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\markboth{\hfil Semi-linearized compressible Navier-Stokes equations  
\hfil EJDE--2003/02}
{EJDE--2003/02\hfil Hakima Bessaih \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 02, pp. 1--18. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Semi-linearized compressible Navier-Stokes equations perturbed by noise
 %
\thanks{ {\em Mathematics Subject Classifications:} 35Q30, 76N10, 60G99.
\hfil\break\indent
{\em Key words:} Compressible Navier-Stokes equations, noise.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Submitted October 16. 2002. Published January 02, 2003.} }
\date{}
%
\author{Hakima Bessaih}
\maketitle

\begin{abstract}
  In this paper, we study semi-linearized compressible barotropic
  Navier-Stokes equations perturbed by noise in a 2-dimensional
  domain. We prove the existence and uniqueness of solutions in a
  class of potential flows.
\end{abstract}

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

\section{Introduction}

We consider the following system of equations with a stochastic
perturbation
\begin{equation} \label{e1.1}
\begin{gathered}
 {\bar{\rho}}u_{t} + \nabla p(\rho) = \mu\Delta u
 + (\mu+\lambda)\nabla\mathop{\rm div}u + G_{t}\quad
 \mbox{\rm in } Q_{T},\\
 \rho_{t} +\mathop{\rm div}(\rho u) = 0\quad \mbox{\rm in }Q_{T},
 \end{gathered}
\end{equation}
where $Q_{T} = (0,T)\times D$, $D = (0,1)^2$),  $\bar{\rho}, \lambda, \mu$
are constants such that $\bar{\rho} > 0$, $\mu > 0$, $\mu+\lambda\geq 0$;
while $G$ is a stochastic process in a function space, which we will
precise below, and $u_{t}$ and $G_{t}$ denote the derivative with respect
to $t$ in the distribution sense. $\nabla$ and $\mathop{\rm div}$ are the
gradient and divergence operators with respect to the space variables,
$\Delta$ is the Laplace operator. The space variables are denoted by
$x=(x_1,x_2)$ and the time by $t$.

In absence of the random perturbation $G_{t}$, (1.1) is reduced to
the system
\begin{equation} \label{e1.2}
\begin{gathered}
 {\bar{\rho}}u_{t} + \nabla p(\rho) = \mu\Delta u
 + (\lambda+\mu)\nabla\mathop{\rm div}u ,\\
 \rho_{t} +\mathop{\rm div}(\rho u) = 0.
 \end{gathered}
\end{equation}
This system can be considered as a semi-linearized approximation of the
compressible Navier-Stokes equations of a barotropic viscous fluid
\begin{equation} \label{e1.3}
\begin{gathered}
 \rho\left(\partial_{t}u_{i} + (u\cdot\nabla)u_{i}\right)- \partial_{i}p(\rho)
 = \sum_{j=1}^{n}\partial_{j}\left(\mu(\partial_{j}u_{i} + \partial_{i}u_{j})\right)
 + \partial_{i}(\lambda\mathop{\rm div}u), \\
 \rho_{t} +\mathop{\rm div}(\rho u) = 0,
 \end{gathered}
\end{equation}
where $i=1,\dots ,n$; $u, \rho, p(\rho)$ represent respectively the velocity
vector, the density, and the pressure; while $\mu, \lambda$ are viscosity
coefficients which according to the thermodynamic principles should satisfy
the inequalities $\mu>0$ and $3\lambda+2\mu\geq 0$.

System (1.3) has been investigated mostly for one-dimensional
flows ($n=1$). For many-dimensional flows, considerably less is known
except for  small initial data or in small time interval. A
global existence theorem for the model (1.3) has been proved
by P.L.Lions [9, 10] and Vaigant-Kazhikhov [16].
Notice that in the first $\mu$ and $\lambda$ are considered constants
while some particular requirements on the growth of the
viscosity coefficient $\lambda$ and the pressure as functions of the
density $\rho$ are imposed for the last result. The semi-linearized
system (1.2) is studied in [15], which proves the existence and uniqueness
of the strong solution. As far as the stochastic
equations for incompressible viscous fluids are concerned, some existence
theorems and some results on various aspects are known see [2, 5, 6] etc\dots
But in the compressible case, the variation of the fluid density
gives some difficulties. For this reason, only the two dimensional space is
considered here with some other restrictions. In the one dimensional case,
the full equation (1.3) subject to a perturbation is studied in [13] and
[14].

We use the standard notation $W^{l,p}$ for the Sobolev spaces consisting in the
functions which are integrable in power $p$ as well as their derivatives
up to the order $l$ and $H^{l}=W^{l,2}$; $C([0,T];X)$ denotes the space
of the continuous
functions with values in a Banach space $X$. In this paper, we use the
Orlicz space $L_{\phi}(D)$ associated to the convex function
$\phi(r) = (1+r)\log(1+r) - r,\ r\geq 0$.
We denote by $\langle .,.\rangle$ the inner product in $L^2$ and by $\|.\|$ the
corresponding norm.
We use the abbreviated notation
$$
\partial_{j}=\frac{\partial}{\partial x_{j}},\quad
\partial^2_{j}=\frac{\partial^2}{\partial x^2_{j}}\,.
$$
The propose of the present paper is to prove
the existence (and uniqueness) of a global solution to (1.1).
The solution will be constructed in the class of periodic and potential flows
as in [15], i.e., in the case where $u$ has the
form
$$ u = \nabla\varphi, $$
with some function $\varphi$, which is periodic in $x_1$ and $x_2$.
More precisely we suppose that every
function appearing in (1.1) is periodic of period 1 in $x_1$ and $x_2$
and take the equation of state
$p(\rho) = c\rho$, $c = const > 0$. We also suppose that the perturbation
$G$ is the gradient of a potential i.e. $G = \nabla W$.
For simplicity, we assume that
the constants $c$ and ${\bar{\rho}}$ are equal to 1 and the constants $\lambda$
and $\mu$ are respectively equal to 1/4 and 1/2 and impose
$\int_{D}\varphi(t,x){\rm d}x = 0$.
When
$$ \int_{D}W(t,x){\rm d}x = 0 $$
which will follows from the assumptions of section 2, integrating the
momentum equation
$(1.1)_1$, the system acquires the form
\begin{equation}
\begin{gathered}
 {\rm d}\varphi = (\Delta\varphi + 1 -\rho){\rm d}t + {\rm d}W
\quad \mbox{\rm in } Q_{T},\\
 \rho_{t} +\mathop{\rm div}(\rho\nabla\varphi) = 0.
 \end{gathered}
\end{equation}
Below, $W$ will be a Wiener process taking values in a particular Hilbert
space.
The unknown functions are assumed to take prescribed values at the initial
time,
\begin{gather*}
\rho|_{t=0} = \rho_{0}(x) \geq 0,\quad \int_{D}\rho_{0}(x)dx = 1, \\
\varphi|_{t=0} = \varphi_{0}(x),\ \ \ \int_{D}\varphi_{0}(x)dx = 0.
\end{gather*}
In addition, we impose the following natural requirement on the solution,
$$\rho(x,t) \geq 0\quad \mbox{\rm in } Q_{T}.
$$

\section{Main result}  %sec 2

Before stating the existence results, we have to precise some conditions
on the noise term appearing in (1.1). We set
$$
D(A) = \big\{ u\in H^2(D) : \mbox{$u$ is periodic of period 1 in
$x_1$ and $x_2$}, \int_{D}u{\rm d}x = 0\big\}.
$$
and define a linear operator
$$
A: D(A)\to \big\{ u\in L^2(D) :
\mbox{$u$ is periodic of period 1 in $x_1$ and $x_2$}\big\},
$$
as $Au = - \Delta u$. The operator $A$ is self-adjoint with compact resolvent.
We denote by  $0 <\lambda_1\leq\lambda_2\dots $ ($\lim\lambda_{j} = \infty$)
the eigenvalues of $A$ and by $e_1, e_2\dots $ the corresponding complete
orthonormal system of eigenvectors. As well known, for the space of periodic
functions the eigenvectors are trigonometric functions and we see easily
that
$\int_{D}e_{j}(x){\rm d}x = 0$, $j=1,2\dots $.
Let
\begin{equation} \label{e2.1}
W(t) = \sum_{j=1}^{\infty}\sigma_{j}\beta_{j}(t)e_{j}(x).
\end{equation}
where $\{\sigma_{j}\}_{j=1}^{\infty}$ is a sequence of constants
satisfying the condition
\begin{equation} \label{e2.2}
\sum_{j=1}^{\infty}\lambda_{j}^{\delta+2}\sigma_{j}^2 < \infty,
\end{equation}
with some $\delta > 0$ while
$\beta_1, \beta_2,\dots$ are independent standard
1-dimensional Brownian motions defined on a complete
probability space $(\Omega, {\cal F}, {\bf P})$ adapted to a filtration
$\left\{{\cal F}_{t}\right\}_{t\geq 0}$.
We denote by ${\bf E}$ the expectation relative to
$(\Omega, {\cal F}, {\bf P})$.

Now, we state the main theorem of this paper.

\begin{theorem}
Let $(\Omega, {\cal F}, {\bf P})$ be a probability space and $T$ a positive
number. Suppose that $W$ is a Wiener process satisfying (2.1) and
the condition (2.2), and that $\rho_{0}$ and $\varphi_{0}$
are two Random variables with values respectively in $L^{\infty}(D)$
and $W^{1,q}(D)\cap H^2(D)$ ($q\geq 1$) satisfying respectively
the conditions (1.5) and (1.6) {\bf P}-a.s. and
$\ \inf_{x\in D}\rho_{0}(x)>0\ $ and
$\ \sup_{x\in D}\rho_{0}(x)<\infty\ $ {\bf P}-a.s.
Then there exists a unique solution to (1.4) up to a modification.
Besides $\rho$ satisfies $\ \inf_{Q_{T}}\rho(x,t)>0\ $ and
$\ \sup_{Q_{T}}\rho(x,t)<\infty\ $ {\bf P}-a.s.
\end{theorem}

\section{Reduction of the problem via the Ornstein-Uhlenbeck equation}
%sec 3

Let us consider an auxiliary problem, the Ornstein-Uhlenbeck equation,
\begin{equation} \label{e3.1}
\begin{gathered}
      dz(t) + Az(t)dt = dW(t),\\
       z(0) = 0.
\end{gathered}
\end{equation}
This equation has a solution given by the process (see [4])
\begin{equation} \label{e3.2}
z(t) = \int_{0}^{t} e^{-(t-s)A}dW(s),
\end{equation}
where ${\rm e}^{-tA}$ denotes a $C_{0}$-semigroup generated by $A$.
The regularity of $z(t)$ depends on the regularity of $W(t)$. Indeed, we
have for an arbitrary $k >0$
$$
A^{k}z(t) = \sum_{j=1}^{\infty}\int_{0}^{t}\lambda_{j}^{k}
{\rm e}^{-(t-s)\lambda_{j}}\sigma_{j}{\rm d}\beta_{j}(s)e_{j}.
$$
\begin{eqnarray*}
{\bf E}\| z(t)\|_{D(A^{k})}^2 &=& {\bf E}\| A^{k}z(t)\|^2 \\
&=&{\bf E}\Big(\sum_{j=1}^{\infty}\int_{0}^{t}\lambda_{j}^{k}
{\rm e}^{-(t-s)\lambda_{j}}\sigma_{j}{\rm d}\beta_{j}(s)\Big)^2\\
&=&  \sum_{j=1}^{\infty}\int_{0}^{t}|\lambda_{j}^{k}
{\rm e}^{-(t-s)\lambda_{j}}\sigma_{j}|^2{\rm d}s \\
&=&  \sum_{j=1}^{\infty}\frac{\lambda_{j}^{2k}\sigma_{j}^2}{2\lambda_{j}}
(1-{\rm e}^{-2t\lambda_{j}}).
\end{eqnarray*}
According to (2.2), $W(t)$ belongs in $D(A^{(\delta+2)/2})$ for
some $\delta>0$ which yields, for $k=(\delta+3)/2$ in the above equality,
that $z(t)$ has continuous trajectories taking values in
$D(A^{(3+\delta)/2})$ (as we will need in the next
sections), i.e. $z(t)\in C([0,T];D(A^{(3+\delta/2}))$
{\bf P}-a.s. for some $\delta >0$.

Following the idea of Bensoussan-Temam [2], we set
\begin{equation} \label{e3.3}
 y(t) = \varphi(t) - z(t).
\end{equation}
Using this change of variable in (1.4) and equation (3.1), one obtains
the system
\begin{equation} \label{e3.4}
\begin{gathered}
 y_{t} - \Delta y = 1 - \rho \quad \mbox{\rm in } Q_{T},\\
 \rho_{t} +\mathop{\rm div}(\rho\nabla(y+z)) = 0\quad \mbox{\rm in } Q_{T}.
\end{gathered}
\end{equation}

\section{Reduced deterministic problem}

In this section, we study the following reduced deterministic problem
\begin{equation} \label{e4.1}
\begin{gathered}
 y_{t} - \Delta y = 1 - \rho \quad \mbox{\rm in } Q_{T},\\
 \rho_{t} +\mathop{\rm div}(\rho\nabla(y+z)) = 0\quad \mbox{\rm in } Q_{T},
 \end{gathered}
\end{equation}
where $z(t)$ is a continuous function taking values in $H^{3+\delta}(D)$,
$\delta>0$.
For this problem, we state the following existence  and uniqueness theorem.

\begin{theorem} \label{thm4.1}
Let $T$ be positive number and suppose that $y_{0}\in W^{2,s}(D)$ and
$\rho_{0}\in L^{s}(D)$, $s\geq 2$. We suppose also that
$z\in C^{0}([0,T];H^{3+\delta}(D))$ ($\delta > 0 )$). Then
there exists at least one solution $(y,\rho)$ to Problem (4.1) which
satisfies
\begin{gather*}
y\in L^{\infty}(0,T;W^{1.q}(D))\cap L^2(0,T;H^2(D)),\\
y_{t}\in L^{\infty}(0,T;L^{s}(D))\cap L^2(0,T;H^{1}(D)),\\
\rho\in L^{\infty}(0,T;L_{\phi}(D))\cap L^{s}(Q_{T}),
\end{gather*}
where $q\geq 2$.
Moreover, if $\inf_{x\in D}\rho_{0}(x)>0$ and
$\sup_{x\in D}\rho_{0}(x)<\infty$,
then $\inf_{Q_{T}}\rho(x,t)>0$,  $\sup_{Q_{T}}\rho(x,t)<\infty$,
and (4.1) is uniquely solvable.
\end{theorem}

The proof follows the lines of Vaigant-Kazhikhov [15],
of which we will use the ideas without quote them explicitly.

\subsection{A priori estimates and existence of solutions} %4.1

In this section, we obtain {\it a}\ {\it priori} estimates that permit us to
prove the existence of a solution.
The first energy estimate is obtained by multiplying the first equation
in (4.1) by $\Delta y$ and the second equation by $\log\rho$, followed by
integrating over $D$. More precisely, we obtain
\begin{equation}
\frac{d}{dt}\int_{D}\Big(\frac{1}{2}|\nabla y|^2 + \rho\log\rho
-\rho + 1\Big)dx + \int_{D}|\Delta y|^2 \leq\|\Delta z\|_{L^\infty}.
\end{equation}
This relation implies that the solution is bounded in the norms of the
spaces
$$
y\in L^{\infty}(0,T;H^{1}(D)),\quad \Delta y\in L^2(0,T;L^2(D)),\quad
\rho\in L^{\infty}(0,T;L_{\phi}(D)).
$$
The following lemma may be derived from the second equation in system (4.1).

\begin{lemma} \label{lm4.2}
If $\rho_{0}\in L^{p-1}(D)$ then there exists a constant $C$ depending
on $p$ such that the inequality
\begin{equation} \label{e4.3}
\begin{aligned}
&\|\rho(t)\|_{L^{p-1}(D)}^{p-1}
+ \int_{0}^{t}\|\rho(\tau)\|_{L^{p}(D)}^{p}d\tau \\
&\leq C\Big(\|\rho_{0}\|_{L^{p-1}(D)}^{p-1}
+ \int_{0}^{t}\| y_{\tau}(\tau)\|_{L^{p}(D)}^{p}d\tau
+ \int_{0}^{t}\|\Delta z(\tau)\|_{L^{p}(D)}^{p}d\tau\Big)
\end{aligned}
\end{equation}
holds for any exponent $p$, $2<p<\infty$ and any $t\in [0,T]$.
\end{lemma}

\paragraph{Proof:}
For $r>1$ using $(4.1)_1$, the expression $(4.1)_2$ can be rewritten
in the form
\begin{equation}
\frac{\partial\rho^{r}}{\partial t} + \nabla\cdot(\rho^{r}\nabla(y+z))
+ (r-1)\rho^{r+1} = (r-1)\rho^{r}(y_{t}-1- \Delta z).
\end{equation}
Integrating over $D$ and estimating $\rho^{r}|y_{t}|$ and $\rho^{r}|\Delta z|$
by the Young inequality,
\begin{gather*}
\rho^{r}|y_{t}| \leq \epsilon_1\rho^{r+1} + C_1|y_{t}|^{r+1},\\
\rho^{r}|\Delta z| \leq \epsilon_2\rho^{r+1} + C_2|\Delta z|^{r+1},
\end{gather*}
with convenient small numbers $\epsilon_1,\epsilon_2$, we obtain (4.3) for
$p = r + 1$. \hfill$\square$

\begin{lemma} \label{lm4.3}
If $\rho_{0}\in L^{p-1}(D)$, then there exists a constant $C$ depending
on $p$ such that the inequality
\begin{equation} \label{e4.5}
\| y\| _{W^{2,p}(Q_{T})}\leq C(\|\rho_{0}\|_{L^{p-1}(D)}
+ \| y_{t}\|_{L^p(Q_{T})} + \|\Delta z\|_{L^p(Q_{T})})
\end{equation}
holds for $2<p<\infty$.
\end{lemma}
The proof of this lemma is is a consequence of (4.3) and the first
equation of (4.1).

Now, to obtain additional a priori estimates, we differentiate
$(4.1)_1$ with respect to $x_1, x_2, t$ so that we obtain
\begin{gather}
\nabla y_{t} - \Delta\nabla y = - \nabla\rho, \label{e4.6}\\
y_{tt} - \Delta y_{t} = - \rho_{t}
= \mathop{\rm div}(\rho\nabla(y+z)). \label{e4.7}
\end{gather}

\begin{lemma} \label{lm4.4}
If $\rho_{0}\in L^{3}(D)$, $y_{0}\in W^{1,q}(D)\cap H^2(D)$,
where $q\geq 4$ then there exists a constant C depending on $q$ such that
the inequality
\begin{equation}
\sup_{0<\tau<t}\Big(\int_{D}|\nabla y|^{q} + \int_{D}|y_{t}|^2\Big)
+ \int_{0}^{t}\int_{D}|\nabla y_{t}|^2\leq C
\end{equation}
holds for all $t\in [0,T]$.
\end{lemma}
\paragraph{Proof:}
For arbitrary $q\geq 2$ and $s\geq 2$, we multiply (4.6) by
$q|\nabla y|^{q-2}\nabla y$ and (4.7) by $s|y_{t}|^{s-2}y_{t}$, sum these
equations and integrate over $D$ to obtain
\begin{equation} \label{e4.9}
\begin{aligned}
&\frac{d}{dt} \int_{D}\left(|\nabla y|^{q} + |y_{t}|^{s}\right)
+ q\sum_{j,k}^2\int_{D}(\partial_{j}\partial_{k}y)^2|\nabla y|^{q-2}\\
&+q(q-2)\sum_{j,k,l}^2\int_{D}(\partial_{j}\partial_{k}y)(\partial_{j}\partial_{l}y)
(\partial_{k}y)(\partial_{l}y)|\nabla y|^{q-4}
+ s(s-1)\int_{D}|\nabla y_{t}|^2|y_{t}|^{s-2} \\
&= q\int_{D}\rho\Delta y|\nabla y|^{q-2}
+q(q-2)\sum_{j,k}^2\int_{D}\rho(\partial_{j}y)(\partial_{k}y)(\partial_{j}\partial_{k}y)
|\nabla y|^{q-4}\\
&\quad - s(s-1)\int_{D}\rho\nabla(y+z)\cdot\nabla y_{t}|y_{t}|^{s-2}.
\end{aligned}
\end{equation}
By taking $q=4$ and $s=2$ in (4.9) and substituting $\rho = 1+\Delta y-y_{t}$,
we have
\begin{align*}
&\frac{d}{dt} \int_{D}\left(|\nabla y|^{4} + |y_{t}|^2\right)
+ 4\sum_{j,k}^2\int_{D}(\partial_{j}\partial_{k}y)^2|\nabla y|^2
+ 2\int_{D}|\nabla y_{t}|^2 \\
&= 4\int_{D}\Delta y|\nabla y|^2 + 4\int_{D}(\Delta y)^2|\nabla y|^2
- 4\int_{D}(\Delta y)y_{t}|\nabla y|^2 \\
&\quad+ 8\sum_{j,k}^2\int_{D}(\partial_{j}y)(\partial_{k}y)(\partial_{j}\partial_{k}y)
+ 8\sum_{j,k}^2\int_{D}\Delta y(\partial_{j}y)(\partial_{k}y)(\partial_{j}\partial_{k}y)
\\
&\quad- 8\sum_{j,k}^2\int_{D}y_{t}(\partial_{j}y)(\partial_{k}y)(\partial_{j}\partial_{k}y)
- 2\int_{D}\nabla(y+z).\nabla y_{t} - 2\int_{D}\Delta y\nabla(y+z).\nabla y_{t}
\\
&\quad+ 2\int_{D}\nabla(y+z).\nabla y_{t}y_{t}
- 8\sum_{j,k,l}^2\int_{D}(\partial_{j}\partial_{k}y)(\partial_{j}\partial_{l}y)
(\partial_{k}y)(\partial_{l}y)\\
&= I_1+\dots +I_{10}\,.
\end{align*}
The assumption that $D$ is a two-dimensional region is essential for
this estimate.
$$ I_2 = 4\int_{D}|\nabla y|^2|\Delta y|^2\leq
\|\Delta y\|\|\Delta y\|_{L^4(D)}\||\nabla y|^2\|_{L^4(D)}.
$$
On the other hand
$$\||\nabla y|^2\|_{L^4(D)} \leq \||\nabla y|^2\|^{1/2}\|\nabla|\nabla y|^2\|^{1/2}.
$$
Then using Young's inequality twice, we obtain for arbitrary positive
small numbers $\epsilon_1$ and $\epsilon_2$ such that
$$
I_2 \leq \epsilon_1\|\Delta y\|\|\Delta y\|_{L^4(D)}^2 + \epsilon_2\|\nabla|\nabla y|^2\|^2
+ C\|\Delta y\|^2\||\nabla y|^2\|^2.$$
Since
$$\|\nabla|\nabla y|^2\|^2 = \int_{D}|\nabla|\nabla y|^2|^2
= 2\sum_{j,k}^2\int_{D}|\nabla y|^2(\partial_{j}\partial_{k}y)^2
$$
and
$$\||\nabla y|^2\|^2 = \int_{D}(|\nabla y|^2)^2
= \int_{D}|\nabla y|^{4} ,$$
one obtains
$$I_2 \leq \epsilon_1\|\Delta y\|\|\Delta y\|_{L^4(D)}^2
+ \epsilon_2\sum_{j,k}^2\int_{D}|\nabla y|^2(\partial_{j}\partial_{k}y)^2
+ C\|\nabla y\|_{L^4(D)}^{4}\|\Delta y\|^2.
$$
By Young's inequality and for arbitrary small positive constant $\epsilon$,
$\epsilon_1$ and $\epsilon_2$ we can estimate $I_1$, $I_{3}$, $I_{4}$, $I_{5}$
and $I_{6}$ as follows:
$$
I_1 = 4\int_{D}|\nabla y|^2\Delta y \leq \epsilon I_2 + C\int_{D}|\nabla y|^2,
$$
\begin{align*}
I_{3} &= -4\int_{D}\Delta y|\nabla y|^2y_{t}\\
&\leq \epsilon_1\int_{D}|\nabla y|^2||\Delta y|^2 + \epsilon_2\|\nabla y_{t}\|^2
+ C\| y_{t}\|^2\|\nabla y\|^2\|\Delta y\|^2,
\end{align*}
$$
I_{4} = 8\sum_{j,k}^2\int_{D}(\partial_{j}y)(\partial_{k}y)(\partial_{j}\partial_{k}y)
\leq\epsilon\sum_{j,k}^2\int_{D}(\partial_{k}y)^2(\partial_{j}\partial_{k}y)^2
+ C\int_{D}|\nabla y|^2,
$$
$$
I_{5} = 8\sum_{j,k}^2\int_{D}\Delta y(\partial_{j}y)(\partial_{k}y)(\partial_{j}\partial_{k}y)
\leq \epsilon\sum_{j,k}^2\int_{D}(\partial_{k}y)^2(\partial_{j}\partial_{k}y)^2+ C I_2.
$$
\begin{align*}
I_{6} &= -8\sum_{j,k}^2\int_{D}y_{t}(\partial_{j}y)(\partial_{k}y)(\partial_{j}\partial_{k}y)\\
&\leq \epsilon_1\sum_{j,k}^2\int_{D}(\partial_{k}y)^2(\partial_{j}\partial_{k}y)^2
+ \epsilon_2\|\nabla y_{t}\|^2
+ C\sum_{j,k}^2\| y_{t}\|^2\|\nabla y\|^2 \|\partial_{j}\partial_{k}y\|^2.
\end{align*}
$$
I_{7} = -2\int_{D}\nabla(y+z)\cdot\nabla y_{t} \leq \epsilon\|\nabla y_{t}\|^2
+ C\int_{D}|\nabla(y+z)|^2.
$$
$$
I_{8} = -2\int_{D}\Delta y\nabla(y+z)\cdot\nabla y_{t}
\leq \epsilon\|\nabla y_{t}\|^2 + C I_2
+ \|\nabla z\|_{L^{\infty}(D)}^2\int_{D}|\Delta y|^2.
$$
$$
I_{9} = 2\int_{D}\nabla(y+z)\cdot\nabla y_{t} y_{t}
\leq \epsilon\|\nabla y_{t}\|^2
+ C\| y_{t}\|^2\|\Delta(y+z)\|^2\|\nabla(y+z)\|^2.
$$
\begin{align*}
I_{10} \leq& \sum_{j,k,l}^2\Big(\int_{D}(\partial_{j}\partial_{k}y)^2
\Big)^{1/2}\Big(\int_{D}(\partial_{j}\partial_{l}y)^{4}\Big)^{1/4}
\Big(\int_{D}(\partial_{k}y)^{8}\Big)^{1/8}\Big(\int_{D}(\partial_{l}y)^{8}
\Big)^{1/8}\\
\leq& \epsilon_1\sum_{j,k,l}^2 \|\partial_{j}\partial_{k}y\|\|\partial_{j}\partial_{l}y\|_{L^{4}(D)}^2
+ \epsilon_2\|\nabla|\nabla y|^2\|^2\\
&+ C_{\epsilon_1,\epsilon_2}\sum_{j,k}\||\nabla y|^2\|^2
\|\partial_{j}\partial_{k}y\|^2.
\end{align*}
We set
\begin{gather*}
 \alpha(t) = \sup_{0<t<T}\int_{D}|\nabla y|^{4} + |y_{t}|^2, \\
\beta(t) = \sum_{j,k}^2\int_{D}(\partial_{j}\partial_{k}y)^2|\nabla y|^2
+ \int_{D}|\nabla y_{t}|^2.
\end{gather*}
On the other hand from (4.5) in particular for $p=4$ and
$\rho_{0}\in L^{3}(D)$, we have
$$
\| y\|_{W^{2,4}(D)}^2\leq C\big(\|\rho_{0}\|_{L^{3}(D)}
+ \| y_{t}\|_{L^4(D)}^2 + \|\Delta z\|_{L^4(D)}^2\big).
$$
Consequently, by using Cauchy's inequality and the energy estimate (4.2)
we obtain
$$
\epsilon\int_{0}^{t}\|\Delta y\|\|\Delta y\|_{L^4(D)}^2\leq
C\epsilon\Big(\int_{0}^{t}\|\Delta y\|_{L^4(D)}^{4}\Big)^{1/2}.
$$
But in the two-dimensional space we have
\begin{equation} \label{*}
\| y_{t}\|_{L^4(D)}^{4} \leq C\| y_{t}\|^2\|\nabla y_{t}\|^2, %(*)
\end{equation}
so that
$$
\epsilon\int_{0}^{t}\|\Delta y\|\|\Delta y\|_{L^4(D)}^2\leq
C\epsilon\Big(\alpha(t)^{1/2}\Big(\int_{0}^{t}\beta(\tau)d\tau\Big)^{1/2}
+ \|\Delta z\|^2_{L^{4}(Q_{T})}\Big).
$$
It is the same for $\epsilon_1\sum_{j,k,l}^2
\int_{0}^{t}\|\partial_{j}\partial_{k}y\|\|\partial_{j}\partial_{l}y\|_{L^{4}(D)}^2$.


Set $\Lambda(t) = \|\Delta y(t)\|^2$. Since this is integrable
on $[0,T]$, we have
$$
\alpha(t) + \int_{0}^{t}\beta(\tau)d\tau
\leq \alpha(0) + \epsilon\alpha(t)^{1/2}\big(\int_{0}^{t}\beta(\tau)d\tau\big)^{1/2}
+ C\big(1 + \int_{0}^{t}(\Lambda(\tau) + 1)\alpha(\tau)d\tau\big).
$$
Using Gronwall's lemma we obtain,
\begin{equation} \label{e4.11}
\sup_{0<\tau<t)}\int_{D}\left(|\nabla y|^{4} + |y_{\tau}|^2\right)
+  \int_{0}^{t}\int_{D}|\nabla y_{\tau}|^2\leq C.
\end{equation}
According to this equation,
\begin{equation} \label{e4.12}
\int_{0}^{T}\int_{D}|y_{t}|^{4}\leq
\int_{0}^{T}\| y_{t}\|^2\|\nabla y_{t}\|^2\leq C.
\end{equation}
Hence, using this inequality in (4.3) and (4.5) for $p=4$, it follows that
\begin{equation} \label{e4.13}
\int_{0}^{t}\int_{D}\rho^{4} + \int_{0}^{t}\int_{D}|\Delta y|^{4}\leq C.
\end{equation}
Now, let us consider the equation (4.9) for $q\geq 4$ and $s=2$. We obtain
the equality
\begin{equation} \label{e4.14}
\begin{aligned}
&\frac{d}{dt}\int_{D}\left(|\nabla y|^{q} + |y_{t}|^2\right)
+ q\sum_{j,k}^2\int_{D}(\partial_{j}\partial_{k}y)^2|\nabla y|^{q-2}
+ 2\int_{D}|\nabla y_{t}|^2 \\
&= - 2\int_{D}\rho\nabla(y+z)\cdot\nabla y_{t}
+ q\int_{D}\rho\Delta y|\nabla y|^{q-2}\\
&\quad+ q(q-2)\sum_{j,k}^2\int_{D}\rho(\partial_{j}y)(\partial_{k}y)
(\partial_{k}\partial_{j}y)|\nabla y|^{q-4} \\
&\quad - q(q-2)\sum_{j,k,l}^2\int_{D}(\partial_{j}\partial_{k}y)(\partial_{j}\partial_{l}y)
(\partial_{k}y)(\partial_{l}y)|\nabla y|^{q-4}.
\end{aligned}
\end{equation}
The first term on the right hand side of \eqref{e4.14} by Young's inequality is
bounded by
$$
\int_{0}^{T}\int_{D}\rho\nabla(y+z)\cdot\nabla y_{t}\leq
\epsilon\int_{0}^{T}\|\nabla y_{t}\|^2
+ C \int_{0}^{T}\|\rho\|_{L^4(D)}^2\|\nabla(y+z)\|_{L^4(D)}^2.
$$
In view of \eqref{e4.11} and \eqref{e4.13}, $\rho$ and $\nabla y$ are bounded respectively
in $L^{4}(Q_{T})$ and in $L^{\infty}(0,T;L^{4}(D))$. Hence
$$
\int_{0}^{T}\int_{D}\rho\nabla(y+z)\cdot\nabla y_{t}\leq
\epsilon\int_{0}^{T}\|\nabla y_{t}\|^2 + C\int_{0}^{T}\|\nabla z\|^{4}_{L^4(D)}.
$$
Using the Young and the H\"{o}lder's inequality for the second term on the
right hand side of \eqref{e4.14}, one obtains
\begin{eqnarray*}
\int_{D}\rho\Delta y|\nabla y|^{q-2}&\leq& \epsilon\int_{D}|\Delta y|^2|\nabla y|^{q-2}
+ C\int_{D}\rho^2|\nabla y|^{q-2} \\
&\leq& \epsilon\int_{D}|\Delta y|^2|\nabla y|^{q-2}
+ C\|\rho\|_{L^4(D)}^2\Big(\int_{D}|\nabla y|^{2(q-2)}\Big)^{1/2}.
\end{eqnarray*}
We write the last integral in the above inequality as
$$
\Big(\int_{D}|\nabla y|^{2(q-2)}\Big)^{1/2}
= \||\nabla y|^{q/2}\|_{L^{4(q-2)/q}}^{2(q-2)/q}.
$$
Using the embedding inequality (see [8], pp 62)
\begin{equation} \label{**}
\| f\|_{L^{4(q-2)/q}}\leq C\|\nabla f\|^{a}\| f\|^{1-a},
\end{equation}
for the function $f = |\nabla u|^{q/2}$, where $a = (q-4)/(2(q-2))$,
yields
\begin{align*}
&\int_{D}\rho\Delta y|\nabla y|^{q-2}\\
&\leq C\|\rho\|_{L^4}^2\|\nabla(|\nabla y|^{q/2}|)\|^{(q-4)/q}
\||\nabla y|^{q/2}\| + \epsilon\int_{D}|\Delta y|^2|\nabla y|^{q-2}.
\end{align*}
Since $\||\nabla y|^{q/2}\| = \Big(\int_{D}|\nabla y|^{q}\Big)^{1/2}$,
and
$$
\|\nabla(|\nabla y|^{q/2})\| = \frac{q}{2}\Big(\sum_{j,k}\int_{D}
(\partial_{j}\partial_{k})^2|\nabla y|^{q-2}\Big)^{1/2},
$$
and in virtue of the Young's inequality with $p=2q/(q-4)$ and
$p'=2q/(q+4)$ we obtain
$$
\int_{D}\rho\Delta y|\nabla y|^{q-2}\leq
\epsilon\sum_{j,k}\int_{D}(\partial_{j}\partial_{k})^2|\nabla y|^{q-2}
+ C\|\rho\|_{L^4(D)}^{4q/(q+4)}\Big(\int_{D}|\nabla y|^{q}\Big)^{q/(q+4)}.
$$
By using the same argument for the third term on the right hand side of \eqref{e4.14}
we obtain, for an arbitrary $\epsilon$,
\begin{align*}
&\sum_{j,k}^2\int_{D}\rho(\partial_{j}y)(\partial_{k}y)(\partial_{k}
\partial_{j}y)|\nabla y|^{q-4}\\
&\leq \epsilon\sum_{j,k}^2\int_{D}|\nabla y|^{q-2}(\partial_{j}\partial_{k}y)
+ \|\rho\|_{L^4(D)}^{4q/(q+4)}
\Big(\int_{D}|\nabla y|^{q}\Big)^{q/(q+4)}.
\end{align*}
For the last term on the right hand side of \eqref{e4.14}, we use the same arguments
as before, we have
\begin{align*}
& \sum_{j,k,l}^2\int_{D}(\partial_{j}\partial_{k}y)(\partial_{j}\partial_{l}y)(\partial_{k}y)
(\partial_{l}y)|\nabla y|^{q-4}\\
&\leq \epsilon\sum_{j,k,l}^2\int_{D}|\nabla y|^{q-2}(\partial_{j}\partial_{k}y)^2
+C_{\epsilon}\sum_{j,l}^2\|\partial_{j}\partial_{l}y\|_{L^{4}(D)}^{4q/(q+4)}
\Big(\int_{D}|\nabla y|^{q}\Big)^{q/(q+4)}
\end{align*}
Consequently, using (4.5) for $p=4$ and (*), we have
\begin{align*}
& \int_{D}\left(|\nabla y|^{q} + |y_{t}|^2\right)
+ \int_{0}^{T}\int_{D}|\nabla y_{t}|^2\\
&\leq C\big(\|\nabla y_{0}\|_{L^q(D)}^{q} + \|\rho_{0}\|^2
+ \|\Delta y_{0}\|^2 + \|\Delta z\|^2_{L^4(Q_{T})}\big) \\
&\quad+ \Big(\int_{0}^{T}\|\rho\|_{L^4(D)}^{4q/(q+4)}
+\| y_{t}\|^{2q/(q+4)}\|\nabla y_{t}\|^{2q/(q+4)}\\
&\quad+ \|\Delta z\|^{4q/(q+4)}_{L^4(Q_{T})}
+ \|\rho_{0}\|^{4q/(q+4)}_{L^3(D)}\Big)
\Big(\int_{D}|\nabla y|^{q}\Big)^{q/(q+4)}
\end{align*}
Now using Gronwall's lemma, \eqref{e4.12} and \eqref{e4.13}, we obtain (4.8).
\hfill$\square$

\begin{lemma} \label{lm4.5}
If $\rho_{0}\in L^{s}(D)$, $y_{0}\in W^{2,s}(D)$
then the inequality
\begin{equation} \label{e4.16}
\sup_{0<\tau<t}\int_{D}|y_{\tau}|^{s}dx \leq C
\end{equation}
holds for $s>2$ and $t\in [0,T]$.
\end{lemma}

\paragraph{Proof:}
We multiply  (4.7) by $s|y_{t}|^{s-2}y_{t}$, $s>2$ and then we
integrate over $D$, to obtain
\begin{equation} \label{e4.17}
\frac{d}{dt}\int_{D}|y_{t}|^{s}
+ s(s-1)\int_{D}|y_{t}|^{s-2}|\nabla y_{t}|^2 =
-s(s-1)\int_{D}\rho\nabla(y+z)\cdot\nabla y_{t}|y_{t}|^{s-2}.
\end{equation}
Cauchy's inequality applied to the right hand side of the above quality gives
$$
\big|\int_{D}\rho\nabla(y+z)\cdot\nabla y_{t}|y_{t}|^{s-2}\big|
\leq \epsilon\int_{D}|y_{t}|^{s-2}|\nabla y_{t}|^2
+ C\int_{D}\rho^2|\nabla(y+z)|^2|y_{t}|^{s-2}.
$$
According to $(4.1)_1$, with $\rho = 1 + \Delta y - y_{t}$, \eqref{e4.17} yields
\begin{equation} \label{e4.18}
\begin{aligned}
&\frac{d}{dt}\int_{D}|y_{t}|^{s}dx +\int_{D}|y_{t}|^{s-2}|\nabla y_{t}|^2\\
&\leq C(\int_{D}|\nabla(y+z)|^2|y_{t}|^{s-2}
+\|\nabla(y+z)\|_{C(D)}^2\int_{D}\left(|y_{t}|^{s}
+ |\Delta y|^2|y_{t}|^{s-2}\right)).
\end{aligned}
\end{equation}
The first term on the right hand side of \eqref{e4.18}, using Young's inequality
with $p=s/2$ and $p'=s/(s-2)$, is estimated as
$$
\int_{D}|\nabla(y+z)|^2|y_{t}|^{s-2}\leq
\big(\|\nabla y\|_{L^s(D)}^2 + \|\nabla z\|_{L^s(D)}^2\big)
\| y_{t}\|_{L^s(D)}^{s-2}.$$
According to (4.5)
$$
\int_{D}|\nabla(y+z)|^2|y_{t}|^{s-2}\leq C\big(\| y_{t}\|_{L^s(D)}^{s}
+ \| y_{t}\|_{L^s(D)}^{s-2}\|\nabla z\|_{L^s(D)}^2\big).
$$
Using H\"{o}lder's inequality, the third term on the right hand side
of \eqref{e4.18} yields
$$
 \int_{D}|\Delta y|^2|y_{t}|^{s-2}\leq \| y_{t}\|_{L^s(D)}^{s-2}
\|\Delta y\|_{L^s(D)}^2.
$$
On the other hand, from the Gagliardo-Nirenberg's inequality it
follows
$$
\|\nabla y\|_{C(D)} \leq\|\Delta y\|_{L^4(D)}^{b}\|\nabla y\|_{L^q(D)}^{1-b}.
$$
For $b = 4/(q+4)$ and by (4.8) and \eqref{e4.13}, we obtain
\begin{equation} \label{e4.19}
\int_{0}^{t}\|\nabla y\|_{C(D)}^{q+4} \leq
\int_{0}^{t}\|\Delta y\|_{L^4(D)}^{4}\|\nabla y\|_{L^q(D)}^{q}
\leq C.
\end{equation}
We set $ \beta_{s}(t) = \sup_{0<\tau<t}\| y_{\tau}\|_{L^s}^{s}$. Then the
expression on the right-hand side of \eqref{e4.18}, after integrating on (0,t),
can be estimated as
\begin{equation} \label{e4.20}
\begin{aligned}
&\| y_{\tau}\|_{L^s(D)}^{s} + \int_{0}^{t}\int_{D}|y_{\tau}|^{s-2}|
\nabla y_{\tau}|^2\\
&\leq C(\| y_{\tau}(0)\|_{L^s(D)}^{s}
+ \int_{0}^{t}\| y_{\tau}\|_{L^s(D)}^{s-2}\|\nabla z\|_{L^s(D)}^2\\
&\quad + \int_{0}^{t}\left(1 + \|\nabla(y+z)\|_{C(D)}^2\right)
\| y_{\tau}\|_{L^s(D)}^{s}\\
&\quad + \int_{0}^{t}\|\nabla(y+z)\|_{C(D)}^2
 \| y_{\tau}\|_{L^s(D)}^{s-2}\|\Delta y\|_{L^s(D)}^{s}).
\end{aligned}
\end{equation}
By Young's inequality ($p=s/(s-2), p'=s/2$), the last term on
the right-hand side of the above
inequality can be estimated as
\begin{align*}
& \int_{0}^{t}\|\nabla(y+z)\|_{C(D)}^2
\| y_{\tau}\|_{L^s(D)}^{s-2}\|\Delta y\|_{L^s(D)}^2\\
&\leq \sup_{0<\tau<t}\| y_{\tau}\|_{L^s(D)}^{s-2}
\int_{0}^{t}\|\nabla(y+z)\|_{C(D)}^2\|\Delta y\|_{L^s(D)}^2\\
&\leq (\beta_{s}(t))^{(s-2)/s}
\Big(\int_{0}^{t}\|\nabla(y+z)\|_{C(D)}^{2s/(s-2)}\Big)^{(s-2)/s}
\Big(\int_{0}^{t}\|\Delta y\|_{L^s(D)}^{s}\Big)^{2/s}.
\end{align*}
 From (4.5), we have
$$
\int_{0}^{t}\|\Delta y\|_{L^s(D)}^{s}\leq C\Big(1 +
\int_{0}^{t}\beta_{s}(\tau)d\tau\Big).
$$
In view of \eqref{e4.19} and Young's inequality it follows that
$$
\int_{0}^{t}\|\nabla(y+z)\|_{C(D)}^2\| y_{\tau}\|_{L^s(D)}^{s-2}
\|\Delta y\|_{L^s(D)}^2d\tau \leq \epsilon\beta_{s}(t) + \Big(1+\int_{0}^{t}
\beta_{s}(\tau)d\tau\Big).
$$
On the other hand the second term of \eqref{e4.20} can be estimated as
\begin{align*}
\int_{0}^{t}\| y_{\tau}\|_{L^s(D)}^{s-2}\|\nabla z\|_{L^s(D)}^2
&\leq \beta_{s}(t)^{(s-2)/2}\int_{0}^{t}\|\nabla z\|_{L^s(D)}^2\\
&\leq \epsilon\beta_{s}(t)+ C_{\epsilon}\Big(\int_{0}^{t}\|\nabla
z\|_{L^s(D)}^2\Big)^{s/2}.
\end{align*}
We conclude \eqref{e4.16} by using Gronwall's lemma. \hfill$\square$

These estimates are
sufficient for proving the existence of solutions. One can use the scheme of
constructing solutions given in [1]. According to this scheme, approximate
solutions $(y_{k},\rho_{k})$ are found by the Galerkin method; $y_{k}$ is
sought as a finite sum of basis and $\rho_{k}$ is determined from the transport
equation. In particular $\nabla y_{k}$ is compact in $L^2(Q_{T})$. Thus,
the passage to the limit in the nonlinear terms in equations (4.1) is
justified.

\subsection{Upper and lower bounds for the density, and
 uniqueness of the solution}

The estimates obtained in the preceding section permit us to establish that
the density $\rho$ is bounded provided that the initial density $\rho_{0}$
is bounded. To this end, we write out a special equation for $\log(\rho)$.
This idea has been used in  [8].

\begin{lemma} \label{lm4.6}
If $y_{0}\in W^{2,s}(D)$, $s\geq 2$ and $\rho_{0}\in L^{\infty}(D)$
then
\begin{equation} \label{e4.21}
\|\rho(t)\|_{L^\infty(D)}\leq M,\quad  \forall t\in [0,T].
\end{equation}
\end{lemma}

\paragraph{Proof:}
Assuming that $\rho(x,t) > 0$, let us rewrite the second equation in
(4.1) in the form
$$
\frac{\partial\log\rho}{\partial t} + \nabla(y+z)\cdot\nabla\log\rho
+ \Delta(y+z) = 0.
$$
By adding the above equation to $(4.1)_1$, we obtain
\begin{equation} \label{e4.22}
\frac{\partial}{\partial t}(\log\rho + y) + \nabla(y+z)\cdot\nabla(\log\rho + y)
+ \rho = 1 - \Delta z + \nabla y\cdot\nabla(y+z).
\end{equation}
Set $\gamma = \log\rho + y$, $\gamma_{+} = \max\{0,\gamma(x,t)\}$.
Considering \eqref{e4.22} as the transport equation for $\gamma$ and taking
into account the fact that $\rho$ is nonnegative, we conclude that
\begin{equation} \label{e4.23}
\gamma_{+}(x,t) \leq \|\gamma_{+}|_{t=0}\|_{L^\infty(D)}
+ \int_{0}^{t}\left(1 + \|\Delta z\|_{C(D)}^2 + \|\nabla y\|_{C(D)}^2\right).
\end{equation}
According to (4.8), $\| y\|_{L^\infty(Q_{T})}$ is bounded; indeed
\begin{equation} \label{e4.24}
\| y\|_{L^\infty(Q_{T})}\leq C\sup_{0<t<T}\|\nabla y\|_{L^q(D)}
\leq C,\quad q>2.
\end{equation}
Hence \eqref{e4.21} follows by the hypotheses of the lemma and from
\eqref{e4.24} and \eqref{e4.19} with the constant
\begin{align*}
M =& \exp \Big(\| y\|_{L^\infty(Q_{T})} + \|\gamma_{+}|_{t=0}\|_{L^\infty(D)}\\
&+ \int_{0}^{t}\big(1 + \|\Delta z\|_{C(D)}^2
+ \|\nabla y\|_{C(D)}^2\big)\Big).
\end{align*}
\hfill$\square$

\begin{lemma} \label{lm4.7}
If the initial density $\rho_{0}(x)$ is strictly positive under the
hypotheses of the lemma 4.6, then $\rho(x,t)$ remains a strictly positive
function in $Q_{T}$ i.e.
\begin{equation} \label{e4.25}
\rho(x,t) \geq m > 0 \quad\mbox{a.e. in } Q_{T}.
\end{equation}
\end{lemma}

\paragraph{Proof:}
Let us change the sign in equation \eqref{e4.22} and rewrite it for $\gamma$.
We wish to find an upper bound for the function
$\gamma_{-} = \max\left\{0,-\gamma\right\}$.
By analogy, we obtain
\begin{equation} \label{e4.26}
\gamma_{-}(x,t)\leq \|\gamma_{-}|_{t=0}\|_{L^\infty(D)}
+ \int_{0}^{t}\left(\|\Delta z\|_{L^\infty(D)}^2 + \|\nabla y\|_{C(D)}^2
\|\rho\|_{L^{\infty}(D)}\right).
\end{equation}
Hence \eqref{e4.25} follows with the constant
\begin{align*}
m =&\exp \Big(- (\| y\|_{L^{\infty}(D)}
+ \|\gamma_{-}|_{t=0}\|_{L^{\infty}(D)} \\
&+\int_{0}^{t}\big(\|\Delta z\|_{\infty}^2 + \|\nabla y\|_{C(D)}^2\big)
+ MT)\Big).
\end{align*}
\hfill$\square$

If the density $\rho$ is bounded, then the solution of (4.1) is unique.
Indeed, if $(\rho',y')$ and $(\rho'',y'')$ are two solutions, then their
difference $\rho = \rho' - \rho''$, $y = y' - y''$ is a solution to the linear
problem
\begin{equation} \label{e4.27}
\begin{gathered}
 y_{t} - \Delta y = - \rho,\\
 \rho_{t} + \mathop{\rm div}(\rho\nabla(y'+z))
 + \mathop{\rm div}(\rho''\nabla y) = 0,
\end{gathered}
\end{equation}
with the zero initial conditions.
Let us introduce an auxiliary function $\psi$, which is a solution to periodic
Neumann problem
\begin{equation} \label{e4.28}
\Delta\psi = \rho,\quad \int_{D}\psi dx = 0.
\end{equation}
Since $\rho$ is bounded, $|\nabla\psi|$ is also bounded i.e.
\begin{equation} \label{e4.29}
|\nabla\psi| \leq M\quad\mbox{a.e. in } Q_{T}.
\end{equation}
We multiply $\eqref{e4.27}_1$ by $y$ and $\eqref{e4.27}_2$ by $\psi$, we sum
these equations and integrate over $D$, we have
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\left(\| y\|^2 + \|\nabla\psi\|^2\right)
+ \|\nabla y\|^2\\
&= \langle \nabla y,\nabla\psi\rangle - \langle \rho''\nabla y,\nabla\psi\rangle
- \langle \Delta\psi\nabla(y'+z),\nabla\psi\rangle .
\end{align*}
The first two terms on the right-hand side of this equation are bounded by
$2^{-1}\|\nabla y\|_2^2 + C\|\nabla\psi\|_2^2$ in view of the Cauchy's
inequality, since $\rho'$ is bounded. The last term can be trasformed
by integrating by parts as follows
$$
\langle \Delta\psi\nabla(y'+z),\nabla\psi\rangle
= \langle (\nabla\psi\nabla)\cdot\nabla(y'+z),\nabla\psi\rangle
-\frac{1}{2}\langle \Delta(y'+z),|\nabla\psi|^2\rangle.
$$
For the other part, by H\"{o}lder's inequality
$$
\int_{D}|\nabla\psi|^2D^2(y'+z)\leq \Big(\int_{D}|D^2(y'+z)|^{p}\Big)
^{1/p}\Big(\int_{D}|\nabla\psi|^{2p'}\Big)^{1/p'},
$$
with $p = \epsilon^{-1} >1$ and $p' = 1/(1-\epsilon)$ where $0<\epsilon<1$ is arbitrary.
\[
\int_{D}|\nabla\psi|^2D^2(y'+z)\leq
M^{2\epsilon}\|\nabla\psi\|^{2(1-\epsilon)}\| D^2(y'+z)\|_{L^{1/\epsilon}}.
\]
Thus, for the nonnegative function
\[
Y(t) = \| y\|^2 + \|\nabla\psi\|^2,\quad Y(0) = 0\,,
\]
expression \eqref{e4.29} yields the inequality
$$\frac{dY(t)}{dt}\leq C_1Y(t)
+ M^{2\epsilon}\| D^2(y'+z)\|_{L^{1/\epsilon}}Y(t)^{1-\epsilon}.$$
Consequently,
\begin{align*}
Y^{\epsilon}(t)&\leq \epsilon M^{2\epsilon}{\rm e}^{(C_1+C)\epsilon t}
\int_{0}^{t}{\rm e}^{-C\epsilon\tau}\| D^2(y'+z)\|_{L^{1/\epsilon}}d\tau\\
&\leq \epsilon C M^{2\epsilon}{\rm e}^{(C_1+C)\epsilon t}
\int_{0}^{t}{\rm e}^{-C\epsilon\tau}\left(\| D^2y'\|_{L^{1/\epsilon}} + 1\right)
d\tau
\end{align*}
We apply H\"{o}lder inequality to the integral with the exponent
$p=1/\epsilon$ and $p'=1/(1-\epsilon)$;  we obtain
\begin{equation} \label{e4.34}
\begin{aligned}
Y^{\epsilon}(t)&\leq\epsilon CM^{2\epsilon}{\rm e}^{(C_1+C)\epsilon t}
\Big(\int_{0}^{t}({\rm e}^{-C\epsilon\tau})^{p'}\Big)^{1/p'}
\Big(\int_{0}^{t}(\| D^2y'\|_{L^{1/\epsilon}} + 1)^{p}d\tau\Big)^{p}\\
&\leq \epsilon CM^{2\epsilon}\Big(\| D^2y'\|_{L^{1/\epsilon}(Q_{T})} + 1\Big)
\Big(\frac{1-\epsilon}{C\epsilon}\big({\rm e}^{C\epsilon t/(1-\epsilon)}-1\big)\Big)^{1-\epsilon}.
\end{aligned}
\end{equation}
Considering $y'$ as a solution to the parabolic equation
$y'_{t} - \Delta y' = 1 - \rho'$ with bounded right-hand side and using
estimates for the higher derivatives in the $L^{1/\epsilon}$-norm (see [9]), we
obtain
$$
\| D^2y'\|_{L^{1/\epsilon}(Q_{T})}
\leq C\epsilon^{-1}\| 1 - \rho'\|_{L^{1/\epsilon}(Q_{T})}\leq C\epsilon^{-1}.
$$
Therefore, from \eqref{e4.34} follows that
\[
Y(t)\leq C^{1/\epsilon}M^2
\Big(\frac{1-\epsilon}{C\epsilon}\big({\rm e}^{C\epsilon t/(1-\epsilon)}-1\big)
\Big)^{(1-\epsilon)/\epsilon}.
\]
It is easy to check that if $t\in [0,\tau]$,
where $C\tau < 1$ on $[0,\tau]$, then the right-hand side of the
last inequality vanishes as $\epsilon\to 0$. Hence $Y(t) = 0$
on $[0,\tau]$.
Repeating the argument for the interval $[\tau,2\tau]$ and so on,
we obtain $Y(t) = 0$, which proves the uniqueness of the solution.

\subsection*{Conclusion}
The solution of deterministic system of equations (4.1) will allow us
constructing the solution of the stochastic system (1.4).

\paragraph{Proof of Theorem 2.1}
 We can apply Theorem 4.1
to obtain existence and uniqueness of solution for problem (3.4) for fixed
$\omega$. The measurability is an obvious fact using the uniqueness of the
solution (see [17]). As a consequence, using (3.3) and the properties
(measurability) of $z$, this theorem is proved.\hfill$\square$

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\end{thebibliography}

\noindent
Current address:\\
Dipartimento di matematica applicata, Univ. Dini \\
Via Bonanno 25/B, 56126 Pisa, Italy\\
e-mail: bessaih@dma.unipi.it\\

\noindent
Permanent address:\\ 
Faculty of Mathematics\\
U.S.T.H.B\\
16111 Algiers, Algeria

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