t
\Big(|Da|_{L^\infty }^2\|\bar{\rho} \|_{k}^2+\|\mathbf{f}\|_{k}^2 \\
&\quad+\|D\mathbf{v}\|_{k_1}^2\|D\mathbf{u}\|_{k-1}^2
 +\|D a\|_{k_1}^2\|\nabla\rho \|_{k-1}^2\Big)d\tau ,
\end{aligned}
\end{equation}
where $|\gamma|=1$, and $|\gamma+\alpha|=k \geq 1$, and
$k_1=\max \{k-1,s_0\}$, with $s_0=[ \frac N2]+1$.

After adding the above inequality \eqref{e2.6} over all $\gamma$, where
$|\gamma|=1$, and then adding over all $\alpha$, where $0 \leq
|\alpha|\leq r-1$, we obtain the estimate
\begin{equation} \label{e2.7}
\begin{aligned}
\|D\mathbf{u}\|_{r-1}^2
&\leq Ce^{2t}\|D\mathbf{u}_0\|_{r-1}^2+
Ce^{2t}\int_0^t\Big(|Da|_{L^\infty }^2\|\bar{\rho}
\|_{r}^2+\|\mathbf{f}\|_{r}^2 \\
&\quad+\|D\mathbf{v}\|_{r_1,T}^2\|D\mathbf{u}\|_{r-1}^2
 +\|D a\|_{r_1}^2\|\nabla\rho \|_{r-1}^2 \Big)d\tau ,
\end{aligned}
\end{equation}
where $r_1=\max\{r-1,s_0\}$, and where
 $s_0=[ \frac N2] +1=2$ for $N=2,3$. Here, we used the fact
that $\sum_{0\leq |\alpha| \leq r-1}\sum_{|\gamma|=1}\|\mathbf{u}
_{\gamma+\alpha} \|_0^2=\sum_{0\leq |\alpha| \leq
r-1}\sum_{i=1}^N \int_{\Omega}|\frac {\partial
\mathbf{u}_{\alpha}}{\partial x_i}  |^2
d\mathbf{x}=\sum_{0\leq
|\alpha| \leq r-1}\|D\mathbf{u}
_{\alpha} \|_0^2=\|D\mathbf{u}\|_{r-1}^2 $.
After applying Gronwall's inequality to \eqref{e2.7}, we get
\begin{equation} \label{e2.8}
\begin{aligned}
\|D\mathbf{u}\|_{r-1}^2
& \leq Ce^{2t}(1+te^{2t}e^{\beta (t)}\|D\mathbf{v}\|
_{r_1,T}^2)(\|D\mathbf{u}_0\|_{r-1}^2 \\
&\quad +\int_0^t(|Da|_{L^\infty }^2\|\bar{\rho}
\|_{r}^2+\|\mathbf{f}\|_{r}^2 +\|D
a\|_{r_1}^2\|\nabla\rho \|_{r-1}^2 )d\tau) ,
\end{aligned}
\end{equation}
with $\beta (t)=te^{2t}\|D\mathbf{v}\|_{r_1,T}^2 $.
Adding the estimates \eqref{e2.2}, \eqref{e2.8}, we obtain
\begin{align*}
\|\mathbf{u}\|_r^2 &\leq (\|
\mathbf{u}\|_0^2+C\|D\mathbf{u}\|_{r-1}^2)  \\
&\leq e^{2t}\|\mathbf{u}_0\|_0^2
+Ce^{2t}(1+te^{2t}e^{\beta (t)}\|D\mathbf{v}\|
_{r_1,T}^2)(\|D\mathbf{u}_0\|_{r-1}^2  \\
&\quad +\int_0^t (|Da|_{L^\infty }^2\|\bar{\rho}
\|_{r}^2+\|\mathbf{f}\|_{r}^2 +\|D
a\|_{r_1}^2\|\nabla\rho \|_{r-1}^2 )d\tau)
 \\
 &\leq  e^{2t}\|\mathbf{u}_0\|_0^2
+Ce^{2t}(1+te^{2t}e^{\beta (t)}\|D\mathbf{v}\|
_{r_1,T}^2)(\|D\mathbf{u}_0\|_{r-1}^2  \\
&\quad +\int_0^t (\|\mathbf{f}\|_{r}^2 +\|D a\|
_{r_1}^2\|\nabla\rho \|_{r-1}^2 )d\tau)
\end{align*}
where $r_1=\max\{r-1,s_0\}$, and where $s_0=[ \frac N2]
+1=2$ for $N=2,3$. Here, we used Poincar\'{e}'s inequality to
estimate $\|\bar{\rho} \|_0^2\leq C\|\nabla\rho
\|_0^2$,  and $\|\bar{\rho} \|_r^2\leq
C(\|\nabla\rho \|_{r-1}^2+\|\bar{\rho} \|
_0^2)\leq C\|\nabla\rho \|_{r-1}^2$. We also used
Sobolev's lemma $|h|_{L^{\infty}}\leq C\|h
\|_{s_0}$ where $s_0=[ \frac N2]+1$. Using the
estimate $|Da|_{L^\infty }^2\|\bar{\rho}
\|_{r}^2 \leq C\|D a\|_{r_1}^2\|
\nabla\rho \|_{r-1}^2 $ in the right-hand side of \eqref{e2.8}
completes the proof.
\end{proof}

\begin{lemma}\label{lem2.4}
Let $u$, $w$ be $C^1$ functions on a bounded, open, connected,
convex domain $\Omega$. And let $u(\mathbf{x}_0)=w(\mathbf{x}_0)$
at a single, fixed point $\mathbf{x}_0 \in \Omega$. Then $u-w$ and
$u$ satisfy the estimates
\begin{gather*}
\|u -w\|_{0}^2\leq C\|\nabla (u -w)\|_{2}^2, \\
\|u \|_{0}^2 \leq C_0\|w\|_{0}^2+C_0\|\nabla w\|_{2 }^2+C_0\|\nabla u
\|_{2 }^2
\end{gather*}
Here $C$, $C_0$ are constants which depend only on $\Omega$.
\end{lemma}

\begin{proof}
First, we obtain an estimate for $\|u
-w\|_{0 }^2$. From the mean value theorem, and since $u
-w$ is sufficiently smooth on the convex domain ${\Omega }$, we
have
\[
(u -w)(\mathbf{x})=(u -w)(\mathbf{x}_0)+\nabla (u
-w)(\mathbf{x}_{*})\cdot (\mathbf{x-x}_0)
\]
where $\mathbf{x}_{*}$ is a point on the line segment joining the points
$\mathbf{x}_0\in {\Omega }$ and $\mathbf{x}\in {\Omega }$.

Since we are given that $u (\mathbf{x}_0)=w(\mathbf{x}_0)$, at a
single fixed point $\mathbf{x}_0\in {\Omega }$, it follows that
\[
(u -w)(\mathbf{x})=\nabla (u -w)(\mathbf{x}_{*})\cdot (\mathbf{x-x}_0)
\]
Taking the absolute value of both sides yields
\begin{align*}
|(u -w)(\mathbf{x})|
&=|\nabla (u -w)(\mathbf{x}_{*})\cdot (\mathbf{x-x}_0)|\\
&\leq |\nabla (u -w)|_{L^\infty }|\mathbf{x-x} _0|\\
&\leq C|\nabla (u -w)|_{L^\infty }
\end{align*}
Here $C$ depends only on $\Omega $. Squaring $|(u -w)(
\mathbf{x})|$ and integrating over ${\Omega }$, and using
the above inequality, yields
\[
\int_{{\Omega }}|(u -w)(\mathbf{x})|^2d\mathbf{x}
\leq C\int_{{\Omega }}|\nabla (u -w)|_{L^\infty }^2d
\mathbf{x}
\leq C\|\nabla (u -w)\|_{2 }^2
\]
where we used Sobolev's inequality $|h|_{L^\infty
}\leq C\|h\|_{s_0}$, where $s_0=[ \frac
N2]+1=2$ for $N=2,3$ and $C$ depends on $\Omega$.

Next, we obtain an estimate for $\|u \|_{0}^2$. From
using the triangle inequality and Cauchy's inequality, and from
using the previous estimate for $\|u-w \|_{0}^2 $, we
obtain
\begin{align*}
\|u \|_{0}^2 &\leq  C\|w\|_{0}^2+C\|u-w\|_{0 }^2  \\
&\leq C\|w\|_{0 }^2+C\|\nabla (u-w) \|_{2 }^2  \\
&\leq C_0\|w\|_{0 }^2+C_0\|\nabla w \|_{2 }^2+C_0\|\nabla u \|_{2 }^2
\end{align*}
Here $C_0$ is a constant which depends only on $\Omega $.
\end{proof}

\begin{lemma}\label{lem2.5}
If $g$ is a sufficiently smooth function on the domain
$\Omega=\mathbb{T}^N$, then $g$ satisfies the estimate $\|
\nabla g\|_r^2\leq C\|\Delta g\|_{r-1}^2$
where $r\geq 1$  and $C$ depends on $r$.
\end{lemma}

\begin{proof}
First, we integrate $-\Delta g$ by parts with the function
$\bar{g}=g-\frac 1{|\Omega |}\int_\Omega g \, d\mathbf{x}$
over $\Omega=\mathbb{T}^N$, to obtain
\begin{align*}
(\nabla g,\nabla g) &=-(\Delta g, \bar{g}) \\
&\leq C(\epsilon)\|\Delta g\|_{0 }^2+\epsilon\|
\bar{g} \|_{0 }^2\\
&\leq C(\epsilon)\|\Delta g\|_{0 }^2+\epsilon C\|
\nabla g \|_{0 }^2
\end{align*}
where we used Cauchy's inequality with $\epsilon$. We also used
Poincar\'{e}'s inequality to estimate
$\|\bar{g} \|_0^2\leq C\|\nabla g \|_0^2$.
Then $D^{\alpha}$ is
applied, yielding $-\Delta g_{\alpha}$, which is then integrated
by parts with the function $g_{\alpha}$ over
$\Omega=\mathbb{T}^N$, to obtain
\begin{align*}
(\nabla g_{\alpha},\nabla g_{\alpha})
&=-(\Delta g_{\alpha}, g_{\alpha})\\
&=(\Delta g_{\alpha-\gamma}, g_{\alpha+\gamma})  \\
&\leq C(\epsilon)\|\Delta g_{\alpha-\gamma}\|_{0
}^2+\epsilon\|g_{\alpha+\gamma} \|_{0}^2\\
&\leq C(\epsilon)\|\Delta g\|_{k-1}^2+\epsilon C\|\nabla g \|_{k}^2
\end{align*}
where $|\gamma|=1$, and $|\alpha|=k \geq 1$. Here, we used
Cauchy's inequality with $\epsilon$. Adding these two
inequalities, for $|\alpha|=k \leq r$, and moving the terms
containing $\epsilon$ to the left-hand side, completes the proof.
\end{proof}

\begin{lemma}\label{lem2.6}
If $\mathbf{u}$, $\mathbf{v}$, $a$, $\mathbf{f}$, and $\rho$ are
sufficiently smooth in the equation
\begin{equation} \label{e2.9}
\Delta ^2\rho =\frac 1c\nabla \cdot (a\nabla \rho)
+\frac1c\nabla \cdot (\mathbf{v}\cdot \nabla \mathbf{u}) -\frac 1c
\nabla \cdot \mathbf{f}
\end{equation}
where $c$ is a positive constant, $\nabla \cdot \mathbf{u}=0$,
$\nabla \cdot \mathbf{v}=0$,
$a(\mathbf{x},t)\geq c_1$, with $c_1>0$, and where
$\Omega = \mathbb{T}^N$, $N=2,3$, and $r \geq 1$, we obtain the
following estimates for $\|\Delta \rho \|_0^2+\|\nabla\rho\|_0^2$
and for $\|\nabla\rho\|_{r+1}^2$
\begin{gather*}
\|\Delta \rho \|
_0^2+\|\nabla\rho\|_0^2\leq C|D
\mathbf{v}|_{L^{\infty}}^2\|D\mathbf{u}\|_{0}^2
+C\|\mathbf{f}\|_{0}^2,
\\
\begin{aligned}
\|\nabla\rho\|_{r+1}^2
&\leq C( \|\Delta \rho \|_r^2+\|\nabla\rho\|_r^2)\\
&\leq C\| Da\|_{r_1}^2 \|\nabla \rho \|_{r-1}^2
+C\|\mathbf{f}\|_{r-1}^2+C\|D\mathbf{v}\|
_{r_2+1}^2\|D\mathbf{u}\|_{r-1}^2
\end{aligned}
\end{gather*}
where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$,
with $s_0=[\frac N2 ]+1$, and $N=2,3$.
\end{lemma}

\begin{proof} First, we obtain an $L^2$ estimate. Integrating
equation \eqref{e2.9} by parts with $\bar{\rho} $, where
$\bar{\rho}=\rho-\frac 1{|\Omega |}\int_\Omega \rho  d
\mathbf{x}$, yields
\begin{align*}
(\Delta \rho ,\Delta \rho )
&=(\Delta ^2\rho ,\bar{\rho} )= \frac
1c(\nabla \cdot (a\nabla \rho),\bar{\rho} )+\frac 1c (\nabla \cdot
(\mathbf{v}\cdot \nabla \mathbf{u}),\bar{\rho} ) - \frac 1c(\nabla
\cdot \mathbf{f},\bar{\rho})
   \\
&=-\frac 1c(a\nabla \rho,\nabla \rho )+\frac 1c (\nabla
\mathbf{v}^T: \nabla \mathbf{u} ,\bar{\rho} )+
\frac 1c(\mathbf{f},\nabla \rho) \\
&\leq - \frac {c_1}{c} \|\nabla \rho\|_0^2
+C(\epsilon)|D \mathbf{v}|_{L^\infty }^2\|
D\mathbf{u}\|_0^2+\epsilon\|\bar{\rho} \|_0^2
+C(\epsilon)\|\mathbf{f}\|_0^2+\epsilon\|\nabla
\rho \|_0^2  \\
&\leq - \frac {c_1}{c} \|
\nabla \rho\|_0^2 +C(\epsilon)|D \mathbf{v}|
_{L^\infty }^2\|D\mathbf{u}\|_0^2+\epsilon\|
\nabla \rho \|_0^2
+C(\epsilon)\|\mathbf{f}\|_0^2+\epsilon\|\nabla
\rho \|_0^2
\end{align*}
where we used Cauchy's inequality with $\epsilon$, and where we
used the fact that $a(\mathbf{x},t) \geq c_1$ where $c_1>0$. We
also used Poincar\'{e}'s inequality to estimate
$\|\bar{\rho} \|_0^2\leq C\|\nabla \rho \|_0^2$.
And we used the fact that $\nabla \cdot (\mathbf{v}\cdot \nabla
\mathbf{u})=\nabla \mathbf{v}^T: \nabla \mathbf{u} $, because
$\nabla \cdot \mathbf{u}=0$. Moving the $\|\nabla \rho
\|_0^2$ terms to the left-hand side yields the estimate
\begin{equation} \label{e2.10}
\|\Delta \rho \|
_0^2+\|\nabla\rho\|_0^2\leq C|D
\mathbf{v}|_{L^{\infty}}^2\|D\mathbf{u}\|_{0}^2
+C\|\mathbf{f}\|_{0}^2
\end{equation}
Next, after applying $D^\alpha $ to the equation \eqref{e2.9}, we obtain
the equation:
\begin{equation} \label{e2.11}
\Delta ^2\rho _\alpha =\frac 1c\nabla \cdot (a\nabla
\rho_\alpha)+\frac 1c \nabla \cdot(\mathbf{v}\cdot \nabla
\mathbf{u})_\alpha +F_\alpha
\end{equation}
where $F_\alpha =-\frac 1c\nabla \cdot \mathbf{f}_{\alpha}+\frac
1c[\nabla \cdot (a\nabla \rho)_\alpha -\nabla \cdot (a\nabla
\rho_\alpha )]$. Integrating equation \eqref{e2.11} by parts with $\rho
_\alpha$, when $|\alpha|\geq 1$, yields
\begin{equation} \label{e2.12}
\begin{aligned}
(\Delta \rho _\alpha ,\Delta \rho _\alpha )
&=(\Delta ^2\rho_\alpha ,\rho _\alpha )\\
&=\frac 1c(\nabla \cdot(a\nabla
\rho_\alpha), \rho _\alpha )+\frac 1c(\nabla \cdot(\mathbf{v}\cdot
\nabla \mathbf{u})_{\alpha} , \rho _{\alpha} )+(F_\alpha ,\rho _\alpha )   \\
&= -\frac 1c(a\nabla \rho_\alpha, \nabla \rho _\alpha )+\frac
1c((\nabla \mathbf{v}^T: \nabla \mathbf{u})_{\alpha} ,
\rho _{\alpha}) +(F_{\alpha} ,\rho _{\alpha} )   \\
&= -\frac 1c(a\nabla \rho_\alpha, \nabla \rho _\alpha )-\frac
1c((\nabla \mathbf{v}^T: \nabla \mathbf{u})_{\alpha-\gamma} ,
\rho _{\alpha+\gamma}) +(F_{\alpha} ,\rho _{\alpha} )   \\
&\leq  -\frac {c_1}{c}(\nabla \rho_\alpha ,\nabla \rho _\alpha)
+C\|(\nabla \mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma
}\|_0\|\rho_{\alpha +\gamma }\|_0 +|
(F_{\alpha} ,\rho_{\alpha} )| \\
&\leq  -\frac {c_1}{c}(\nabla \rho_\alpha ,\nabla \rho _\alpha)
+C(\epsilon) \|(\nabla \mathbf{v}^T: \cdot\nabla
\mathbf{u})_{\alpha -\gamma }\|_0^2 +\epsilon
\|\rho_{\alpha +\gamma }\|_0^2+|(F_{\alpha}
,\rho_{\alpha} )|
\end{aligned}
\end{equation}
where $|\gamma|  = 1$. Here, we used Cauchy's inequality with
$\epsilon$, and we used the fact that $a(\mathbf{x},t) \geq c_1$
where $c_1>0$. Next, we estimate the terms on the right-hand side
of the above inequality. First, we estimate
$\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma
}\|_0^2$. When $|\alpha|=1$, we choose $\gamma =\alpha$ and
obtain
\begin{equation} \label{e2.13}
\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma
}\|_0^2=\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})\|_0^2
\leq C|D\mathbf{v}|_{L^{\infty}}^2\|D\mathbf{u}\|_0^2
\end{equation}
Next, we use the triangle inequality and the commutator estimate
from Lemma \ref{lem2.2} to estimate $\|(\nabla\mathbf{v}^T : \nabla
\mathbf{u})_{\alpha -\gamma }\|_0^2$, when
$|\alpha|>|\gamma|=1$, obtaining
\begin{equation} \label{e2.14}
\begin{aligned}
\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma
}\|_0^2
&\leq C( \|\nabla \mathbf{v}^T: \nabla
\mathbf{u}_{\alpha -\gamma }\|_0^2
+\|(\nabla\mathbf{v}^T : \nabla \mathbf{u})_{\alpha -\gamma
}-\nabla\mathbf{v}^T: \nabla
\mathbf{u}_{\alpha -\gamma }\|_0^2)  \\
&\leq  C|D\mathbf{v}|_{L^{\infty}}^2\|
D\mathbf{u}_{\alpha -\gamma} \|_0^2 +C\|D^{2}
\mathbf{v}\|_{k_2}^2\|D\mathbf{u}\|_{k-2}^2
 \\
&\leq  C\|D\mathbf{v}\|_{k_2+1}^2\|
D\mathbf{u}\|_{k-1}^2,
\end{aligned}
\end{equation}
Here $|\gamma|=1$, $k=|\alpha |$, and $k_2=\max
\{k-2,s_0\}$, with $s_0=[\frac N2 ]+1$. Here, we also
used the Sobolev inequality $|h|_{L^{\infty}} \leq
C\|h\|_{s_0}$.

Next, we estimate the following term from the right-hand side of
\eqref{e2.12}, obtaining
\begin{equation} \label{e2.15}
\epsilon \|\rho_{\alpha +\gamma }\|_0^2 \leq \epsilon C
\|\nabla \rho\|_k^2
\end{equation}
where $|\gamma|  = 1$ and $|\alpha|=k$.

Next, we use the commutator estimate from Lemma \ref{lem2.2} to estimate
$|(F_{\alpha} ,\rho _{\alpha} )|$ from the right-hand
side of \eqref{e2.12}, obtaining
\begin{equation} \label{e2.16}
\begin{aligned}
|(F_{\alpha} ,\rho _{\alpha} )|
&\leq |\frac 1c(\nabla \cdot \mathbf{f}_{\alpha},\rho_{\alpha})|
+|\frac 1c([\nabla \cdot (a\nabla \rho)_{\alpha} -\nabla
\cdot (a\nabla \rho_{\alpha} )],\rho _{\alpha} )|
 \\
&= |\frac 1c(\mathbf{f}_{\alpha-\gamma},\nabla
\rho_{\alpha+\gamma})|+|\frac 1c([(a\nabla
\rho)_{\alpha} -a\nabla \rho_{\alpha} ],\nabla \rho
_{\alpha} )| \\
&\leq  C\|\mathbf{f}\|_{k-1}\|\nabla
\rho\|_{k+1}+ C\|Da\|_{k_1}\|\nabla \rho
\|_{k-1}\|\nabla \rho \|_k  \\
&\leq C(\epsilon)\|\mathbf{f}\|_{k-1}^2+\epsilon\|
\nabla \rho \|_{k+1}^2+ C(\epsilon)\|Da\|
_{k_1}^2\|\nabla \rho \|_{k-1}^2+\epsilon\|
\nabla \rho \|_k^2
\end{aligned}
\end{equation}
where $|\gamma|  = 1$, $k=|\alpha |$, and
$k_1=\max \{k-1,s_0\}$, with $s_0=[\frac N2 ]
+1$. Again, we used Cauchy's inequality with $\epsilon$.
Substituting \eqref{e2.13}-\eqref{e2.16} into \eqref{e2.12}, and adding \eqref{e2.12} over
$|\alpha |=k\leq r$, including the $L^2$ estimate
\eqref{e2.10}, we obtain for $r\geq 1$ the estimate
\begin{equation} \label{e2.17}
\begin{aligned}
\|\Delta \rho \|_r^2 +\|\nabla \rho \|_r^2
&\leq  C\|Da\|_{r_1}^2 \|\nabla \rho
\|_{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\|
D\mathbf{v}\|_{r_2+1}^2\|D\mathbf{u}\|_{r-1}^2  \\
&\quad +\epsilon C\|\nabla \rho \|_{r+1}^2+\epsilon
C\|\nabla \rho\|_{r}^2,
\end{aligned}
\end{equation}
where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$,
with $s_0=[\frac N2 ]+1$. Here, we also used the
Sobolev inequality $|h|_{L^{\infty}} \leq
C\|h\|_{s_0}$.

 From Lemma \ref{lem2.5}, we have $\epsilon \|\nabla \rho \|
_{r+1}^2\leq \epsilon C\|\Delta \rho \|_{r}^2$. After
substituting this estimate into the right-hand side of \eqref{e2.17}, we
obtain the estimate
\begin{equation} \label{e2.18}
\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2
\leq C\|Da\|_{r_1}^2 \|\nabla\rho \|
_{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\|
D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2,
\end{equation}
where we have moved the terms $\epsilon C\|\Delta \rho
\|_r^2$ and $\epsilon C\|\nabla \rho \|_r^2$
to the left-hand side. Finally, using the estimate for $\|
\nabla \rho \|_{r+1}^2$ from Lemma \ref{lem2.5}, we obtain from
\eqref{e2.18} the estimate
\begin{align*}
\|\nabla \rho \|_{r+1}^2
&\leq C(\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2) \\
&\leq C\| Da\|_{r_1}^2 \|\nabla\rho \|_{r-1}^2
+C\|\mathbf{f}\|_{r-1}^2 +C\|D\mathbf{v}\|
_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2
\end{align*}
\end{proof}

\begin{lemma}\label{lem2.7}
If $\mathbf{u}$, $\mathbf{v}$, $a$, $\mathbf{f}$, and $\rho$ are
sufficiently smooth in
\begin{equation} \label{e2.19}
\Delta ^2\rho =\frac 1c\nabla \cdot (a\nabla \rho)
+\frac1c\nabla \cdot (\mathbf{v}\cdot \nabla \mathbf{u}) -\frac 1c
\nabla \cdot \mathbf{f}
\end{equation}
where $c$ is a positive constant,  $\nabla \cdot \mathbf{u}=0$,
$\nabla \cdot \mathbf{v}=0$, $a(\mathbf{x},t) \geq c_1$, with
$c_1>0$, and $\Omega = \mathbb{T}^N$, $N=2,3$, then for $r \geq 1$
, $\rho$ satisfies the following estimate
\begin{align*}
\|\nabla \rho \|_{r+1}^2
&\leq C\left(\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2 \right) \\
&\leq C\Big[1+\sum_{j=1}^{r}\|Da\| _{r_1}^{2j}\Big](\|\mathbf{f}\|_{r-1}^2
+\| D\mathbf{v}\|_{r_2+1}^2 \|D \mathbf{u}\|_{r-1}^2 )
\end{align*}
where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$,
with $s_0=[\frac N2 ]+1$, and $C$ depends on $r$.
\end{lemma}

\begin{proof} From Lemma \ref{lem2.6} applied to equation \eqref{e2.19}, we have
the estimate
\begin{equation} \label{e2.20}
\|\Delta \rho \|_s^2+\|\nabla \rho \|_s^2
\leq C\|Da\|_{s_1}^2 \|\nabla\rho \|
_{s-1}^2 +C\|\mathbf{f}\|_{s-1}^2 +C\|
D\mathbf{v}\|_{s_2+1}^2\|D \mathbf{u}\|_{s-1}^2
\end{equation}
where $s \geq 1$, and where $s_1=\max\{s-1,s_0\}$,
$s_2=\max \{s-2,s_0\}$, with $s_0=[\frac N2 ]
+1$. Letting $s=r$ in the estimate \eqref{e2.20} yields
\begin{equation} \label{e2.21}
\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2
\leq C\|Da\|_{r_1}^2 \|\nabla\rho \|
_{r-1}^2 +C\|\mathbf{f}\|_{r-1}^2 +C\|
D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2
\end{equation}
Applying the estimate \eqref{e2.20}, letting $s=r-1$, to the term
$C\|Da\|_{r_1}^2 \|\nabla\rho \|
_{r-1}^2$ which appears on the right-hand side of \eqref{e2.21} yields
\begin{equation} \label{e2.22}
\begin{aligned}
\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2
&\leq C\|Da\|_{r_1}^2\Big[\|Da\|_{r_2}^2 \|\nabla\rho \|_{r-2}^2
+\|\mathbf{f}\|_{r-2}^2  +\|D\mathbf{v}\|
_{r_3+1}^2\|D\mathbf{u}\|_{r-2}^2 \Big]
\\
&\quad +C\|\mathbf{f}\|_{r-1}^2 +C\|D\mathbf{v}\|
_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2  \\
&\leq C\|Da\|_{r_1}^4 \|\nabla\rho \|_{r-2}^2 +C\|Da\|_{r_1}^2
\Big(\|\mathbf{f}\|_{r-2}^2 +\|
D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|
_{r-2}^2\Big)
 \\
&\quad+C\|\mathbf{f}\|_{r-1}^2 +C\|
D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2
\end{aligned}
\end{equation}
where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$,
and $r_3=\max \{r-3,s_0\}$, $ r_3\leq r_2 \leq r_1$, with
$s_0=[\frac N2 ]+1=2$ for $N=2,3$. Similarly, by
repeatedly applying the estimate \eqref{e2.20}, letting $s=r-j$, to the
term $ C\|Da\|_{r_1}^{2j}\|\nabla\rho \|
_{r-j}^2$, for $j=2,3,\dots ,r-1$, which will appear on the
right-hand side of \eqref{e2.22} yields
\begin{equation} \label{e2.23}
\begin{aligned}
\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2
&\leq C\sum_{j=1}^{r-1}\|Da\|_{r_1}^{2j}
\Big(\|\mathbf{f}\|_{r-1-j}^2 +\|
D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|
_{r-1-j}^2 \Big) \\
&\quad  +C\|Da\|_{r_1}^{2r}\|\nabla\rho \|_{0}^2
 +C\|\mathbf{f}\|_{r-1}^2 +C\|D\mathbf{v}\|_{r_2+1}^2
\|D \mathbf{u}\|_{r-1}^2  \\
&\leq C\Big[1+\sum_{j=1}^{r-1}\|Da\|
_{r_1}^{2j}\Big]\Big(\|\mathbf{f}\|_{r-1}^2 +\|
D\mathbf{v}\|_{r_2+1}^2\|D \mathbf{u}\|_{r-1}^2\Big)
 \\
&\quad +C\|Da\|_{r_1}^{2r}\|\nabla\rho \|_{0}^2
\end{aligned}
\end{equation}
Substituting the estimate for $\|\nabla \rho\|_0^2$
from Lemma \ref{lem2.6} into the right-hand side of \eqref{e2.23} yields
\begin{align*}
\|\Delta \rho \|_r^2+\|\nabla \rho \|_r^2
&\leq C\big[1+\sum_{j=1}^{r-1}\|Da\|_{r_1}^{2j}\Big]
\Big(\|\mathbf{f}\|_{r-1}^2 +\| D\mathbf{v}\|_{r_2+1}^2
\|D \mathbf{u}\|_{r-1}^2 \Big) \\
&\quad +C\|Da\|_{r_1}^{2r}\Big(\|\mathbf{f}\|_0^2 +|
D\mathbf{v}|_{L^\infty}^2\|D\mathbf{u}\|_{0}^2 \Big) \\
&\leq C\Big[1+\sum_{j=1}^{r}\|Da\|_{r_1}^{2j}\Big]
\Big(\|\mathbf{f}\|_{r-1}^2 +\| D\mathbf{v}\|_{r_2+1}^2
\|D \mathbf{u}\|_{r-1}^2\Big)
\end{align*}
This completes the proof.
 \end{proof}

\section{Existence theorem}

 In this section, we prove the existence
of a unique classical solution to the initial-value problem for
equations \eqref{e1.2}, \eqref{e1.3}, on any given time interval $0\leq t\leq
T$, with periodic boundary conditions, for sufficiently small
initial velocity gradient.

\begin{theorem}\label{thm3.1}
Suppose $s>\frac N2+3$ and $\Omega=\mathbb{T}^N$, $N=2,3$. For any
given time interval $0 \leq t\leq T$, equations \eqref{e1.2}, \eqref{e1.3} have
a unique classical solution $\rho$, $\mathbf{v}$, for initial data
$\mathbf{v}_0(\mathbf{x})\in H^{s}(\Omega )$, $\nabla \cdot
\mathbf{v}_0=0$, and for given data $\rho^0(\mathbf{x},t)$ and
$\mathbf{x}_0$, provided $D \mathbf{v}_0$ is sufficiently small.
Here $\rho^0(\mathbf{x},t)$ is a given positive function with
sufficiently small gradient $\nabla \rho^0$, and $\mathbf{x}_0$ is
a point in the domain $\Omega$. The regularity of the solution is
\begin{gather*}
\rho \in C([0,T],C^5)\cap L^\infty([0,T],H^{s+2}),  \\
\mathbf{v} \in C([0,T],C^3)\cap  L^\infty([0,T],H^s),  \\
\frac{\partial \mathbf{v}}{\partial t} \in C([0,T],C^2)\cap
L^\infty([0,T],H^{s-1}).
\end{gather*}
and $\rho(\mathbf{x},t) \geq c_1$, and $\rho^{-1} \hat{p}'
(\rho)(\mathbf{x},t) \geq c_1$, for some positive constant $c_1 $,
for $\mathbf{x}\in \Omega$, and $0\leq t \leq T$.
\end{theorem}

\begin{proof} We will construct the solution of the problem for
\eqref{e1.2}, \eqref{e1.3}, with $\Omega = \mathbb{T}^N$, through an iteration
scheme. To define the iteration scheme, we will let the sequence
of approximate solutions be $\mathbf{v}^k$ and $\rho^k$. Set
$\mathbf{v}^0(\mathbf{x},t)=\mathbf{v}_0(\mathbf{x})$, the initial
velocity data, and set $\rho^0(\mathbf{x},t)\in C([0,T],C^5)\cap
L^\infty([0,T],H^{s+2})$ to be a positive function satisfying
$\|\nabla \rho ^{0}\|_{s+1,T} \leq \|D
\mathbf{v}_0\|_{s-1}$.  For $k=0,1,2,\dots $, construct
$\mathbf{v}^{k+1}$, $ \rho^{k+1}$ from the previous iterates
$\mathbf{v}^k$, $\rho ^k$ by solving the linear system of
equations
\begin{gather}
\frac{D^k\mathbf{v}^{k+1}}{Dt}+(\rho^k)^{-1} \hat{p}'
(\rho^k)\nabla \rho ^{k+1} = c\nabla \Delta \rho
^{k+1}, \label{e3.1} \\
\nabla \cdot \mathbf{v}^{k+1} =0, \label{e3.2}
\end{gather}
where $D^k/{Dt}= \partial/{\partial t}+\mathbf{v}^k \cdot\nabla$,
and with initial data
$\mathbf{v}^{k+1}(\mathbf{x}, 0)=\mathbf{v}_0(\mathbf{x})$. Since
the solution $\rho ^{k+1} $ is unique up to an arbitrary function
of $t$, we specify that the solution $\rho ^{k+1} $ satisfy $\rho
^{k+1}(\mathbf{x}_0,t) =\rho ^{k}(\mathbf{x}_0,t) $ at a single,
fixed point $\mathbf{x}_0 \in \Omega$, for all $k\geq 0$. Hence
$\rho ^{k+1}(\mathbf{x}_0,t)  = \rho ^{0}(\mathbf{x}_0,t) $.

Existence of a sufficiently smooth solution to equations
\eqref{e3.1}, \eqref{e3.2} for fixed $k$ follows from the results
stated in Section 4. We proceed now to prove convergence of the
iterates as $k\to \infty $ to a unique classical solution of
\eqref{e1.2}, \eqref{e1.3}. We assume that $p$ is a sufficiently
smooth function of the thermodynamic state variable $\rho$ in an
open interval $G\subset \mathbb{R}$. We fix connected, bounded
open sets $G_0$ and $G_1$ such that $\bar{G}_0\subset G_1$ and
$\bar{G}_1\subset G$, and we require that the initial iterate
$\rho^0$ satisfies $\rho^0 \in G_0$. We fix $\delta
=\hat{\delta}(G_0,G_1)$ so that $0 < \delta <
\mbox{dist}(\bar{G}_0,\partial{G_1})$; therefore if $|\rho^k -\rho
^0|_{L^\infty } \leq \delta $, then $\rho^{k}(\mathbf{x},t)\in
G_1$ for all $\mathbf{x}\in \mathbb{T}^N$, and for $t \in [0,T]$.
The values of $\rho^k \in G_1$ are assumed to be strictly
positive, bounded, and bounded away from zero. And the values of
$(\rho^k)^{-1} \hat{p}' (\rho^k)$ for $\rho^k \in G_1$ are also
assumed to be strictly positive, bounded, and bounded away from
zero. Using a proof by induction on $k$, we assume that $\rho^k
\in G_1$, and then later we will show that $\rho^{k+1} \in G_1$.
First, we proceed with the proof of uniform boundedness of the
approximating sequence in a high Sobolev norm.

\begin{proposition}\label{prop3.1}
Assume that the hypotheses of Theorem \ref{thm3.1} hold. Let $\rho^0$,
$\mathbf{v}_0$ satisfy $C_0\|\rho^0\|_{0
}^2+C_0\|\nabla \rho^0 \|_{2 }^2 \leq L_0^2$ and
$e^{2T}\|\mathbf{v}_0\|_{0 }^2\leq L_0^2$, where
$L_0$ is a constant and where $C_0$ is the constant from Lemma
\ref{lem2.4}. There are constants $\epsilon_0$, $L_1$, $L_2$, $L_3$, where
$0<\epsilon_0<1$, such that the following estimates hold for
$k=1,2,3\dots $, provided $\|D\mathbf{v}_0\|_{s-1}$ is
sufficiently small:
\begin{itemize}
\item[(a)] $\|\nabla \rho^k\|_{s+1,T}^2 \leq
\epsilon_0$, $ \|\rho^k\|_{s+2,T} \leq L_1$,
\item[(b)]
$\|D\mathbf{v}
^k\|_{s-1,T}^2\leq \epsilon_0$, $\|\mathbf{v}
^k\|_{s,T}\leq L_2$,
\item[(c)] $\|{\partial
\mathbf{v}^k}/{\partial t}\|_{s-1,T}^2 \leq \epsilon_0
L_3$.
\end{itemize}
\end{proposition}

\begin{proof} The proof is by induction on $k$. We show only the
inductive step. We will derive estimates for $\rho ^{k+1}$ and
$\mathbf{v}^{k+1}$, and then use these estimates to prescribe
$\epsilon_0$, $L_1$, $L_2$, $L_3$ a priori, independent of $k$, so
that if $\rho ^k$ and $\mathbf{v}^k$
satisfy the estimates in (a)-(c), then $\rho ^{k+1}$ and $\mathbf{v}
^{k+1}$ also satisfy the same estimates. And we will show that
$\epsilon_0$ will be small if $\|D\mathbf{v}
_0\|_{s-1}$ is sufficiently small. In the estimates below,
we use $C$ to denote a generic constant whose value may change
from one instance to the next, but is independent of $\epsilon_0$,
$L_1$, $L_2$, and $L_3$.
\smallskip

\noindent\textbf{Estimate for $\nabla\rho ^{k+1}$:} Applying the
divergence operator to equation \eqref{e3.1}, we obtain
\begin{equation} \label{e3.3}
c\Delta ^2\rho ^{k+1}-\nabla \cdot ((\rho ^k)^{-1}\hat{p}'
(\rho^k)\nabla \rho ^{k+1})=\nabla \cdot (\mathbf{v}^k\cdot \nabla
\mathbf{v} ^{k+1})
\end{equation}
where we used the fact that $\nabla \cdot \mathbf{v}^{k+1}=0$, and
where $c$ is a positive constant. Applying Lemma \ref{lem2.7} we obtain
from the above equation the following estimate for $\nabla \rho
^{k+1}$, for $s>\frac N2+3$
\begin{equation} \label{e3.4}
\begin{aligned}
\|\nabla \rho ^{k+1}\|_{s+1}^2
&\leq C(\| \Delta \rho^{k+1}\|_s^2+\|\nabla \rho ^{k+1}\|_s^2)   \\
&\leq C[1+\sum_{j=1}^s\|D((\rho ^k)^{-1}\hat{p}'(\rho ^k))\|_{s_1}^{2j}]
\|D\mathbf{v}^k\|_{s_2+1}^2\|
D\mathbf{v}^{k+1}\|_{s-1}^2   \\
&\leq C_1\|D\mathbf{v}^k\|_{s-1}^2\|D \mathbf{v}^{k+1}\|_{s-1}^2
\end{aligned}
\end{equation}
where $C_1=\hat{C}_1(L_1)$. Here, $s_1=\max\{s-1,s_0\}=s-1$,
$s_2=\max\{s-2,s_0\}=s-2$, where
$s_0=[ \frac N2]+1=2$, and $s>\frac N2+3$, so $s\geq 5$ for
$N=2,3$. And we used the induction hypothesis for $\rho ^k$,
$\mathbf{v}^k$.
\smallskip

\noindent\textbf{Estimate for $\rho ^{k+1}$:}
To obtain an $L^2$ estimate for $\rho ^{k+1}$, we apply Lemma \ref{lem2.4},
where we define $u =\rho ^{k+1}$ and we define $w=\rho ^0$ , to be
used for the functions $u ,w$ appearing in Lemma \ref{lem2.4}. Note
that since by hypothesis we have $\rho ^{k+1}(
\mathbf{x}_0,t)=\rho ^0(\mathbf{x}_0,t)$ at a single fixed point
$\mathbf{x}_0\in \Omega$, the hypotheses of Lemma \ref{lem2.4} are
satisfied, and so we obtain the estimate
\begin{equation} \label{e3.5}
\begin{aligned}
\|\rho ^{k+1}\|_0^2
&\leq C_0\|\rho ^0\|_0^2+C_0\|\nabla \rho ^0\|_{2 }^2+C_0\|\nabla
\rho ^{k+1}\|_{2}^2  \\
&\leq  L_0^2+C_2\|D\mathbf{v}^k\|_{s_0+1}^2\|D
\mathbf{v}^{k+1}\|_1^2  \\
&\leq  L_0^2+C_2\|D\mathbf{v}^k\|_{s-1}^2\|D
\mathbf{v}^{k+1}\|_1^2
\end{aligned}
\end{equation}
where $C_0$ is the constant from Lemma \ref{lem2.4} (recall $C_0$ depends
only on $\Omega$), and where $C_2=\hat{C}_2(L_1)$. Here we used
the estimate for $\|\Delta \rho ^{k+1}\|_2^2+\|
\nabla \rho ^{k+1}\|_2^2$ from applying Lemma \ref{lem2.7} to \eqref{e3.3}
with $r=2$. We also used the hypothesis that $C_0 \|\rho
^0\|_0^2+C_0\|\nabla \rho ^0\|_2^2\leq L_0^2$.
Adding the estimates \eqref{e3.4}, \eqref{e3.5} yields the following estimate
for $\rho ^{k+1}$,
\begin{equation} \label{e3.6}
\begin{aligned}
\|\rho ^{k+1}\|_{s+2}^2 &\leq \|\rho^{k+1}\|
_0^2+C\|\nabla \rho ^{k+1}\|_{s+1}^2   \\
&\leq  L_0^2+C_3\|D\mathbf{v}^k\|_{s-1}^2\|D
\mathbf{v}^{k+1}\|_{s-1}^2
\end{aligned}
\end{equation}
where $C_3=\hat{C}_3(L_1)$.
\smallskip

\noindent\textbf{Estimate for  $\mathbf{v}^{k+1}$:}
 Applying Lemma \ref{lem2.3} to equations \eqref{e3.1}, \eqref{e3.2},
we obtain for $D\mathbf{v}^{k+1}$ the estimate
\begin{equation} \label{e3.7}
\begin{aligned}
\|D \mathbf{v}^{k+1}\|_{s-1}^2
& \leq Ce^{2T}(1+Te^{2T}e^{\beta (T)}\|D\mathbf{v}^k\| _{s-1,T}^2)
(\|D\mathbf{v}_0\|_{s-1}^2  \\
&\quad +\int_0^t (\|D ((\rho^k)^{-1} \hat{p}'(\rho^k))\|_{s_1}^2
\|\nabla\rho^{k+1} \|_{s-1}^2 )d\tau)  \\
&\leq Ce^{2T}(1+ Te^{2T}e^{ Te^{2T}})(
\|D\mathbf{v}_0\|_{s-1}^2 +C_4\int_0^t \|
\nabla\rho^{k+1} \|_{s-1}^2 d\tau ) \\
&\leq C_5\|D\mathbf{v}_0\|_{s-1}^2+C_5\int_0^t \|
D\mathbf{v}^k\|_{s-1,T}^2\|D \mathbf{v}^{k+1}
\|_{s-1}^2 d\tau  \\
&\leq C_5\|D\mathbf{v}_0\|_{s-1}^2+C_5\int_0^t \|D
\mathbf{v}^{k+1} \|_{s-1}^2 d\tau
\end{aligned}
\end{equation}
where $\beta (T)=Te^{2T}\|D\mathbf{v}^k\|_{s-1,T}^2 $.
Here, we used \eqref{e3.4} to estimate $\|\nabla \rho ^{k+1}\|
_{s-1}^2$, and we used the fact that by the induction hypothesis,
$\|D \mathbf{v}^{k}\|
_{s-1,T}^2 \leq \epsilon_0$, where $0<\epsilon_0 <1$. And here
$C_{4}=\hat{C}_{4}(L_1,T)$, $C_{5}=\hat{C}_{5}(L_1,T)$.

Applying Gronwall's inequality yields the estimate
\begin{equation} \label{e3.8}
\|D \mathbf{v}^{k+1}\|_{s-1}^2   \leq C_6\|D\mathbf{v}_0\|_{s-1}^2
\end{equation}
where $C_{6}=\hat{C}_{6}(L_1,T)$. We now choose $\epsilon_0$
to satisfy $\epsilon_0 =C_6\|D\mathbf{v}_0\|_{s-1}^2$. It follows that
$\|D \mathbf{v}^{k+1}\|_{s-1,T}^2 \leq \epsilon_0$.

Substituting the estimate \eqref{e3.8} for
$\|D\mathbf{v}^{k+1}\|_{s-1}^2$
into the right-hand side of \eqref{e3.4}, \eqref{e3.6} yields the
following estimates for
$\|\nabla \rho ^{k+1}\|_{s+1}^2$ and $\|\rho^{k+1}\|_{s+2}^2$:
\begin{gather*}
\|\nabla \rho ^{k+1}\|_{s+1}^2 \leq
C_1\|D\mathbf{v}^k\|_{s-1}^2\|D \mathbf{v}
^{k+1}\|_{s-1}^2\leq \epsilon_0 C_1C_6\|D\mathbf{v}_0\|_{s-1}^2,
\\
\|\rho ^{k+1}\|_{s+2}^2 \leq  L_0^2
+C_3\|D\mathbf{v}^k\|_{s-1}^2\|D\mathbf{v}^{k+1}\|_{s-1}^2
\leq  L_0^2+\epsilon_0 C_3C_6\|D \mathbf{v}_0\|_{s-1}^2
\end{gather*}
Therefore, we have $\|\rho ^{k+1}\|_{s+2,T}\leq L_1$
and we have $\|\nabla \rho ^{k+1}\|_{s+1,T}^2\leq
\epsilon_0$,  provided that we choose $L_1$ large enough so that
$\frac 12 L_1^2>L_0^2$, and provided that we choose
$\| D\mathbf{v}_0\|_{s-1}$ small enough so that
$\epsilon_0 C_3 C_6\|D \mathbf{v}_0\|_{s-1}^2<\frac 12 L_1^2$, and
provided that we choose $\|D \mathbf{v}_0\|_{s-1}$
small enough so that $C_1 C_6\|D\mathbf{v}_0\|_{s-1}^2<1$.
 We also choose $\|D\mathbf{v}_0\|_{s-1}$
small enough so that $C_6\|D\mathbf{v}_0\|_{s-1}^2<1$;
therefore, we have $0<\epsilon_0<1$, since
$\epsilon_0=C_6\|D\mathbf{v}_0\|_{s-1}^2$. (Note that
$\epsilon_0$ will be small if $\|D\mathbf{v} _0\|
_{s-1}$ is sufficiently small.) This completes the proof of
 part (a).

Next, by applying Lemma \ref{lem2.3} to \eqref{e3.1}, \eqref{e3.2}, and using the
estimates \eqref{e3.4}, \eqref{e3.7}, \eqref{e3.8}, we have the estimate
\begin{align*}
\|\mathbf{v}^{k+1}\|_s^2
& \leq e^{2T}\|\mathbf{v}_0\|_0^2 +Ce^{2T}(1+Te^{2T}e^{\beta
(T)}\|D\mathbf{v}^k\|_{s-1,T}^2)(\|D\mathbf{v}_0\|_{s-1}^2  \\
&\quad +\int_0^t (\|D ((\rho^k)^{-1} \hat{p}'
(\rho^k))\|_{s_1}^2\|\nabla\rho^{k+1} \|_{s-1}^2 )d\tau)  \\
&\leq L_0^2 + C_7\|D\mathbf{v}_0\|_{s-1}^2 +C_7\int_0^t \|D \mathbf{v}^{k+1} \|
_{s-1}^2 d\tau  \\
&\leq L_0^2 + C_7\|D\mathbf{v}_0\|_{s-1}^2 +  C_8 \|D \mathbf{v}_0 \|_{s-1}^2
\end{align*}
with $\beta (T)=Te^{2T}\|D\mathbf{v}^k\|_{s-1,T}^2 $.
Here $s_1=\max\{s-1,s_0\}=s-1$, $s>\frac N2 +3$, and
$s_0=[ \frac N2]+1=2$ for $N=2,3$. We also used the hypothesis
that $e^{2T}\|\mathbf{v}_0\|_0^2 \leq L_0^2 $, and that
$\|D \mathbf{v}^{k}\|_{s-1,T}^2 \leq \epsilon_0$, where $0<\epsilon_0<1$.
And here $C_{7}=\hat{C}_{7}(L_1,T)$, $C_{8}=\hat{C}_{8}(L_1,T)$.

Therefore, we have $\|\mathbf{v}^{k+1}\|_{s,T}\leq L_2$ provided that
 we choose $L_2$ large enough so that $\frac 12L_2^2>L_0^2$, and so
that $\frac 12 L_2^{2}>(C_7+ C_8)\|D\mathbf{v} _0\|_{s-1}^2$.
This completes the proof of part (b). We next consider part (c).
\smallskip

\noindent\textbf{Estimate for $\mathbf{v}_t^{k+1}$:}
 Directly from
equation \eqref{e3.1} for $\mathbf{v}_t^{k+1}$, and the estimates already
derived for $\rho^{k+1}$ and $\mathbf{v}^{k+1}$, we deduce that
$\|\mathbf{v}_t^{k+1}\|_{s-1}^2\leq
C_{9}(\|D\mathbf{v}^{k+1}\|_{s-1}^2 +\|
\nabla\rho^{k+1} \|_{s+1}^2)\leq 2\epsilon_0
C_9$, where $C_{9}=
\hat{C}_{9}(L_1,L_2,T)$. Thus
$\|\mathbf{v}_t^{k+1}\|_{s-1,T}^2 \leq \epsilon_0 L_3$
provided we choose $L_3$ sufficiently large so that $L_3 \geq 2
C_9$.

Summarizing, if we choose $\|D\mathbf{v}_0\|_{s-1} $ to
be sufficiently small, then $\rho ^k$ and $\mathbf{v}^k$ satisfy
the estimates in (a)-(c) for all $k\geq 1$. This completes the
proof of Proposition \ref{prop3.1}.
\end{proof}

Next, we give the proof of contraction in low Sobolev norm.

\begin{proposition}\label{prop3.2}
Assume that the hypotheses of Theorem \ref{thm3.1} hold. Then we have
\[
\|\rho ^{k+1}-\rho ^k\|_{2,T}^2+\|
\mathbf{v}^{k+1}-\mathbf{v}^k\|_{2,T}^2 \leq \zeta
\Big(\|\rho ^{k}-\rho ^{k-1}\|_{2,T}^2+
\|\mathbf{v}^k-\mathbf{v}^{k-1}\|_{2,T}^2 \Big)
\]
for $k=1,2,3\dots $. Here $\zeta$ is a constant such that
$0<\zeta<1$. And $|\rho^{k+1} - \rho^0|_{L^{\infty}}
\leq \delta$; hence, $\rho^k \in G_1$, for all $k$.
\end{proposition}

\begin{proof}
Subtracting the equations \eqref{e3.1}, \eqref{e3.2} for $\rho^k$ and
$\mathbf{v}^k$ from the equations \eqref{e3.1}, \eqref{e3.2} for $\rho^{k+1}$ and $\mathbf{v
}^{k+1}$ yields equations which we write in the form
\begin{gather} \label{e3.9}
\begin{aligned}
&\frac{D^k(\mathbf{v}^{k+1}-\mathbf{v}^k)}{Dt}+ (\rho^k)^{-1} \hat{p}
' (\rho^k)\nabla (\rho ^{k+1}-\rho ^k)\\
& =c\nabla \Delta (\rho ^{k+1}-\rho ^k)
-(\mathbf{v}^k-\mathbf{v}^{k-1})\cdot \nabla \mathbf{v}^k\\
&\quad -(\big((\rho^k)^{-1} \hat{p}' (\rho^k)-(\rho^{k-1})^{-1}
\hat{p}' (\rho^{k-1})\big)\nabla\rho^k ,
\end{aligned}\\
\nabla \cdot (\mathbf{v}^{k+1}-\mathbf{v}^k) =0,  \label{e3.10}
\end{gather}
where $D^k/{Dt}= \partial/{\partial t}+\mathbf{v}^k \cdot\nabla$, and where
$(\mathbf{v}^{k+1}-\mathbf{v}^k)(\mathbf{x},0)=0$. Using a proof by
induction
on $k$, we assume that $\rho^k \in G_1$, and then we will show that
$\rho^{k+1} \in G_1$. First, we obtain estimates for $\rho
^{k+1}-\rho ^k$ and $\mathbf{v}^{k+1}-\mathbf{v}^k$.
\smallskip

\noindent\textbf{Estimate for $\mathbf{\rho }^{k+1}-\rho^{k}$:}
Applying the divergence operator to \eqref{e3.9} yields
\begin{equation} \label{e3.11}
\begin{aligned}
&c\Delta ^2(\rho ^{k+1}-\rho ^k)-\nabla \cdot ((\rho
^k)^{-1}p'(\rho ^k)\nabla (\rho ^{k+1}-\rho ^k)) \\
&=\nabla \cdot (\mathbf{v}^k\cdot \nabla (\mathbf{v}^{k+1}
-\mathbf{v}^k))
+\nabla \cdot ((\mathbf{v}^k-\mathbf{v}^{k-1}) \cdot \nabla
\mathbf{v}^k) \\
&\quad +\nabla \cdot (((\rho ^k)^{-1}p'(\rho
^k)-(\rho ^{k-1})^{-1}p'(\rho ^{k-1}))\nabla \rho ^k)
\end{aligned}
\end{equation}
Applying Lemma \ref{lem2.7} to equation \eqref{e3.11} with $r=2$ yields the
estimate
\begin{equation} \label{e3.12}
\begin{aligned}
&\|\Delta (\rho ^{k+1}-\rho ^k)\|_2^2+\|\nabla
(\rho ^{k+1}-\rho ^k\|_2^2  \\
&\leq C[1+\sum_{j=1}^2\|D((\rho ^k)^{-1}p'(\rho
^k))\|_2^{2j}](\|D\mathbf{v}^k\|
_3^2\|
D(\mathbf{v}^{k+1}-\mathbf{v}^k)\|_1^2  \\
&\quad +\|(\mathbf{v}^k-\mathbf{v}
^{k-1})\cdot \nabla \mathbf{v}^k\|_1^2 +\|((\rho
^k)^{-1}p'(\rho ^k)-(\rho ^{k-1})^{-1}p'(\rho
^{k-1}))\nabla \rho ^k\|_1^2)   \\
&\leq C_1\|D\mathbf{v}^k\|_3^2\|\mathbf{v}^{k+1}-\mathbf{
v}^k\|_2^2+C_1\|D \mathbf{v}^k\|_2^2\|\mathbf{v}^k-
\mathbf{v}^{k-1}\|_2^2+C_1\|\nabla \rho ^k\|
_2^2\|\rho ^k-\rho ^{k-1}\|_2^2
\end{aligned}
\end{equation}
where $C_1=\hat{C}_1(L_1)$ from Proposition \ref{prop3.1}. Here we used the
Sobolev calculus inequality $\|f g \|_r \leq C\|
f\|_r\|g\|_r$ for $r=2>\frac N2$.

To obtain an $L^2$ estimate for $\rho ^{k+1}$, we apply
Lemma \ref{lem2.4}, where we define $u =\rho ^{k+1}$ and we define
$w=\rho ^k$ , to be used for the functions $u ,w$ appearing in
Lemma \ref{lem2.4}. Note that since by hypothesis we have $\rho
^{k+1}(\mathbf{x}_0,t)=\rho ^k(\mathbf{x}_0,t)$ for all $k\geq 0$,
at a single fixed point $\mathbf{x}_0\in \Omega $, the hypotheses
of Lemma \ref{lem2.4} are satisfied, and so we obtain the estimate
\begin{equation} \label{e3.13}
\begin{aligned}
\|\rho ^{k+1}-\rho ^k\|_0^2
&\leq C\|\nabla
(\rho^{k+1}-\rho ^k)\|_2^2   \\
&\leq C_2\|D\mathbf{v}^k\|_3^2\|\mathbf{v}^{k+1}-\mathbf{
v}^k\|_2^2+C_2\|D\mathbf{v}^k\|_2^2\|\mathbf{v}^k-
\mathbf{v}^{k-1}\|_2^2  \\
&\quad +C_2\|\nabla \rho ^k\|_2^2\|\rho ^k-\rho^{k-1}\|_2^2
\end{aligned}
\end{equation}
where we used the estimate for
$\|\Delta (\rho ^{k+1}-\rho ^k)\|_2^2+\|\nabla (\rho ^{k+1}
-\rho ^k)\|_2^2$ from \eqref{e3.12}.
Here $C_2=\hat{C}_2(L_1)$ from Proposition \ref{prop3.1}.

Now, adding \eqref{e3.12}, \eqref{e3.13}, we obtain the estimate
\begin{equation} \label{e3.14}
\begin{aligned}
\|\rho ^{k+1}-\rho ^k\|_2^2
&\leq  \|\rho ^{k+1}-\rho ^k\|_0^2+C\|\nabla (\rho ^{k+1}-\rho
^k)\|_1^2   \\
&\leq  \|\rho ^{k+1}-\rho ^k\|_0^2+C(\|\nabla
(\rho ^{k+1}-\rho ^k)\|_2^2+\|\Delta (\rho
^{k+1}-\rho ^k)\|_2^2)   \\
&\leq  C_3\|D\mathbf{v}^k\|_3^2\|\mathbf{v}^{k+1}-
\mathbf{v}^k\|_2^2+C_3 \|D\mathbf{v}^k\|_2^2\|\mathbf{v}
^k-\mathbf{v}^{k-1}\|_2^2 \\
&\quad +C_3\|\nabla \rho ^k\|_2^2\|\rho ^k-\rho
^{k-1}\|_2^2  \\
&\leq C_3\left(\|\mathbf{v}^{k+1}-\mathbf{v}^k\|
_2^2+\epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\|
_2^2+\epsilon_0 \|\rho ^k-\rho ^{k-1}\|_2^2\right)
\end{aligned}
\end{equation}
where $C_3=\hat{C}_3(L_1)$. Here, we used
the Proposition \ref{prop3.1} estimates
$\|D\mathbf{v}^k\|_2^2\leq \|D\mathbf{v}^k\|_3^2
\leq \|D \mathbf{v}^k\|_{s-1,T}^2\leq \epsilon_0 $, and
$\|\nabla \rho ^k\|_2^2\leq \|\nabla \rho
^k\|_{s+1,T}^2\leq \epsilon_0 $, where $0<\epsilon_0<1$,
and where $\epsilon_0$ is as small as we like if $\|D
\mathbf{v}_0\|_{s-1}$ is sufficiently small.

\noindent\textbf{Estimate for $\mathbf{v}^{k+1}\mathbf{-v}^{k}$:}
 After applying Lemma \ref{lem2.3} to \eqref{e3.9}, using $r=2$, we
obtain
\begin{equation} \label{e3.15}
\begin{aligned}
\|\mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2
&\leq C_4\int_0^t\|D\mathbf{v}^k\|_2^2\|\mathbf{v}^k-
\mathbf{v}^{k-1}\|_2^2d\tau +C_4\int_0^t\|\nabla
(\rho^{k+1}-\rho ^k)\|_1^2d\tau    \\
&\quad +C_4\int_0^t\|\nabla \rho ^k\|_2^2\|(\rho
^k)^{-1}p'(\rho ^k)-(\rho ^{k-1})^{-1}p'(\rho
^{k-1})\|_2^2d\tau    \\
&\leq C_5\int_0^t\left(\|
\mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2+\epsilon_0 \|
\mathbf{v}^k-\mathbf{v}^{k-1}\|_2^2+\epsilon_0 \|\rho
^k-\rho ^{k-1}\|_2^2\right) d\tau
\end{aligned}
\end{equation}
with $C_4=\hat{C}_4(L_1,L_2,T)$ and $C_5=\hat{C}_5(L_1,L_2,T)$,
and where we substituted the estimate for
$\|\nabla (\rho ^{k+1}-\rho ^k)\|_1^2$ from \eqref{e3.14}.
 Here, we used the estimates
$\|D\mathbf{v}^k\|_2^2\leq \|D\mathbf{v} ^k\|_{s-1,T}^2\leq \epsilon_0 $,
and $\|\nabla \rho^k\|_2^2\leq \|\nabla \rho ^k\|_{s+1,T}^2\leq
\epsilon_0 $ from Proposition \ref{prop3.1}.

Next, we apply Gronwall's inequality to \eqref{e3.15}, which yields
\begin{equation} \label{e3.16}
\begin{aligned}
\|\mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2
&\leq C_6\int_0^t(\epsilon_0\|\mathbf{v}^k-\mathbf{v
}^{k-1}\|_2^2+\epsilon_0\|\rho ^k-\rho ^{k-1}\|
_2^2)d\tau    \\
&\leq C_7\Big( \epsilon_0 \|
\mathbf{v}^k-\mathbf{v}^{k-1}\|_{2,T}^2+\epsilon_0 \|
\rho ^k-\rho ^{k-1}\|_{2,T}^2\Big)
\end{aligned}
\end{equation}
where $C_6=\hat{C}_6(L_1,L_2,T)$, $C_7=\hat{C}_7(L_1,L_2,T)$.

Substituting \eqref{e3.16} into the right-hand side of \eqref{e3.14} yields
\begin{align*}
\|\rho ^{k+1}-\rho ^k\|_2^2
&\leq C_3 C_7\left(\epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\|
_{2,T}^2+\epsilon_0 \|
\rho ^k-\rho ^{k-1}\|_{2,T}^2   \right)  \\
&\quad +C_3\left(\epsilon_0 \|
\mathbf{v}^k-\mathbf{v}^{k-1}\|_2^2+\epsilon_0 \|\rho
^k-\rho ^{k-1}\|_2^2\right)
\end{align*}
Adding the above estimate for $\|\rho ^{k+1}-\rho ^k\|
_2^2$ to the estimate \eqref{e3.16} yields
\begin{align*}
\|\rho ^{k+1}-\rho ^k\|_2^2+\|
\mathbf{v}^{k+1}-\mathbf{v}^k\|_2^2
&\leq  C_3 C_7\left(
\epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\|
_{2,T}^2+\epsilon_0 \|
\rho ^k-\rho ^{k-1}\|_{2,T}^2   \right)  \\
&\quad +C_3\left(\epsilon_0 \|
\mathbf{v}^k-\mathbf{v}^{k-1}\|_2^2+\epsilon_0 \|\rho
^k-\rho ^{k-1}\|_2^2\right)  \\
&\quad + C_7( \epsilon_0 \|\mathbf{v}^k-\mathbf{v}^{k-1}\|
_{2,T}^2+\epsilon_0 \|\rho ^k-\rho ^{k-1}\|
_{2,T}^2)
\end{align*}
Finally, after choosing $\epsilon_0$ sufficiently small, we get
\begin{equation} \label{e3.17}
\|\rho ^{k+1}-\rho ^{k}\|_{2,T}^2+\|\mathbf{v} ^{k+1}-
\mathbf{v} ^{k}\|_{2,T}^2 \leq \zeta \left(\|
\rho^k-\rho^{k-1}\|_{2,T}^2 + \|\mathbf{v}^{k}-\mathbf{v}
^{k-1}\|_{2,T}^2 \right)
\end{equation}
where $0<\zeta<1$. This completes the proof of the first part of
Proposition \ref{prop3.2}. It only remains to prove that
$|\rho^{k+1} - \rho^0|_{L^{\infty}} \leq \delta$, so that
 $\rho^{k+1} \in G_1$.
\smallskip

\noindent\textbf{Proof that $|\rho ^{k+1} -\rho ^{0}|_{L^{\infty}}\leq \delta$:}
Repeated application of the estimate \eqref{e3.17} yields
\begin{equation} \label{e3.18}
\|\mathbf{v}^{k+1}-\mathbf{v}^k\|_{2,T}^2+|
|\rho ^{k+1}-\rho ^k\|_{2,T}^2\leq \zeta
^k\left( \|\mathbf{v}^1-\mathbf{v}^0\|
_{2,T}^2+\|\rho ^1-\rho ^0\|_{2,T}^2\right)
\end{equation}
where $0<\zeta <1$. Next, we estimate
\begin{equation} \label{e3.19}
\begin{aligned}
|\rho ^{k+1}-\rho ^0|^2
&\leq C\sum_{j=0}^k| \rho ^{j+1}-\rho ^j|_{L^\infty }^2\\
&\leq C\sum_{j=0}^k\| \rho ^{j+1}-\rho ^j\|_{s_0}^2
  \leq C\sum_{j=0}^k\|\rho^{j+1}-\rho^j\|_{2,T}^2   \\
&\leq C\sum_{j=0}^k{\zeta }^j\left( \|\rho ^1-\rho
^0\|_{2,T}^2+\|\mathbf{v}^1-\mathbf{v}^0\|
_{2,T}^2\right)
\end{aligned}
\end{equation}
where we used \eqref{e3.18} in the last step. We also used the triangle
inequality and Cauchy's inequality and Sobolev's inequality
$|h|_{L^{\infty}} \leq C\|h\|_{s_0}$. Here
$s_0=[\frac N2]+1=2$ for $N=2,3$. Thus, we have
$|\rho ^{k+1}-\rho ^0|\leq \delta $, provided that
$\|\rho ^1-\rho ^0\|_{2,T}$ and $\|
\mathbf{v}^1-\mathbf{v}^0\|_{2,T}$ are sufficiently small.
We now proceed to show that
$\|\rho ^1-\rho ^0\|_{2,T}$ and $\|\mathbf{v}^1-\mathbf{v}^0\|_{2,T}$ can
be made sufficiently small.

 From  equations \eqref{e3.1}, \eqref{e3.2}, we obtain the following
equations for $\mathbf{v}^1-\mathbf{v}^0$ and $\rho ^1-\rho ^0$,
\begin{equation} \label{e3.20}
\begin{gathered}
\frac{D^0(\mathbf{v}^1-\mathbf{v}^0)}{Dt}+(\rho
^0)^{-1}\hat{p}'(\rho ^0)\nabla (\rho ^1-\rho ^0) \\
= c\nabla \Delta (\rho ^1-\rho ^0)
-\mathbf{v}^0\cdot \nabla \mathbf{v}^0
-(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0+c\nabla \Delta \rho^0, \\
\nabla \cdot (\mathbf{v}^1-\mathbf{v}^0) = 0 ,
\end{gathered}
\end{equation}
where $D^0/{Dt}=\partial /{\partial t}+\mathbf{v}^0\cdot \nabla $,
and where
$(\mathbf{v}^1-\mathbf{v}^0)(\mathbf{x},0)=0$. Here, we used the fact that
$\mathbf{v}^0(\mathbf{x},t)=\mathbf{v}_0(\mathbf{x})$. Now we
obtain estimates for $\rho ^1-\rho ^0$ and
$\mathbf{v}^1-\mathbf{v}^0$.
\smallskip

\textbf{Estimate for $\rho ^1-\rho ^0$:}
Applying the divergence operator to \eqref{e3.20}, we obtain
\begin{equation} \label{e3.21}
\begin{aligned}
&c\Delta ^2(\rho ^1-\rho ^0)-\nabla \cdot ((\rho
^0)^{-1}\hat{p}'(\rho ^0)\nabla (\rho ^1-\rho ^0))\\
&=\nabla \cdot (\mathbf{v}^0\cdot \nabla (
\mathbf{v}^1-\mathbf{v}^0))+\nabla \cdot (\mathbf{v}^0\cdot \nabla \mathbf{v}
^0) \\
&\quad +\nabla \cdot ((\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho
^0)-c \Delta^2 \rho^0
\end{aligned}
\end{equation}
Applying Lemma \ref{lem2.7} to \eqref{e3.21} with $r=2$, we obtain the estimate
\begin{equation} \label{e3.22}
\begin{aligned}
\|\Delta (\rho ^1-\rho ^0)\|_2^2+\|\nabla (\rho
^1-\rho ^0)\|_2^2
&\leq C_{8}( \|D\mathbf{v}^0\|
_3^2\|D(\mathbf{v}^1-\mathbf{v}^0)\|_1^2+\|
\mathbf{v}^0\cdot \nabla \mathbf{v}^0\|_1^2 \\
&\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho
^0\|_1^2+\|\nabla \Delta \rho^0\|_1^2)
\end{aligned}
\end{equation}
where $C_{8}=\hat{C}_{8}(L_1)$. Next, we apply Lemma \ref{lem2.4}
and obtain the $L^2$ estimate
\begin{equation} \label{e3.23}
\begin{aligned}
\|\rho ^1-\rho ^0\|_0^2
&\leq C\|\nabla (\rho ^1-\rho ^0)\|_2^2   \\
&\leq C_{9}( \|D\mathbf{v}^0\|_3^2\|D( \mathbf{v}^1-
\mathbf{v}^0)\|_1^2+\|\mathbf{v}^0\cdot \nabla \mathbf{v}
^0\|_1^2 \\
&\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho
^0\|_1^2+\|\nabla \Delta \rho^0\|_1^2)
\end{aligned}
\end{equation}
where $C_{9}=\hat{C}_{9}(L_1)$. And we used estimate \eqref{e3.22} for
$\|\nabla (\rho ^1-\rho ^0)\|_2^2$.

Adding the estimates \eqref{e3.22}, \eqref{e3.23} yields the following
 estimate for $\rho ^1-\rho ^0$:
\begin{equation} \label{e3.24}
\begin{aligned}
\|\rho ^1-\rho ^0\|_2^2
& \leq C\left( \|\rho ^1-\rho ^0\|_0^2+\|
\nabla (\rho ^1-\rho ^0)\|_1^2\right)   \\
&\leq C_{10}( \|D\mathbf{v}^0\|_3^2\|
D(\mathbf{v}^1-\mathbf{v}^0)\|_1^2+\|
\mathbf{v}^0\cdot \nabla \mathbf{v}^0\|_2^2\\
&\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0\|_2^2
+\|\nabla \Delta \rho^0\|_1^2)  \\
&\leq C_{11}( \|D\mathbf{v}^0\|_3^2\|
\mathbf{v}^1-\mathbf{v}^0\|_2^2
+\|\mathbf{v}^0\|_2^2\|D\mathbf{v}^0\|_2^2 \\
&\quad +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\|_2^2\|\nabla \rho ^0\|_2^2+\|\nabla
\rho^0\|_3^2)
\end{aligned}
\end{equation}
where $C_{10}=\hat{C}_{10}(L_1)$, $C_{11}=\hat{C}_{11}(L_1)$. Here
we used the Sobolev calculus inequality $\|f g
\|_r\leq C\|f\|_r\|g\|_r$ for $r=2>\frac N2$.
\smallskip

\textbf{Estimate for $\mathbf{v}^1\mathbf{-v}^0$:}
 After applying Lemma \ref{lem2.3} to \eqref{e3.20}, using $r=2$, we obtain
\begin{equation} \label{e3.25}
\begin{aligned}
\|\mathbf{v}^1-\mathbf{v}^0\|_2^2
& \leq C_{12}\int_0^t\left( \|\mathbf{v}^0\cdot \nabla
\mathbf{v}^0\|_2^2+\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\nabla \rho ^0\|_2^2\right)
 d\tau\\
&\quad +C_{12}\int_0^t\left(\|\nabla \Delta \rho^0\|_2^2
+\|\nabla (\rho ^1-\rho ^0)\|_1^2\right) d\tau   \\
&\leq C_{13}\int_0^t \left(\|\mathbf{v}^0\|_2^2\|
D\mathbf{v}^0\|_2^2 +\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\|_2^2\|\nabla \rho ^0\|
_2^2\right) d\tau  \\
&\quad +C_{13}\int_0^t\left(\|\nabla \rho^0\|_4^2+\|
D\mathbf{v}^0\|_3^2\|
\mathbf{v}^1-\mathbf{v}^0\|_2^2 \right) d\tau
 \end{aligned}
\end{equation}
where $C_{12}=\hat{C}_{12}(L_1,L_2)$,
$C_{13}=\hat{C}_{13}(L_1,L_2)$, and where we substituted the
estimate for $\|\nabla (\rho ^1-\rho ^0)\|_1^2$ from
\eqref{e3.24}. Next, we apply Gronwall's inequality to \eqref{e3.25}, which
yields
\begin{equation} \label{e3.26}
\begin{aligned}
\|\mathbf{v}^1-\mathbf{v}^0\|_2^2
&\leq C_{14}\int_0^t \left(\|\mathbf{v}^0\|_2^2\|D\mathbf{v}^0\|_2^2
+\|(\rho ^0)^{-1}\hat{p}'(\rho ^0)\|_2^2\|\nabla \rho ^0\|
_2^2+\|\nabla \rho^0\|_4^2\right) d\tau
 \\
&\leq C_{15}\left( \|D \mathbf{v}_0\|_{s-1}^2+\|
\nabla \rho ^0\|_{4,T}^2\right) \leq 2 C_{15}\|
D\mathbf{v}_0\|_{s-1}^2
\end{aligned}
\end{equation}
where we used the fact that
$\mathbf{v}^0(\mathbf{x},t)=\mathbf{v}_0(\mathbf{x})$, and we used the
fact that $\|\nabla \rho ^0\|_{4,T}^2\leq \|\nabla \rho ^0\|_{s+1,T}^2\leq
\|D\mathbf{v}_0\|_{s-1}^2$, by hypothesis. Here
$C_{14}=\hat{C}_{14}(L_1,L_2,T) $ and $C_{15}=\hat{C}_{15}(L_1,L_2,T)$.
Finally,
we substitute the above estimate \eqref{e3.26} into the right-hand side
of the estimate \eqref{e3.24} for $\|\rho ^1-\rho ^0\|_2^2$,
which yields
\begin{equation} \label{e3.27}
\|\rho ^1-\rho ^0\|_2^2\leq C_{16}\|D\mathbf{v }_0\|_{s-1}^2
\end{equation}
where $C_{16}=\hat{C}_{16}(L_1,L_2,T)$ and where we used the fact
that $\|\nabla \rho ^0\|_2^2 \leq \|\nabla \rho
^0\|_3^2\leq \|\nabla \rho ^0\|_{s+1,T}^2\leq
\|D\mathbf{v}_0\|_{s-1}^2$, by hypothesis. From
\eqref{e3.19}, \eqref{e3.26}, and \eqref{e3.27}, we
see that $|\rho ^{k+1}-\rho ^0|_{L^\infty }\leq \delta $,  for
$\|D \mathbf{v}_0\|_{s-1}$ sufficiently small. And so
$\rho ^{k+1}\in G_1$. It thus follows from the proof by induction
on $k$ that $\rho ^k\in G_1 $ for all $k$. This completes the
proof of Proposition \ref{prop3.2}.
\end{proof}

Using Propositions \ref{prop3.1} and \ref{prop3.2}, we now complete the proof of
Theorem \ref{thm3.1}. From \eqref{e3.18}, it follows that
\[
\sum_{k=1}^\infty \Big(\|
\mathbf{v}^{k+1}-\mathbf{v}^k\|_{2,T}^2 +\|\rho
^{k+1}-\rho ^k\|_{2,T}^2 \Big) < \infty .
\]
Hence, $\|\mathbf{v}^{k+1}-\mathbf{v}^k\|
_{2,T}^2\to 0$ and $\|\rho ^{k+1} -\rho
^{k}\|_{2,T}^2\to 0 $ as $k\to \infty$.
Therefore, we conclude that there exist
 $\mathbf{v}\in C([0,T],H^2(\Omega ))$ and
$\rho \in C([0,T],H^{2}(\Omega ))$ so
that $\|\mathbf{v}^k-\mathbf{v} \|_{2,T}\to 0$, and
 $\|\rho ^k-\rho \|_{2,T}\to 0$,  as $k\to \infty $.
Using the interpolation inequalities
$\|f \|_{r'+2} \leq C\|f \|_2^\alpha \|f\|_{r+2}^{1-\alpha } $,
 with $\alpha =(r-r')/r$,
and $\|g \|_{r'} \leq C\|g \|_2^\beta \|g \|_{r}^{1-\beta } $,
with $\beta =(r-r')/(r-2)$, with $r'<r$, where $r\geq 5$,  and
using Proposition \ref{prop3.1}, we can conclude that
$\|\rho ^k-\rho \|_{s'+2,T}\to 0 $ and
$\| \mathbf{v}^k-\mathbf{v} \|_{s',T}\to 0$,
 as $k\to \infty $ for any $s'<s$.
For $s'>\frac N2+3$, Sobolev's lemma implies that
$\mathbf{v}^k\to \mathbf{v}$ in $C([0,T],C^3)$ and $\rho
^k\to \rho $ in $C([0,T],C^5)$. From the linear system of
equations \eqref{e3.1}, \eqref{e3.2} it follows that
$\| \mathbf{v}_t^k-\mathbf{v}_t \|_{s'-1,T}\to 0$, as $k\to \infty $,
so that $\mathbf{v}_t^k\to \mathbf{v}_t$ in $C([0,T],C^2)$, and
$\rho$, $\mathbf{v}$, is a classical solution of the system of
equations \eqref{e1.2}, \eqref{e1.3}. The additional facts that
$\mathbf{v}\in L^\infty([0,T],H^s)$, and
$ \rho \in L^\infty([0,T],H^{s+2})$ can
be deduced using boundedness in high norm and a standard
compactness argument (see, for example, \cite{e1,m1}).

To prove uniqueness, let $\rho^1$, $\mathbf{v}^1$, and let
$\rho^2$, $\mathbf{v}^2$, be any solutions of \eqref{e1.2},
\eqref{e1.3} having the regularity of solutions from Theorem
\ref{thm3.1}, and satisfying $\rho ^{1}(\mathbf{x}_0,t)=\rho
^2(\mathbf{x}_0,t)=\rho ^0(\mathbf{x}_0,t)$ at a single fixed
point $\mathbf{x}_0\in \Omega $. Here, $\rho^0$ is the initial
iterate from Theorem \ref{thm3.1}. And we assume that
$\|D\mathbf{v}_0\|_{s-1}$ is sufficiently small. We next show that
$\rho^1=\rho^2$ and $\mathbf{v}^1=\mathbf{v}^2$.

Let $\rho = \lim_{k\to \infty} \rho^{k+1}$, $\mathbf{v} =
\lim_{k\to \infty} \mathbf{v}^{k+1}$, where $\{\rho^{k+1}$,
$\mathbf{v}^{k+1}\}$, $k\geq 0$, is a sequence of solutions to the
linear equations \eqref{e3.1}, \eqref{e3.2}, so that $\rho$,
$\mathbf{v}$ is a solution of \eqref{e1.2}, \eqref{e1.3} from
Theorem \ref{thm3.1}. We next show that $\rho^1=\rho$ and
$\mathbf{v}^1=\mathbf{v}$. It then follows that the solution to
\eqref{e1.2}, \eqref{e1.3} is unique, because repeating the proof
given below will also show that $\rho^2=\rho$ and
$\mathbf{v}^2=\mathbf{v}$, and so we will obtain
\begin{align*}
&\|\rho^2-\rho^1\|_{2,T} +\|\mathbf{v}^2-\mathbf{v}^1\|_{2,T} \\
&\leq \|\rho^2-\rho\|_{2,T}+ \|\rho^1-\rho\|_{2,T}
+\|\mathbf{v}^2-\mathbf{v}\|_{2,T}+\|\mathbf{v}^1-\mathbf{v}\|_{2,T}
= 0
\end{align*}
Therefore, we now show that $\rho^1=\rho$ and
$\mathbf{v}^1=\mathbf{v}$.

Subtracting the equations \eqref{e1.2}, \eqref{e1.3} for $\rho^{1}$ and
$\mathbf{v}^{1}$  from the equations \eqref{e1.2}, \eqref{e1.3} for $\rho$ and
$\mathbf{v}$ yields equations which we write in the form
\begin{gather*}
\begin{aligned}
&\frac{D^1(\mathbf{v}-\mathbf{v}^{1})}{Dt}+ (\rho^1)^{-1}
\hat{p}' (\rho^1)\nabla (\rho -\rho ^{1}) \\
&= c\nabla \Delta (\rho -\rho ^{1}) -(\mathbf{v}-\mathbf{v}^{1})\cdot \nabla
\mathbf{v}
-((\rho)^{-1} \hat{p}' (\rho)-(\rho^{1})^{-1}
\hat{p}' (\rho^{1}))\nabla \rho,
\end{aligned} \\
\nabla \cdot (\mathbf{v}-\mathbf{v}^{1}) = 0,
\end{gather*}
where $D^1/{Dt}= \partial/{\partial t}+\mathbf{v}^1 \cdot\nabla$,
where $(\mathbf{v}-\mathbf{v}^{1})(\mathbf{x},0)=0$. Repeating the
proof of Proposition \ref{prop3.2} yields the estimate
\[
\|\rho -\rho ^{1}\|_{2,T}^2+\|\mathbf{v}
-\mathbf{v} ^{1}\|_{2,T}^2 \leq \zeta \left(\|
\rho-\rho^{1}\|_{2,T}^2 + \|
\mathbf{v}-\mathbf{v}^{1}\|_{2,T}^2 \right)
\]
where $0<\zeta<1$. Note that in repeating the proof of the
estimates \eqref{e3.14}, \eqref{e3.15} in the proof of Proposition
\ref{prop3.2}, we use the fact that $\|D\mathbf{v}\|_{s-1,T}^2\leq
\epsilon_0 $ and $\|\nabla \rho\|_{s+1,T}^2\leq \epsilon_0$, which
holds since $\rho$, $\mathbf{v}$ is a solution from Theorem
\ref{thm3.1}. Finally, after moving the right-hand side of the
above inequality to the left-hand side, we obtain $\|\rho -\rho
^{1}\|_{2,T}^2+\|\mathbf{v}-\mathbf{v} ^{1}\|_{2,T}^2=0$ and so
the solution is unique. \end{proof}

\subsection*{A Final Remark}
We sketch a proof that $\partial \rho/\partial t$ is small. Then,
since we already know that
$\nabla \rho$ is small, it follows that the conservation of mass
equation is approximately satisfied. Therefore, the model
equations \eqref{e1.2}, \eqref{e1.3} might be useful as an approximation
in the case of almost constant density, and nearly incompressible fluid
flow with a small velocity gradient.

Proceeding formally, we differentiate equation \eqref{e1.3} with respect
to $t$, obtaining
\[
\frac{\partial ^2 \mathbf{v}}{\partial t ^2}+\mathbf{v}\cdot
\nabla \mathbf{v}_t+\mathbf{v}_t\cdot \nabla \mathbf{v} +
\rho^{-1} \hat{p}' (\rho)\nabla \rho_t +\frac{\partial}{\partial
\rho}(\rho^{-1} \hat{p}' (\rho))\rho_t\nabla \rho = c\nabla \Delta
\rho_t
\]
Applying the divergence operator to this equation yields
\[
c\Delta^2 \rho_t -\nabla \cdot ((\rho)^{-1} \hat{p}'
(\rho)\nabla \rho_t)=\nabla \cdot (\frac{\partial}{\partial
\rho}(\rho^{-1} \hat{p}' (\rho))\rho_t\nabla \rho )+
\nabla \cdot (\mathbf{v}\cdot \nabla \mathbf{v}_t)+\nabla \cdot
(\mathbf{v}_t\cdot \nabla \mathbf{v})\quad
\]
where we used the fact that $\nabla \cdot \mathbf{v}=0$, and where
$c$ is a positive constant. Applying Lemma \ref{lem2.7} we obtain from the
above equation the following estimate for $ \nabla \rho_t$,
\begin{equation} \label{e3.28}
\begin{aligned}
\|\nabla \rho_t \|_{3}^2
&\leq C(\|\Delta \rho_t \|_2^2+\|\nabla \rho_t \|_2^2)
 \\
&\leq C[1+\sum_{j=1}^{2}\|D((\rho)^{-1} \hat{p}'
(\rho))\|_{2}^{2j}](\|D \mathbf{v}\|
_{3}^2\|D\mathbf{v}_t\|_{1}^2+\|\mathbf{v}_t
\cdot \nabla\mathbf{v}\|_{1}^2  \\
&\quad + \|\frac{\partial}{\partial \rho}(\rho^{-1}
\hat{p}' (\rho))\rho_t\nabla \rho\|_{1}^2)  \\
&\leq  C_{17}(\|D\mathbf{v}\|_{3}^2\|\mathbf{v}_t\|
_{2}^2+\|\rho_t\|_2^2\|\nabla \rho\|_{2}^2)
\end{aligned}
\end{equation}
where $C_{17}=\hat{C}_{17}(L_1,T)$. Next, by Lemma \ref{lem2.4} we have the
$L^2$ estimate
\begin{equation} \label{e3.29}
\begin{aligned}
\|\rho_t\|_{0}^2
&\leq C_0\|\rho_{t}^{0}\|_0^2+C_0\|\nabla\rho_{t}^{0}\|_2^2
+C_0\|\nabla \rho_t\|_2^2\\
&\leq C_{18}(\|\rho_{t}^{0}\|_0^2+\|\nabla\rho_{t}^{0}\|_2^2
+\|D\mathbf{v}\|_{3}^2\|\mathbf{v}_t\|
_{2}^2+\|\rho_t\|_2^2\|\nabla \rho\|
_{2}^2)\end{aligned}
\end{equation}
where $C_{18}=\hat{C}_{18}(L_1,T)$  and where we used the estimate
for $\|\Delta \rho_t \|_2^2+\|\nabla \rho_t\|_2^2$ from \eqref{e3.28}.
Here, $\rho^0$ is the initial iterate
as defined in Theorem \ref{thm3.1}. Note that by definition of $\rho^0$, we
have $\rho_t(\mathbf{x}_0,t)=\rho_t^0(\mathbf{x}_0,t)$ at the
single fixed point $\mathbf{x}_0 \in \Omega$. Also, we now specify
that $\|\rho_{t}^{0}\|_{0,T}\leq \|D\mathbf{v}_0\|_{s-1}$ and
$\|\nabla\rho_{t}^{0}\|_{2,T} \leq \|D\mathbf{v}_0\|_{s-1}$,
as an additional hypothesis for $\rho^0$. Recall that
$\epsilon_0=C_6\|D\mathbf{v}_0\|_{s-1}^2$ where $C_6=\hat{C}_6(L_1,T)$,
from Proposition \ref{prop3.1}. Then adding the estimates \eqref{e3.28}, \eqref{e3.29}
yields the following estimate for $\rho_t$:
\begin{align*}
\|\rho_t \|_{3}^2 &\leq C(\|\rho_{t}\|_0^2+\|\nabla \rho_t\|
_{2}^2) \leq C_{19} (\|\rho_{t}^{0}\|_0^2+\|\nabla\rho_{t}^{0}\|_2^2
+\|D\mathbf{v}\|_{3}^2\|\mathbf{v}_t\|_{2}^2+\|\rho_t\|_2^2\|
\nabla \rho\|_{2}^2)
\\
&\leq C_{20} (\epsilon_0+\epsilon_0\|\mathbf{v}_t\|_{2}^2
+\epsilon_0 \|\rho_t\|_3^2)
\end{align*}
where $C_{19}=\hat{C}_{19}(L_1,T)$, $C_{20}=\hat{C}_{20}(L_1,T)$
and where we used the facts that $\|D\mathbf{v}\|
_{3}^2 \leq \epsilon_0$, and $\|\nabla \rho\|_{2}^2
\leq \epsilon_0$ from Proposition \ref{prop3.1}. After moving the term
$\epsilon_0 \|\rho_t\|_3^2 $ to the left-hand side, and
substituting the estimate for $\mathbf{v}_t$ from Proposition \ref{prop3.1},
we then obtain the estimate
\[
\|\rho_t \|_{3}^2 \leq \epsilon_0 C_{21}
\]
where $C_{21}=\hat{C}_{21}(L_1,L_3,T)$. Therefore, $\rho_t$ is
small, since $\epsilon_0$ will be small if
$\|D\mathbf{v}_0\|_{s-1} $ is sufficiently small.


\section{Existence for the linear case}

 In this section, we sketch the proof of
the existence of a solution to the linear equations \eqref{e3.1}, \eqref{e3.2}.
Before going into the existence proof, which appears in Lemma \ref{lemA.5},
we first collect some preliminary technical facts. We recall that
every vector field $\mathbf{u}\in L^2(\mathbb{T}^N)$ admits a
unique orthogonal decomposition in terms of a solenoidal vector
field $\mathbf{w}$ and a potential $\nabla \phi $, so that
$\mathbf{u}=\mathbf{w}+\nabla \phi $,
where $\mathbf{w}=P\mathbf{u}$ and $\nabla \phi =Q\mathbf{u.}$ Moreover, if
$\mathbf{u}\in H^s(\mathbb{T}^N)$, with $s\geq 1$, then
$\mathbf{w}$ and $\phi $ satisfy $\nabla \cdot \mathbf{w=}0$,
$\Delta \phi =\nabla \cdot \mathbf{u.}$ The next lemma, due to
Embid \cite{e1}, provides us with an estimate that will be useful for
the existence proof.

\begin{lemma}\label{lemA.1}
If $\mathbf{u}$, $\mathbf{v}$ $\in H^r(\Omega )$, $r>\frac N2+3$,
$\Omega =\mathbb{T}^N$, then $Q(\mathbf{v\cdot }\nabla
P\mathbf{u})\in H^r(\Omega )$ and  $\|Q(\mathbf{v\cdot
}\nabla P\mathbf{u})\|_r\leq C\|\mathbf{v}\|
_r\|\mathbf{u}\|_r$.
\end{lemma}

 For a proof of this lemma, see Embid \cite{e1}.
To prove existence of a solution to \eqref{e3.1}, \eqref{e3.2},
we need the following lemmas.

\begin{lemma} \label{lemA.2}
Given $a\in H^r(\Omega )$, $f\in H^r(\Omega )$, $r>\frac N2+3$,
$\Omega =\mathbb{T}^N$, then $P(a\nabla f)\in H^r(\Omega )$, and
$\|P(a\nabla f)\|_r\leq C\|Da \|
_{r-1}\|\nabla f\|_{r-1}$, where $P$ is the
projection onto the solenoidal vector field.
\end{lemma}

\begin{proof}
 First, from the orthogonality property of the
projection $P$, we obtain an $L^2$ estimate as follows:
\[
\|P(a\nabla f)\|_0=\|P(\bar{f} \nabla a)\|_0
\leq \|\bar{f} \nabla a\|_0 \leq C|Da|
_{L^\infty }\|\bar{f}\|_0\leq C|Da|
_{L^\infty }\|\nabla f\|_0.
\]
where $\bar{f}=f-\frac {1}{|\Omega|} \int_{\Omega}fdx$.
Here, we used Poincar\'{e}'s inequality to estimate
$\|\bar{f}\|_0\leq C\|\nabla f\|_0$.

Next, from the orthogonality property of the projection $P$, and
using the commutator estimate from Lemma \ref{lem2.2}, as well as using the
triangle inequality, we obtain for $|\alpha|\geq 1$:
\begin{align*}
\|P(a\nabla f)_\alpha\|_0
&\leq \|P(a\nabla f_\alpha)\|_0
+\|P\big[ (a\nabla f)_\alpha -a\nabla f_\alpha \big]\|_0 \\
&= \|P(\bar{f}_\alpha \nabla a)\|_0+\|P[(a\nabla f)_\alpha
-a\nabla f_\alpha ]\|_0\\
&\leq C|Da|_{L^\infty }\|\bar{f}\|_k
+C\|Da\|_{k_1}\|\nabla f\|_{k-1} \\
&\leq C\|Da\|_{k_1}\|\nabla f\|_{k-1},
\end{align*}
where $|\alpha|=k$, where $k_1=\max (k-1,s_0)$,
and where $s_0=[ \frac N2]+1$. And we used the facts
that $\|\bar{f}\|_k\leq C(\|\bar{f}\|
_0+\|\nabla f\|_{k-1})$, and $\|\bar{f}\|
_0\leq C\|\nabla f\|_0$, by  Poincar\'{e}'s
inequality.

Finally, by adding the above estimates over all $0\leq |
\alpha |\leq r$, we obtain the estimate $\|P(a\nabla
f)\|_r \leq C\|Da\|_{r-1}\|\nabla
f\|_{r-1}$.
\end{proof}

\begin{lemma}\label{lemA.3}
Given $\mathbf{v}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^r)$,
and $\mathbf{f}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$, with
$r>\frac N2+3$, for $\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$,
$0\leq t\leq T$, there is a unique, classical solution
$\mathbf{u}\in C([0,T],C^3)\cap L^\infty ([0,T],H^r)$ of
\[
{D\mathbf{u}}/{Dt} =\mathbf{f},
\]
with $\mathbf{u}(\mathbf{x},0) =\mathbf{u}_0(\mathbf{x})\in H^r$,
where $D/{Dt}=\partial/ {\partial t}+\mathbf{v\cdot }\nabla $.
Moreover, $\mathbf{u}$ satisfies the estimate $\|
\mathbf{u}\|_r^2\leq e^{\alpha (t)}\left(\|
\mathbf{u}_0\|_r^2+\int_0^tC\|\mathbf{f}\|
_r^2d\tau \right)$, where $\alpha (t)=\int_0^tC(1+\|
D\mathbf{v}\|_{r-1})d\tau$ and $C$ depends on $r$.
\end{lemma}

 The proof of the above lemma is standard; see, for example, \cite{e1}.

\begin{lemma}\label{lemA.4}
Given $a$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$ and
$\mathbf{f}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r-1})$,
where $r>\frac N2+3$, $a(\mathbf{x,}t)\geq c_1$, with $c_1>0$ for
$\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$, $0\leq t\leq T$,
there is a classical solution $\rho \in C([0,T],H^0)\cap L^\infty
([0,T],H^{r+2})$ of
\begin{equation} \label{A.1}
c\Delta^2 \rho-\nabla\cdot (a\nabla \rho)=\nabla \cdot \mathbf{f}
\end{equation}
Here, $c$ is a positive constant. And $\rho$ satisfies the
estimate
\[
\|\nabla\rho\|_{r+1}^2
\leq C\left( \|\Delta \rho \|_r^2+\|\nabla\rho\|_r^2\right)
\leq C\Big[1+\sum_{j=1}^{r}\|Da\|
_{r-1}^{2j}\Big]\|\mathbf{f}\|_{r-1}^2
\leq C\|\mathbf{f}\|_{r-1}^2
\]
\end{lemma}

\begin{proof} The operator in \eqref{A.1} is linear with
$a(\mathbf{x,}t)\geq c_1$, where $c_1>0$ for
$\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$, and $c$ is a
positive constant, and the compatibility condition
$\int_{\Omega}\nabla \cdot \mathbf{f}dx=0$ is satisfied. The
existence of a solution $\rho$ follows from the standard theory
for elliptic equations, specifically, the Lax-Milgram Lemma (see,
for example, \cite{e2}). The a priori estimate follows from Lemma \ref{lem2.7}.
\end{proof}

Now we prove the existence of a solution to \eqref{e3.1}, \eqref{e3.2}.

\begin{lemma}\label{lemA.5}
Given $\mathbf{v}$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^r)$ and
 $ a$ in $C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$,  and given $r>\frac
N2+3$, $a(\mathbf{x},t)\geq c_1$, with $c_1>0$ for
$\mathbf{x}\in\Omega$, $\Omega=\mathbb{T}^N$, $0\leq t\leq T$,
there exists a classical solution $\rho$, $\mathbf{w}$ of
\begin{gather}
\frac{D\mathbf{w}}{Dt} =-a\nabla \rho+c\nabla \Delta \rho, \label{A.2}\\
\nabla \cdot \mathbf{w} = 0, \label{A.3} \\
\mathbf{w}(\mathbf{x},0) = \mathbf{w}_0(\mathbf{x}),\quad
\nabla \cdot \mathbf{w}_0 = 0, \label{A.4}
\end{gather}
with $\mathbf{w}\in C([0,T],C^3)\cap L^\infty ([0,T],H^r)$, and
$\rho \in C([0,T],C^5)\cap L^\infty ([0,T],H^{r+2})$. Here, $c$ is
a positive constant, and $D/{Dt}=\partial/ {\partial
t}+\mathbf{v\cdot }\nabla $, and $\nabla \cdot \mathbf{v}=0$.
\end{lemma}

\begin{proof} First, we reduce the equations to an equivalent
system by employing the projections $P$ and $Q=I-P$, where $P$ is
the orthogonal projection of $L^2$ onto the solenoidal vector
field and $Q$ is the orthogonal projection of $L^2$ onto the
gradient field. Applying the operator $P$ to \eqref{A.2}, and using the
fact that $P\mathbf{w=w}$, we obtain the equation
\begin{equation} \label{A.5}
\frac{D\mathbf{w}}{Dt}=J:= Q(\mathbf{v}\cdot \nabla
P\mathbf{w})-P(a\nabla\rho).
\end{equation}
Applying the operator $Q$ to \eqref{A.2}, we obtain the equation
\begin{equation} \label{A.6}
Q(\mathbf{v}\cdot \nabla P\mathbf{w}+ a\nabla \rho -c\nabla \Delta
\rho)=0.
\end{equation}
This equation is equivalent to the equation
\begin{equation} \label{A.7}
Q(Q(\mathbf{v}\cdot \nabla P\mathbf{w})+a\nabla \rho -c\nabla
\Delta \rho)=0.
\end{equation}
From the definition of $Q$, we observe that equation \eqref{A.7} is
equivalent to
\begin{equation} \label{A.8}
c\Delta^2 \rho -\nabla \cdot (a\nabla \rho) =\nabla \cdot
Q(\mathbf{v}\cdot \nabla P \mathbf{w}).
\end{equation}
With the given initial data \eqref{A.4}, the two systems of equations
\eqref{A.2}, \eqref{A.3}, and \eqref{A.5}, \eqref{A.8} are equivalent.
(The proof is standard; see, for example, \cite{e1}).

Next, we construct the solution of the system \eqref{A.5}, \eqref{A.8} using
the method of successive approximation as follows: Set
$\mathbf{w}^0=$ $\mathbf{w}_0(\mathbf{x})$ and set $\rho^0$ to be
the initial iterate from Theorem \ref{thm3.1}. For $k=0,1,2,\dots $ define
$\mathbf{w}^{k+1}$, $\rho ^{k+1}$ as the solution of the equations
\begin{gather}
\frac{D\mathbf{w}^{k+1}}{Dt} =J^k, \label{A.9} \\
c\Delta \rho^{k+1}-\nabla \cdot (a\nabla \rho^{k+1})
= \nabla \cdot Q(\mathbf{v}\cdot \nabla (P\mathbf{w}^{k+1})), \label{A.10}
\end{gather}
where $J^k=Q(\mathbf{v}\cdot \nabla P\mathbf{w}^{k}) -P(a\nabla
\rho ^{k})$. And $\mathbf{w}^{k+1}(\mathbf{x},0)=\mathbf{w}_0\in
H^r$, with $\nabla \cdot \mathbf{w}_0=0$.

The first step is to solve \eqref{A.9} for $\mathbf{w}^{k+1}$. By Lemma
\ref{lemA.1} and by the induction hypothesis we have $Q(\mathbf{v}\cdot
\nabla P\mathbf{w}^k)\in C([0,T],H^0)\cap L^\infty ([0,T],H^r)$.
Furthermore, from Lemma \ref{lemA.2} and the induction hypothesis we have
$P(a\nabla \rho ^{k})\in C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$.
Therefore, the existence of a solution $\mathbf{w}^{k+1}\in
C([0,T],H^0)\cap L^\infty ([0,T],H^{r})$ to \eqref{A.9} follows from
Lemma \ref{lemA.3}.

The next step is, given the solution $\mathbf{w}^{k+1}$ just
obtained, to solve \eqref{A.10} for $\rho^{k+1}$. From Lemma \ref{lemA.1} we have
$Q(\mathbf{v}\cdot \nabla (P\mathbf{w}^{k+1}))\in C([0,T],H^0)\cap
L^\infty ([0,T],H^r)$, and therefore it follows from Lemma \ref{lemA.4}
that there is a solution $\rho^{k+1} \in C([0,T],H^0)\cap L^\infty
([0,T],H^{r+2})$ to \eqref{A.10}. And since $\rho^{k+1}$ is unique up to
an arbitrary function of $t$, we specify that
$\rho^{k+1}(\mathbf{x}_0,t)=\rho^k(\mathbf{x}_0,t)$ for all $k\geq
0$, at the single fixed point $\mathbf{x}_0 \in \Omega$, so
$\rho^{k+1}(\mathbf{x}_0,t)=\rho^0(\mathbf{x}_0,t)$.

Next, we derive estimates for $\|\rho^{k+1}-\rho^k
\|_{r+2,T}$ and $\|\mathbf{w}^{k+1}\mathbf{-w}^k\|_{r,T}$. We see that
$\rho^{k+1}-\rho ^k$ and $\mathbf{w}^{k+1}\mathbf{-w}^k$ solve the
system of equations
\begin{gather}
\frac{D(\mathbf{w}^{k+1}-\mathbf{w}^k)}{Dt}
=J^k-J^{k-1}, \label{A.11} \\
c\Delta^2(\rho^{k+1}-\rho^k) -\nabla \cdot (a\nabla(\rho
^{k+1}-\rho ^k)) = \nabla \cdot Q(\mathbf{v}\cdot
\nabla(P(\mathbf{w}^{k+1}-\mathbf{w}^{k}))), \label{A.12}
\end{gather}
where $J^k-J^{k-1}=Q(\mathbf{v}\cdot \nabla P(\mathbf{w}^k-\mathbf{w}
^{k-1}))$ $ -P(a\nabla(\rho^k-\rho^{k-1}))$. Initially we have
$(\mathbf{w}^{k+1}-\mathbf{w}^k)(0)=0 $.
First, we derive the estimate for $\|\mathbf{w}^{k+1}-\mathbf{w}
^k\|_{r,T}$. From \eqref{A.11} and from Lemmas \ref{lemA.1},
\ref{lemA.2}, and \ref{lemA.3},
we obtain the estimate
\begin{equation} \label{A.13}
\begin{aligned}
\|\mathbf{w}^{k+1}-\mathbf{w}^k\|_r^2
&\leq Ce^{\alpha(t)}\int_0^t \|Q(\mathbf{v\cdot }\nabla
P(\mathbf{w}^k -\mathbf{w}^{k-1}))\|_r^2+\|P(a\nabla
(\rho ^{k}-\rho ^{k-1}))\|_{r}^2 d\tau  \\
&\leq C_1\int_0^t \|\mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2
+\|\nabla(\rho ^{k}-\rho ^{k-1})\|_{r-1}^2 d\tau
\end{aligned}
\end{equation}
Here $\alpha (t)=C\int_0^t(1+\|D\mathbf{v}\|
_{r-1})d\tau $, $C$ depends on $r$, and $C_1$ depends on $r$,
$\|\mathbf{v}\|_{r,T}$ and $\|a\|
_{r,T}$. Next, from \eqref{A.12}, and from Lemmas \ref{lemA.1}
and \ref{lemA.4}, we
obtain the estimate
\begin{equation} \label{A.14}
\begin{aligned}
\|\nabla( \rho ^{k+1}-\rho ^k)\|_{r+1}^2
&\leq C[1+\sum_{j=1}^{r}\|Da\|_{r-1}^{2j}]\|
Q(\mathbf{v}\cdot \nabla (P(\mathbf{w}^{k+1}-\mathbf{w}^{k})))
\|_{r-1}^2 \\
&\leq C_2 \|\mathbf{w}^{k+1}-\mathbf{w}^{k}\|_{r}^2  \\
&\leq C_3\int_0^t \left(\|
\mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2+\|\nabla( \rho
^{k}-\rho ^{k-1})\|_{r-1}^2 \right)d\tau
\end{aligned}
\end{equation}
where $C_2$ and $C_3$ depend on $r$, $\|a\|_{r,T}$,
and $\|\mathbf{v}\|_{r,T}$. Here, we used the
estimate \eqref{A.13} in the last step. From Lemma \ref{lem2.4} and \eqref{A.14}, we
have the estimate
\begin{equation} \label{A.15}
\begin{aligned}
\|\rho ^{k+1}-\rho ^k\|_{0}^2
&\leq C\|\nabla(\rho^{k+1} -\rho^{k})\|_2^2 \\
&\leq C_4 \|\mathbf{w}^{k+1}-\mathbf{w}^{k}\|_{r}^2  \\
&\leq  C_5 \int_0^t \left(\|\mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2
  +\|\nabla( \rho^{k}-\rho ^{k-1})\|_{r-1}^2 \right)d\tau
\end{aligned}
\end{equation}
where $C_4$ and $C_5$ depend on $r$, $\|a\|_{r,T}$,
and $\|\mathbf{v}\|_{r,T}$, and where
$\rho^{k+1}(\mathbf{x}_0,t)=\rho^k(\mathbf{x}_0,t)$ for all
$k\geq 0$, at the single fixed point $\mathbf{x}_0 \in \Omega$. Adding
the estimates \eqref{A.13}, \eqref{A.14}, \eqref{A.15}, we obtain
\begin{equation} \label{A.16}
\|\mathbf{w}^{k+1}-\mathbf{w}^k\|_r^2+\|\rho
^{k+1}-\rho ^k\|_{r+2}^2 \leq C_6 \int_0^t \left(\|
\mathbf{w}^k-\mathbf{w}^{k-1}\|_r^2+\|\rho ^{k}-\rho
^{k-1}\|_{r+2}^2\right)d\tau
\end{equation}
where $C_6$ depends on $r$, $\|\mathbf{v}\|_{r,T}$,
$\|a\|_{r,T}$. Repeated application of \eqref{A.16} yields
\[
\|\mathbf{w}^{k+1}-\mathbf{w}^k\|_{r,T}^2+\|
\rho
^{k+1}-\rho ^k\|_{r+2,T}^2 \leq \frac{(C_6T)^k}{k!}
\left(\|\mathbf{w}^1-\mathbf{w}^0\|_{r,T}^2+ \|
\rho ^{1}-\rho ^{0}\|_{r+2,T}^2\right)
\]
from which it follows that
\[
\sum_{k=1}^\infty \left(\|
\mathbf{w}^{k+1}-\mathbf{w}^k\|_{r,T}^2+\|\rho
^{k+1}-\rho ^k\|_{r+2,T}^2 \right)<\infty .
\]
Hence, $\|\mathbf{w}^{k+1}-\mathbf{w}^k\|
_{r,T}^2\to 0$ as $k\to \infty$ and
$\|\mathbf{\rho }^{k+1} -\mathbf{\rho
}^{k}\|_{r+2,T}^2\to 0 $ as $k\to \infty$.
Therefore, we deduce that there exist $\mathbf{w}$ $\in
C([0,T],H^0)\cap
L^\infty ([0,T],H^r)$ such that $\mathbf{w}^k\to \mathbf{w}$ as
$k\to \infty $ strongly in $C([0,T],H^0)\cap $ $L^\infty
([0,T],H^r)$, and there exists $ \rho \in
C([0,T],H^0)\cap L^\infty ([0,T],H^{r+2})$ such that
$\rho ^k\to  \rho $ as $k\to \infty $ strongly in
$C([0,T],H^0)\cap L^\infty ([0,T],H^{r+2})$. The fact that
$\mathbf{w}$ and $\rho$ is a solution to the system of equations
\eqref{A.5}, \eqref{A.8} follows by a standard argument. And therefore
$\mathbf{w}$, $\rho$ is a solution to the equivalent system \eqref{A.2},
\eqref{A.3}.
\end{proof}

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\end{document}