\documentclass[twoside]{article} \usepackage{amsfonts} \pagestyle{myheadings} \markboth{\hfil Boussinesq models in exterior domains\hfil EJDE--1998/22} {EJDE--1998/22\hfil E.A. Notte-Cuello \& M.A. Rojas-Medar \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1998}(1998), No.~22, pp. 1--9. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{% \bigskipamount} \\ Stationary solutions for generalized Boussinesq models in exterior domains \thanks{\emph{1991 Mathematics Subject Classifications:} 35Q10. \hfil\break \indent {\em Key words and phrases:} Boussinesq, thermally driven, temperature dependent viscosity, \hfil\break \indent exterior domain. \hfil\break \indent \copyright 1998 Southwest Texas State University and University of North Texas. \hfil\break \indent Submitted July 3, 1998. Published October 1, 1998. \hfil\break \indent M.A.R. was partially supported by grant 300116/93(RN), CNPq. \hfil\break \indent E.A.N. and M.A.R. were partially supported by grant 1998/00619-9 FAPESP. } } \date{} \author{E.A. Notte-Cuello \& M.A. Rojas-Medar} \maketitle \begin{abstract} We establish the existence of a stationary weak solution of a generalized Boussinesq model for thermally driven convection in exterior domains. We use the fact that the exterior domain can be approximated by interior domains. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \section{Introduction} We study the stationary problem for equations governing a coupled mass and heat flow of a viscous incompressible fluid in generalized Boussinesq approximations. Assuming that the viscosity and the heat conductivity are temperature dependent in an exterior domain $\Omega \subset {\mathbb R}^{3}$, we study the equation \begin{eqnarray} &-\mathop{\rm div}(\nu (T)\nabla u)+u\cdot \nabla u-\alpha Tg+\nabla p =0& \nonumber \\ &\mathop{\rm div} u =0& \label{A1} \\ &-\mathop{\rm div}(\kappa (T)\nabla T)+u\cdot \nabla T =0\,.& \nonumber \end{eqnarray} Here $u(x)\in {\mathbb R}^{3}$ denotes the velocity of the fluid at a point $x\in \Omega $; $p(x)\in {\mathbb R}$ is the hydrostatic pressure; $T(x)\in {\mathbb R}$ is the temperature; $g(x)$ is the external force per unit of mass; $\nu(\cdot )>0$ and $\kappa (\cdot )>0$ are kinematic viscosity and thermal conductivity, respectively; and $\alpha $ is a positive constant associated to the coefficient of volume expansion. Without loss of generality, we have taken the reference temperature as zero. For a derivation of the above equations, see Drazin and Reid \cite{Drazin}. The expressions $\nabla$, $\Delta $, and $\mathop{\rm div}$ denote the gradient, Laplace, and divergence operators, respectively. The gradient is also denoted by grad. The i-th component of $u\cdot \nabla u$ is given by \[ (u\cdot\nabla u)_i=\sum_{j=1}^{3}u_j(\partial u_i/\partial x_j)\,; \quad u\cdot \nabla T=\sum_{j=1}^{3}u_j(\partial T/\partial x_j)\,. \] The boundary conditions and conditions at infinity are \begin{eqnarray} &u\left| _{\Gamma }\right. = 0\,, \quad T\left|_{\Gamma}\right. =T_0>0\,; & \label{A2} \\ &\lim_{| x| \rightarrow \infty }u(x) = 0\,, \quad \lim_{| x| \rightarrow \infty }T(x)=0\,,& \label{A3} \end{eqnarray} where $\Gamma $ is the boundary of $\Omega $. Problem (\ref{A1}) was considered by Lorca and Boldrini \cite{Lorca} in a bounded domain with Dirichlet's conditions; while the reduced model, where $\nu $ and $\kappa $ are positive constants, was studied by Morimoto \cite {Morimoto} (in a bounded domain) and recently by Oeda \cite{Oeda1} (in an exterior domain). The evolution problem corresponding to (\ref{A1}) was analyzed by Lorca and Boldrini \cite{Lorca1} in a bounded domain; when $\nu $ and $\kappa $ are positive constants was discussed by many authors, see for instance, Korenev \cite{Korenev}, Rojas-Medar and Lorca \cite{Lorca3, Lorca4} (in a bounded domain) and Hishida \cite{Hishida}, Oeda \cite{Oeda2}, \cite{Oeda3} (in an exterior domain). In another publication we will study the evolution problem corresponding to (\ref{A1}). \section{Preliminaries} Functions in this paper are either ${\mathbb R}$ or ${\mathbb R}^{3}$ valued, and we will not distinguish these two situations in our notation. To which case we refer to will be clear from the context. Now, we give the precise definition of the exterior domain, $\Omega $, where our boundary-value problem associated to the problem (\ref{A1})-(\ref{A3}) has been formulated. Let $K$ be a compact subset of ${\mathbb R}^{3}$, whose boundary $\partial K$ is of class $C^{2}$. The exterior domain is $\Omega =K^{c}$, and $\Gamma =\partial \Omega =\partial K$. The extending domain method was introduced by Ladyzhenskaya \cite {Ladyzhenskaya} to study the Navier-Stokes equations in unbounded domains. As observed by Heywood \cite{Heywood} the method is useful in certain class of unbounded domains. Certainly, our domain is in this class. The basic idea is the following: The exterior domain $\Omega $ can be approximated by interior domains $\Omega _m=B_m\cap \Omega $, where $B_m$ is a ball with radius $m$ and center at $0$, as $m\rightarrow \infty $. In each interior domain $\Omega _m$, we will prove the existence of a weak solution, by using the Galerkin method together with the Brouwer's fixed point theorem as in Heywood \cite{Heywood}. Next, by using the estimates given in Ladyzhenskaya's book \cite{Ladyzhenskaya} together with diagonal argument and Rellich's compactness theorem, we obtain the desirable weak solution to problem (\ref{A1})-(\ref{A3}). Let $D$ denote $\Omega $ or $\Omega _m$. Define function spaces as follows: \begin{eqnarray*} &W^{r,p}(D) =\left\{ u;D^{\alpha }u\in L^{p}(D),| \alpha | \leq r\right\} & \\ &W_0^{r,p}(D) =\mbox{ completion of }C_0^\infty (D)\mbox{ in }W^{r,p}(D) & \\ &C_{0,\sigma }^\infty (D) =\left\{ \varphi \in C_0^\infty (D);% \mathop{\rm div}\varphi =0\right\} & \\ &J(D) =\mbox{ completion of }C_{0,\sigma }^\infty (D)\mbox{ in norm }\| \nabla \phi \| & \\ &H(D) =\mbox{ completion of }C_{0,\sigma }^\infty (D)\mbox{ in norm }\| \phi \| \,.& \end{eqnarray*} Here $\| \cdot \| $ denotes the $L^{2}$-norm, $\| \cdot \| _p$ denotes the $L^{p}$-norm. We note that $J(D)$ can be characterized as \[ J(D)=\left\{ \phi \in W^{1,2}(D);\phi \left| _{\Gamma }\right. =0,% \mathop{\rm div}\phi =0\right\}\,, \] as was proved by Heywood \cite{Heywood}. When $p=2$, we write $W^{r,p}(D)\equiv H^{r}(D)$ and $W_0^{r,p}(D)\equiv H_0^{r}(D)$. We make use of some inequalities with constants that depend only on the dimension and are independent of the domain (see \cite{Ladyzhenskaya} chapter I). \begin{lemma} Suppose the space dimension is 3, with $D$ bounded or unbounded. Then (a) For $u\in W_0^{1,2}(D)$ $($ or $J(D)$ or $H_0^1(D))$, we have \[ \| u\| _{L^6(D)}\leq C_L\| \nabla u\| _{L^2(D)} \] where $C_L=(48)^{1/6}$. \newline (b) (H\"older's inequality). If each integral makes sense. Then we have \[ | ((u\cdot \nabla )v,w)| \leq 3^{\frac 1p+\frac 1r}\| u\| _{L^p\left( D\right) }\| \nabla v\| _{L^q\left( D\right) }\| w\| _{L^r\left( D\right) } \] where $p,q,r>0$ and $\frac 1p+\frac 1q+\frac 1r=1$. \end{lemma} The following assumptions will be needed throughout this paper. \begin{description} \item {(S1)} $w_0\subset K$ ( $w_0$ is a neighborhood of the origin 0) and $K\subseteq B=B(0,d)$ which is a ball with radius $d$ and center at 0. \item {(S2)} $\partial \Omega =\Gamma =\partial K\in C^{2}$. \item {(S3)} $g(x)$ is a bounded and continuous vector function in ${\mathbb R}^{3}\backslash w_0$. Moreover $g\in L^{p}(\Omega )$ for $p\geq 6/5$. \end{description} We assume that the functions $\nu (\cdot )$ and $\kappa (\cdot )$ satisfy \begin{eqnarray*} &0 <\nu _0(T_0)\leq \nu (\tau )\leq \nu_1(T_0)& \\ &0 <\kappa _0(T_0)\leq \kappa (\tau )\leq \kappa_1(T_0)& \end{eqnarray*} for all $\tau \in {\mathbb R}$, where \[ \nu _0(T_0)=\inf \{\nu (t); |t| \leq \sup_{\partial \Omega }| T_0| \}/2, \nu_{1}(T_0)=\sup \{\nu (t);| t| \leq \sup_{\partial \Omega}| T_0| \}\,, \] with analogous definitions for $\kappa _0(T_0)$ and $\kappa_1(T_0)$, and $\nu ,\kappa $, are continuous functions. To transform the boundary condition on $T$ to a homogeneous boundary condition, we introduce an auxiliary function $S$ (see Gilbarg and Trudinger \cite{Gilbarg} p. 137). \begin{lemma} \label{L2.2} There exists a function $S$ which satisfies the following properties (i) $S(\Gamma )=T_{0.}$ (ii) $S\in C_0^2({\mathbb R}^3)$. (iii) for any $\epsilon >0$ and $p\geq 1$, we can redefine $S$, if necessary, such that $\| S\| _{L^p}<\epsilon $. \end{lemma} Now we make a change of variable: $\varphi =T-S$ to obtain \begin{eqnarray} &-\mathop{\rm div}(\nu (\varphi +S)\nabla u)+u\cdot \nabla u-\alpha \varphi g-\alpha Sg+\nabla p =0 & \nonumber \\ &\mathop{\rm div} u = 0 & \label{A4} \\ &-\mathop{\rm div}(\kappa (\varphi +S)\nabla \varphi )+u\cdot \nabla \varphi -\mathop{\rm div}(\kappa (\varphi +S)\nabla S)+u\cdot \nabla S = 0 & \nonumber \end{eqnarray} in $\Omega$, with boundary conditions \begin{eqnarray} &u=0\quad \mbox{and} \quad \varphi =0\quad \mbox{on }\partial \Omega& \label{A5} \\ &\lim_{| x| \rightarrow \infty }u(x)=0\, ;\quad \lim_{| x| \rightarrow \infty }\varphi (x)=0 \,.& \label{A6} \end{eqnarray} \paragraph{Definition} $(u,\varphi )\in J(\Omega )\times H_0^{1}(\Omega )$ is called a stationary weak solution of (\ref{A4})-(\ref{A6}) if it satisfies \begin{eqnarray} &(\nu (\varphi +S)\nabla u,\nabla v)+B(u,u,v)-\alpha (\varphi g,v)-\alpha (Sg,v) =0\,& \label{A6.5} \\ &(\kappa (\varphi +S)\nabla \varphi ,\nabla \psi )+b(u,\varphi ,\psi )+(\kappa (\varphi +S)\nabla S,\nabla \psi )+b(u,S,\psi ) = 0\,,& \nonumber \end{eqnarray} for all $v \in J(\Omega )$ and all $\psi \in H_0^{1}(\Omega )$. Where \begin{eqnarray*} &B(u,v,w)=(u\cdot \nabla v,w)=\int\!\int_{\Omega }\sum_{i,j=1}^{3}u_j(x)(\partial v_i/\partial x_j)(x)w_i(x)\,dx\,,& \\ &b(u,\varphi ,\psi )=(u\cdot \nabla \varphi ,\psi )=\int\!\int_{\Omega }\sum_{i,j=1}^{3}u_j(x)(\partial \varphi_i/\partial x_j)(x)\psi _i(x)\,dx\,. & \end{eqnarray*} \begin{theorem} (Existence) Under Assumptions (S1), (S2) and (S3), there exists a stationary weak solution of (\ref{A6.5}). \end{theorem} \section{Auxiliary problem.} Following the extending domain method, we first present a lemma which ensures the existence of weak solutions of interior problems in domains $\Omega _m=B_m\cap \Omega $. The interior problem is stated as follows: $$ \begin{array}{c} -\mathop{\rm div}(\nu (\varphi +S)\nabla u)+u\cdot \nabla u-\alpha \varphi g-\alpha Sg+\nabla p=0 \\ \mathop{\rm div} u=0 \\ -\mathop{\rm div}(\kappa (\varphi +S)\nabla \varphi )+u\cdot \nabla \varphi - \mathop{\rm div}(\kappa (\varphi +S)\nabla S)+u\cdot \nabla S=0 \\ u=0,\ \varphi =0\mbox{ on }\partial \Omega _m=\partial \Omega \cap\partial B_m \end{array} \eqno{(P_m)} $$ \paragraph{Definition} $(u,\varphi )\in J(\Omega _m)\times H_0^{1}(\Omega _m)$ is called a stationary weak solution for $(P_m)$ if it satisfies \begin{eqnarray} &(\nu (\varphi +S)\nabla u,\nabla v)+B(u,u,v)-\alpha (\varphi g,v)-\alpha (Sg,v) = 0 & \label{A7} \\ &(\kappa (\varphi +S)\nabla \varphi ,\nabla \psi )+b(u,\varphi ,\psi )+(\kappa (\varphi +S)\nabla S,\nabla \psi )+b(u,S,\psi ) = 0\,,& \nonumber \end{eqnarray} for all $v \in J(\Omega _m)$, and for all $\psi \in H_0^{1}(\Omega _m)$. \begin{lemma} Under Assumptions (S1), (S2), and (S3) we can construct a weak solution $(\overline{u}^m,\overline{\varphi }^m)$ of $(P_m)$. \end{lemma} \paragraph{Proof} Let $m$ be an arbitrary fixed number. Let $\left\{ v_{j}\right\} _{j=1}^\infty \subset J(\Omega _m)$ and $\left\{ \psi _{j}\right\} _{j=1}^\infty \subset H_{0}^{1}(\Omega _m)$ be a sequences of functions, linearly independent and such that the linear span of the $v_{j}$ and $\psi _{j}$ are dense in $J(\Omega _m)$ and $H_{0}^{1}(\Omega _m)$ respectively. Since $\Omega _m$ is bounded, we can choose them such that \begin{eqnarray*} &(\nabla v_j,\nabla v_k)=\delta _{ik}\,,\quad (\nabla \psi_j,\nabla \psi _k)=\delta _{jk} & \\ &u^n(x)=\sum_{k=1}^nc_{n,k}v_k(x)\,,\quad \varphi^n(x)=\sum_{k=1}^nd_{n,k}\psi _k(x)\,. & \end{eqnarray*} Then we consider the system of equations \begin{eqnarray} (\nu (\varphi ^n+S)\nabla u^n,\nabla v_j)+B(u^n,u^n,v_j)-\alpha (\varphi ^ng,v_j)-\alpha (Sg,v_j)&=&0 \nonumber \\ (\kappa (\varphi ^n+S)\nabla \varphi ^n,\nabla \psi _j)+b(u^n,\varphi ^n,\psi _j)&& \label{A8} \\ +(\kappa (\varphi ^n+S)\nabla S,\nabla \psi _j)+b(u^n,S,\psi _j)&=&0\,, \nonumber \end{eqnarray} where $1\leq j\leq n$. Using the representations of $u^n,\varphi ^n$, we have \begin{eqnarray} \sum_{k=1}^nc_k(\nu (\varphi ^n+S)\nabla v_k,\nabla v_j)+\sum_{k,l}^nc_kd_lB(v_k,v_l,v_j) & & \nonumber \\ -\sum_{k=1}^n\alpha d_k(g\psi _k,v_j)-\alpha (Sg,v_j) & = & 0\,, \label{A9} \\ \sum_{k=1}^nd_k(\kappa (\varphi ^n+S)\nabla \psi _k,\nabla \psi _j)+\sum_{k,l}^nc_kd_lb(v_k,\psi _l,\psi _j) & & \nonumber \\ +(\kappa (\varphi ^n+S)\nabla S,\nabla \psi _j)+\sum_{k=1}^nc_kb(v_k,S,\psi _j) & = & 0\,, \nonumber \end{eqnarray} where $1\leq j\leq n$. Put $(c;d)=(c_{1},\dots ,c_{n},d_{1},\dots ,d_{n})$, and \newline $P(c;d)=(P_{1}(c;d),\dots ,P_{2n}(c;d))$. Then, from (\ref{A9}) we obtain \begin{eqnarray} \lefteqn{\sum_{k=1}^nc_k\nu _0(T_0)(\nabla v_k,\nabla v_j)} \nonumber \\ &\leq&| \sum_{k,l}c_kd_lB(v_k,v_j,v_l)| +|\sum_k\alpha d_k(g\psi _k,v_j)|\,, +| \alpha (Sg,v_j)| \nonumber \\ \lefteqn{\sum_{k=1}^nd_k\kappa _0(T_0)(\nabla \psi _k,\nabla \psi_j) } \label{A10} \\ &\leq &| \sum_{k,l}c_kd_lb(v_k,\psi _j,\psi _l)| +\kappa_1(T_0)| (\nabla S,\nabla \psi _j)| +| \sum_kc_kb(v_k,S,\psi _j)|\,; \nonumber \end{eqnarray} thus \begin{eqnarray} \lefteqn{P_j(c;d) } \nonumber \\ &\leq &\frac{1}{\nu _0(T_0)}\left\{ |\sum_{k,l}c_kd_lB(v_k,v_j,v_l)| +| \sum_k\alpha d_k(g\psi _k,v_j)| +| \alpha(Sg,v_j)| \right\} \,, \nonumber \\ \lefteqn{P_{n+j}(c;d)} \label{A11} \\ &\leq &\frac{1}{\kappa _0(T_0)}\left\{ | \sum_{k,l}c_kd_lb(v_k,\psi _j,\psi _l)| +\kappa_1(T_0)| (\nabla S,\nabla \psi _j)| +| \sum_kc_kb(v_k,S,\psi _j)| \right\} \nonumber \end{eqnarray} where $1\leq j\leq n$. Then our problem is reduced to obtaining a fixed point of $P:{\mathbb R}^{2n}\rightarrow {\mathbb R}^{2n}$. Now we use Brouwer's fixed point theorem. Namely, if all possible solutions $(c;d)$ of the equation $(c;d)=\lambda P(c;d)$ for $\lambda \in [0,1]$ stay in a same ball $\| (c;d)\| \leq r$, then there exists a fixed point of $P$. By multiplying $($\ref{A10}$)_i$ (respectively. $($\ref{A10}$)_{ii}$ ) by $c_j$ (respectively. $d_j$ ), summing up with respect to $j$ and noting $B(u^n,u^n,u^n)=0$, $b(u^n,\varphi ^n,\varphi ^n)=0$ we have \begin{eqnarray*} \nu _0(T_0)\sum_{j=1}^n| c_j| ^{2} &=&\nu _0(T_0)| \nabla u^n| ^{2}=\nu _0(T_0)\lambda \sum_{j=1}^nP_j(c;d)c_j \\ &\leq &\lambda \alpha | (g\varphi ^n,u^n)| +| (Sg,u^n)| \\ &\leq &\lambda \alpha \left\{ | g| _{3/2}| \varphi ^n| _{6}| u^n| _{6}+| g| _{3/2}| S| _{6}| u^n| _{6}\right\} \\ &\leq &\lambda \alpha \left\{ | g| _{3/2}\left( | \nabla \varphi ^n| +| S| _{6}\right) | \nabla u^n| \right\} \end{eqnarray*} then \begin{equation} | \nabla u^n| ^{2}\leq \frac{\lambda \alpha }{\nu _0(T_{o})}| g| _{3/2}\left\{ | \nabla \varphi ^n| +| \nabla S| \right\} . \label{A12} \end{equation} In the same manner, we find \begin{equation} | \nabla \varphi ^n| \leq \frac{\lambda \kappa_1(T_0)}{\kappa _0(T_0)}| \nabla S| +\frac{\lambda }{\kappa _0(T_0)}| \nabla u^n| | S| _{3} \label{A13} \end{equation} by substituting (\ref{A13}) into (\ref{A12}), we obtain \[ | \nabla u^n| \leq \frac{\lambda \alpha }{\nu _0(T_{o})}| g| _{3/2}\left\{ \frac{\lambda \kappa_1(T_0)}{\kappa _0(T_0)}| \nabla S| +\frac{\lambda }{% \kappa _0(T_0)}| \nabla u^n| | S| _{3}\right\} +\frac{\lambda \alpha }{\nu _0(T_{o})}| g| _{3/2}| \nabla S|\,; \] therefore, \[ \left( 1-\frac{\lambda ^{2}\alpha }{\nu _0(T_{o})\kappa _0(T_0)}| g| _{3/2}| S| _{3}\right) | \nabla u^n| \leq \frac{\lambda \alpha }{\nu _0(T_{o})}| g| _{3/2}| \nabla S| \left( \frac{\kappa_1(T_0)}{\kappa _0(T_0)}+1\right) . \] According to Lemma \ref{L2.2}, with $p=3$ , we can choose an extension $S$ of $T_0$ such that \[ \gamma \equiv \frac \alpha {\nu _0(T_o)\kappa _0(T_0)}| g| _{3/2}| S| _3<1/2\,. \] Then we have \begin{equation} | \nabla u^n| \leq \frac{\lambda \alpha }{(1-\lambda ^2\gamma )\nu _0(T_o)}| g| _{3/2}| \nabla S| \left( \frac{\kappa _1(T_0)}{\kappa _0(T_0)}+1\right)\,. \label{A14} \end{equation} By substituting the previous inequality in (\ref{A13}), we obtain \begin{equation} | \nabla \varphi ^n| \leq \frac{\lambda | \nabla S| }{\kappa _0(T_{o})}% \left( \kappa_1(T_0)+\frac{\lambda \alpha }{(1-\lambda ^{2}\gamma )\nu _0(T_{o})}| g| _{3/2}\left( \frac{\kappa_1(T_0)}{\kappa _0(T_0)}+1\right) | S| _{3}\right) . \label{A15} \end{equation} Since $0\leq \lambda \leq 1$ and $\frac{1}{1-\lambda ^{2}\gamma }\leq \frac{1% }{1-\gamma }$, from (\ref{A14}) and (\ref{A15}) we have \begin{eqnarray} &| \nabla u^n| \leq \frac{\alpha }{(1-\gamma )\nu _0(T_{o})}| g| _{3/2}| \nabla S| \left( \frac{\kappa_1(T_0)}{\kappa _0(T_0)}+1\right) \equiv r_{1}& \label{A16} \\ &| \nabla \varphi ^n| \leq \frac{| \nabla S| }{\kappa _0(T_{o})}\left( \kappa_1(T_0)+\frac{\lambda \alpha }{(1-\gamma )\nu _0(T_{o})}| g| _{3/2}\left( \frac{\kappa_1(T_0)}{\kappa _0(T_0)}+1\right) | S| _{3}\right) \equiv r_{2} & \label{A17} \end{eqnarray} Therefore we have uniform estimates on $u^n$ and $\varphi ^n$. Indeed, $r_{1} $ and $r_{2}$ are both independent of $\lambda ,n,m$. Hence solutions of $(c;d)=\lambda P(c;d)$ for $\lambda \in [0,1]$ lie in a ${\mathbb R}^{2n}- $ball $\left\{ \sum_{j=1}^n\left( | c_j| ^{2}+| d_j| ^{2}\right) \leq r_{1}^{2}+r_{2}^{2}\right\} $. Therefore, due to Brouwer's fixed point theorem, we have obtained a solution $(u^n,\varphi ^n)$ of the equations (% \ref{A7}) with the property (after getting the fixed point, repeat the same calculation as $\lambda =1)$ \begin{equation} | \nabla u^n| \leq r_{1}\,, \quad | \nabla \varphi ^n| \leq r_{2}\,. \label{A18} \end{equation} Since $J(\Omega _m)$ (respectively. $H_0^{1}(\Omega _m)$ ) is compactly imbedded in $H(\Omega _m)$ (respectively. $L^{2}(\Omega _m))$ we can choose subsequences, which we again denote by $(u^n,\varphi ^n)$, and elements $\overline{u}^{m}\in J(\Omega _m)$, $\overline{\varphi }^{m}\in H_0^{1}(\Omega _m)$ such that $u^n\rightarrow \overline{u}^{m}$ weakly in $J(\Omega _m)$ and strongly in $H(\Omega _m)$ and also $\varphi ^n\rightarrow \overline{\varphi }^{m}$ weakly in $H_0^{1}(\Omega _m)$, and strongly in $L^{2}(\Omega _m)$ and also everywhere in $\Omega _m$. Passing to the limit in (\ref{A9}) as $n\rightarrow \infty $, we find that $(% \overline{u}^{m},\overline{\varphi }^{m})$ is a desired weak solution of $(P_m)$. \begin{lemma} Let us $(\overline{u}^{m},\overline{\varphi }^{m})$ be a weak solution for $(P_m)$ obtained in the previous lemma. Put \[ u^{m}(x)=\left\{ \begin{array}{l} \overline{u}^{m}(x)\mbox{ if }x\in \Omega _m \\ 0\mbox{ \quad \quad if }x\in \Omega \backslash \Omega _m \end{array} \right. \] \[ \varphi ^{m}(x)=\left\{ \begin{array}{l} \overline{\varphi }^{m}(x)\mbox{ if }x\in \Omega _m \\ 0\mbox{ \quad \quad if }x\in \Omega \backslash \Omega _m. \end{array} \right. \] Then it holds that $(u^{m},\varphi ^{m})\in J(\Omega )\times H_0^{1}(\Omega )$ and furthermore \begin{equation} | \nabla u^{m}| \leq r_{1}\,, \quad | \nabla \varphi ^{m}| \leq r_{2} \label{A18.5} \end{equation} where $r_{1}$ and $r_{2}$ be taken uniformly in $m$. \end{lemma} \paragraph{Proof} It is easy to show $(u^{m},\varphi ^{m})\in J(\Omega )\times H_0^{1}(\Omega ) $. The estimates (\ref{A18.5}) are directly deduced from the (\ref{A18}) and the lower semi-continuity of the norm. \section{Proof of main theorem} Using the previous lemma, applying Rellich's compactness theorem, and the diagonal argument, we can choose subsequences which we again denote by $(u^{m},\varphi ^{m})$ and $u\in J(\Omega ),\varphi \in H_0^{1}(\Omega )$ such that \begin{eqnarray*} &u^{m} \rightarrow u\mbox{ weakly in }J(\Omega )\mbox{ and strongly in }% L_{loc}^{2}(\Omega ) & \\ &\varphi ^{m} \rightarrow \varphi \mbox{ weakly in }H_0^{1}(\Omega )% \mbox{ and strongly in }L_{loc}^{2}(\Omega )\,.& \end{eqnarray*} Once we get such subsequences and limits, we can show that $(u,\varphi ) $ becomes a stationary weak solution of (\ref{A6.5}). In fact, let us $(\xi ,\psi )$ be an arbitrary given test function. Then we find a bounded domain $\Omega ^{\prime}$ and a number $m_0$ such that supp$\xi $, supp $\psi\subset\Omega ^{\prime}$ and $\Omega^{\prime}\subset \Omega_{m_0}\subset \Omega _m$ for all $m\geq m_0$. Then \begin{eqnarray*} \lefteqn{| \left( \nu (\varphi ^{m}+S)\nabla \xi ,\nabla u^{m}\right) _{\Omega }-\left( \nu (\varphi +S)\nabla \xi ,\nabla u\right) _{\Omega }|} \\ &\leq &| \left( (\nu (\varphi ^{m}+S)-\nu (\varphi +S))\nabla \xi ,\nabla u^{m}\right) _{\Omega ^{\prime}}| +| \left( \nu (\varphi +S)\nabla \xi ,\nabla (u^{m}-u)\right) _{\Omega ^{\prime}}| \\ &\leq &| \nu (\varphi ^{m}+S)-\nu (\varphi +S)| _{\infty }| \nabla \xi | | \nabla u^{m}| +| \left( \nu (\varphi +S)\nabla \xi ,\nabla (u^{m}-u)\right) _{\Omega ^{\prime}}| \end{eqnarray*} because the function $\nu $ is continuous and $\varphi ^{m}\rightarrow \varphi $ strongly in $L_{loc}^{2}(\Omega )$, it is now immediate that $\nu (\varphi ^{m}+S)$ converges strongly towards $\nu (\varphi +S)$. This, together with the weak convergence $u^{m}\rightarrow u$ in $J(\Omega )$, yields the convergence \[ | \left( \nu (\varphi ^{m}+S)\nabla \xi ,\nabla u^{m}\right) _{\Omega }-\left( \nu (\varphi +S)\nabla \xi ,\nabla u\right) _{\Omega }| \rightarrow 0 \] as $m\rightarrow \infty $. The other convergences are analogously established. Thus, we see $(u,\varphi )$ is a stationary weak solution for (% \ref{A6.5}) \paragraph{Acknowledgments} The first author would like to express his deepest gratitude to FAPESP (Project 1998/00619-9) for their support during the author's stay at the Departamento de Matem\'{a}tica Aplicada of UNICAMP in May of 1998 where this paper was completed. \begin{thebibliography}{99} \bibitem{Drazin} \textit{\ }P.G. Drazin and W.H. Reid, \textit{Hydrodynamic Stability, }Cambridge Univ. Press, 1981. \bibitem{Gilbarg} D. Gilbarg and N.S. 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