\documentclass[twoside]{article} \usepackage{amsfonts,amsmath} % font used for R in Real numbers \pagestyle{myheadings} \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 $$\label{e1.1} \begin{gathered} {\bar{\rho}}u_{t} + \nabla p(\rho) = \mu\Delta u + (\mu+\lambda)\nabla\mathop{\rm div}u + G_{t}\quad \mbox{\rm in } Q_{T},\\ \rho_{t} +\mathop{\rm div}(\rho u) = 0\quad \mbox{\rm in }Q_{T}, \end{gathered}$$ 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 $$\label{e1.2} \begin{gathered} {\bar{\rho}}u_{t} + \nabla p(\rho) = \mu\Delta u + (\lambda+\mu)\nabla\mathop{\rm div}u ,\\ \rho_{t} +\mathop{\rm div}(\rho u) = 0. \end{gathered}$$ This system can be considered as a semi-linearized approximation of the compressible Navier-Stokes equations of a barotropic viscous fluid $$\label{e1.3} \begin{gathered} \rho\left(\partial_{t}u_{i} + (u\cdot\nabla)u_{i}\right)- \partial_{i}p(\rho) = \sum_{j=1}^{n}\partial_{j}\left(\mu(\partial_{j}u_{i} + \partial_{i}u_{j})\right) + \partial_{i}(\lambda\mathop{\rm div}u), \\ \rho_{t} +\mathop{\rm div}(\rho u) = 0, \end{gathered}$$ 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 $$\begin{gathered} {\rm d}\varphi = (\Delta\varphi + 1 -\rho){\rm d}t + {\rm d}W \quad \mbox{\rm in } Q_{T},\\ \rho_{t} +\mathop{\rm div}(\rho\nabla\varphi) = 0. \end{gathered}$$ 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 $$\label{e2.1} W(t) = \sum_{j=1}^{\infty}\sigma_{j}\beta_{j}(t)e_{j}(x).$$ where $\{\sigma_{j}\}_{j=1}^{\infty}$ is a sequence of constants satisfying the condition $$\label{e2.2} \sum_{j=1}^{\infty}\lambda_{j}^{\delta+2}\sigma_{j}^2 < \infty,$$ 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, $$\label{e3.1} \begin{gathered} dz(t) + Az(t)dt = dW(t),\\ z(0) = 0. \end{gathered}$$ This equation has a solution given by the process (see [4]) $$\label{e3.2} z(t) = \int_{0}^{t} e^{-(t-s)A}dW(s),$$ 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 $$\label{e3.3} y(t) = \varphi(t) - z(t).$$ Using this change of variable in (1.4) and equation (3.1), one obtains the system $$\label{e3.4} \begin{gathered} y_{t} - \Delta y = 1 - \rho \quad \mbox{\rm in } Q_{T},\\ \rho_{t} +\mathop{\rm div}(\rho\nabla(y+z)) = 0\quad \mbox{\rm in } Q_{T}. \end{gathered}$$ \section{Reduced deterministic problem} In this section, we study the following reduced deterministic problem $$\label{e4.1} \begin{gathered} y_{t} - \Delta y = 1 - \rho \quad \mbox{\rm in } Q_{T},\\ \rho_{t} +\mathop{\rm div}(\rho\nabla(y+z)) = 0\quad \mbox{\rm in } Q_{T}, \end{gathered}$$ 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 $$\frac{d}{dt}\int_{D}\Big(\frac{1}{2}|\nabla y|^2 + \rho\log\rho -\rho + 1\Big)dx + \int_{D}|\Delta y|^2 \leq\|\Delta z\|_{L^\infty}.$$ 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 \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} holds for any exponent $p$, $21$ using $(4.1)_1$, the expression $(4.1)_2$ can be rewritten in the form $$\frac{\partial\rho^{r}}{\partial t} + \nabla\cdot(\rho^{r}\nabla(y+z)) + (r-1)\rho^{r+1} = (r-1)\rho^{r}(y_{t}-1- \Delta z).$$ 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 $$\label{e4.5} \| y\| _{W^{2,p}(Q_{T})}\leq C(\|\rho_{0}\|_{L^{p-1}(D)} + \| y_{t}\|_{L^p(Q_{T})} + \|\Delta z\|_{L^p(Q_{T})})$$ holds for $22$ 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 $$\label{e4.17} \frac{d}{dt}\int_{D}|y_{t}|^{s} + s(s-1)\int_{D}|y_{t}|^{s-2}|\nabla y_{t}|^2 = -s(s-1)\int_{D}\rho\nabla(y+z)\cdot\nabla y_{t}|y_{t}|^{s-2}.$$ 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 \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} 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 $$\label{e4.19} \int_{0}^{t}\|\nabla y\|_{C(D)}^{q+4} \leq \int_{0}^{t}\|\Delta y\|_{L^4(D)}^{4}\|\nabla y\|_{L^q(D)}^{q} \leq C.$$ We set $\beta_{s}(t) = \sup_{0<\tau 0$, 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 $$\label{e4.22} \frac{\partial}{\partial t}(\log\rho + y) + \nabla(y+z)\cdot\nabla(\log\rho + y) + \rho = 1 - \Delta z + \nabla y\cdot\nabla(y+z).$$ 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 $$\label{e4.23} \gamma_{+}(x,t) \leq \|\gamma_{+}|_{t=0}\|_{L^\infty(D)} + \int_{0}^{t}\left(1 + \|\Delta z\|_{C(D)}^2 + \|\nabla y\|_{C(D)}^2\right).$$ According to (4.8), $\| y\|_{L^\infty(Q_{T})}$ is bounded; indeed \label{e4.24} \| y\|_{L^\infty(Q_{T})}\leq C\sup_{02. 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. $$\label{e4.25} \rho(x,t) \geq m > 0 \quad\mbox{a.e. in } Q_{T}.$$ \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 $$\label{e4.26} \gamma_{-}(x,t)\leq \|\gamma_{-}|_{t=0}\|_{L^\infty(D)} + \int_{0}^{t}\left(\|\Delta z\|_{L^\infty(D)}^2 + \|\nabla y\|_{C(D)}^2 \|\rho\|_{L^{\infty}(D)}\right).$$ 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 $$\label{e4.27} \begin{gathered} y_{t} - \Delta y = - \rho,\\ \rho_{t} + \mathop{\rm div}(\rho\nabla(y'+z)) + \mathop{\rm div}(\rho''\nabla y) = 0, \end{gathered}$$ with the zero initial conditions. Let us introduce an auxiliary function $\psi$, which is a solution to periodic Neumann problem $$\label{e4.28} \Delta\psi = \rho,\quad \int_{D}\psi dx = 0.$$ Since $\rho$ is bounded, $|\nabla\psi|$ is also bounded i.e. $$\label{e4.29} |\nabla\psi| \leq M\quad\mbox{a.e. in } Q_{T}.$$ 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 \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} 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$\square$ \begin{thebibliography}{00} \frenchspacing \bibitem{1}{S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov}, {\it Boundary value problems in mechanics of nonhomogeneous fluids}, 1990. \bibitem{2}{A. Bensoussan, R. Temam}, {\it Equations stochastiques du type Navier-Stokes}, J. Func. Anal, {\bf 13}, 195--222, 1973. \bibitem{3}{H. 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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 \end{document} \end