\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. Trudinger, \textit{Elliptic Partial
Differential Equations of Second Order, }Springer, Berlin-Heidelberg (1983).

\bibitem{Heywood}  J.G. Heywood, \textit{The Navier-Stokes equations: on the
existence, regularity and decay of solutions, }Indiana Univ. Math. J. 
\textbf{29 }(1980), 639-181.

\bibitem{Heywoodj}  J.G. Heywood, \textit{On uniqueness questions in the
theory of viscous flow, }Acta Math. 136 (1976), 61-102.

\bibitem{Hishida}  T. Hishida, \textit{Asymptotic behavior and stability of
solutions to exterior convection problem}, Nonlinear Anal. 22, 1994, 895-925.

\bibitem{Korenev}  N.K. Korenev, \textit{On some problems of convection in a
viscous incompressible fluid}, Vestnik Leningrad. Univ. 1971, No. 7; English
Trans.. Vestnik Leningrad Univ. Math., Vol. 4, 1977, pp. 125-137.

\bibitem{Ladyzhenskaya}  O.A. Ladyzhenskaya, \textit{The Mathematical theory
of Viscous Incompressible Flow, }Gordon and Breach, New York-London (1963).

\bibitem{Lorca}  S.A. Lorca, J.L. Boldrini, \textit{Stationary solutions for
Generalized Boussinesq models}, J.Diff Eq., 124, 1996, 389-406.

\bibitem{Lorca1}  S.A. Lorca, J.L. Boldrini, \textit{The initial value
Problem for generalized Boussinesq model}, to appear in Nonlinear Anal.

\bibitem{Morimoto}  H. Morimoto, \textit{On the existence and uniqueness of
the stationary solution to the equations of natural convection}, Tokyo J.
Math. , 14, 1991, 220-226.

\bibitem{Oeda1}  K. Oeda, \textit{Stationary solutions of the heat
convection equations in exterior domains}, Proc. Japan Acad. 73, Ser. A,
1997, 111-115.

\bibitem{Oeda2}  K. Oeda, \textit{Remarks on the periodic solution of the
heat convection equation in a perturbed annulus domain}, Proc. Japan Acad.,
73 Ser. A, 1997, 21-25.

\bibitem{Oeda3}  K. Oeda, \textit{Periodic solutions of the heat convection
equation in exterior domain}, Proc. Japan Acad. 73, Ser. A, 1997, 49-54.

\bibitem{Lorca3}  M.A. Rojas-Medar, S.A. Lorca, \textit{The equations of a
viscous incompressible chemical active fluid I: uniqueness and existence of
the local solutions}, Rev. Mat. Apl., 16, 1995, 57-80.

\bibitem{Lorca4}  M.A. Rojas-Medar, S.A. Lorca, \textit{The equations of a
viscous incompressible chemical active fluid II: regularity of solutions},
Rev. Mat. Apl., 16, 1995, 81-95.
\end{thebibliography}

\bigskip

\textsc{E.A. Notte-Cuello}\newline
Dpto. de Matem\'{a}tica, U. de Antofagasta\newline
Casilla 170, Antofagasta, Chile.\newline
Email address: enotte@uantof.cl \medskip

\textsc{M.A. Rojas-Medar}\newline
UNICAMP-IMECC, C.P.6065,13081-970, Campinas,\newline
S\~{a}o Paulo, Brazil \newline
Email address: marko@ime.unicamp.br

\end{document}