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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 92, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/92\hfil Continuous dependence]
{Continuous dependence for the Brinkman
equations of flow in  double-diffusive  convection}

\author[H. Tu, C. Lin \hfil EJDE-2007/92\hfilneg]
{Hongliang Tu,   Changhao Lin}  

\address{Hongliang Tu \newline
Department of Applied Mathematics,
 Zhuhai college of Jilin University, 519041, China}
 \email{tuhongliang00@126.com}

\address{Changhao  Lin \newline
 School of Mathematical  Sciences,
 South China Normal University, 510631, China}
\email{linchh@scnu.edu.cn}


\thanks{Submitted April 9, 2007. Published June 16, 2007.}
\subjclass[2000]{35B30, 35K55, 35Q35}
\keywords{Continuous dependence; structural stability;
\hfill\break\indent gravity coefficients;  Soret coefficient;  Brinkman
equations}

\begin{abstract}
 This paper  concerns the  structural  stability for
 convective motion in a fluid-saturated porous medium
 under the Brinkman scheme. Continuous dependence for the
 solutions on the gravity coefficients and the Soret coefficient
 are proved. First of all,  an a priori bound in
 $L^2$ norm is derived whereby we show the solution depends
 continuously in $L^2$ norm on changes in the gravity coefficients
 and the Soret coefficient. This estimate also implies that the solutions
 decay exponentially.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\section{Introduction}


Within the context of fluid flow in porous media, or simply within
theory of fluid flow, there has been substantial recent interest in
deriving stability estimates where changes in coefficient are
allowed, or even the model (the equations themselves) changes. This
type of stability has earned the name structural stability, and is
different from continuous dependence on the initial data. The
concept of structural stability in which the study of continuous
dependence (or stability) is on changes in the model itself rather
than the initial data. Thus structural stability constitutes a class
of stability problem every bit important. Structural stability is
focus of attention now,  and for the relevant results,  the reader
is referred to \cite{a1,f1,g1,p1,p2,p3,p4,p5,s1}.

 The Brinkman  model is believed accurate when the flow
velocity is too large for Darcy's law to be valid,  and
additionally the porosity is not too small.  In this article,  we
are concerned with structural stability for the Brinkman equations
modeling the double diffusive convection.   The temperature field
is non-constant and a solute is also diffused throughout the
porous body.

 The Brinkman  equations governing the flow of fluid  in double-diffusive
  convection   are
\begin{equation}
   \begin{gathered}
     u_{i,t}=\nu \Delta u_i -au_i-p_{,i}+g_iT+h_iC\\
      T_{,t}+u_iT_{,i}=\Delta T\\
     C_{,t}+u_iC_{,i}=\Delta C+\rho\Delta T\quad \mbox{in }\Omega\times{t>0}\\
     u_{i,i}=0
\end{gathered} \label{11}
\end{equation}
where    $u_i$,  $T$,  $C$   and   $p$   represent fluid velocity,
temperature,  salt concentration and pressure,  respectively. The
quantities   $g_i(x)$   and  $h_i(x)$   are gravity vector terms,
and  positive constants   $\rho$,  $a$ are known the Soret
coefficient and the  Darcy coefficient,  respectively. $\Omega$ is
a bounded domain of of  $R^3$,  and with sufficiently smooth
boundary $\partial\Omega $.  $\Delta$ is the Laplace
operator,   $\|\cdot\|$   and  $\langle u,v\rangle$ denote the
norm  and inner product on $L^2(\Omega)$ .

Associated with   (\ref{11}),  we impose the boundary data
\begin{equation}
 u_i=0 ; \quad  T=0; C=0\quad \mbox{on }   \partial \Omega\label{12}
\end{equation}
and the initial data
 \begin{equation}
u_i(x,0)=z_i(x);  T(x,0)=T_0(x) ;   C(x,0)=C_0(x);
x\in\Omega\label{13}
\end{equation}
  In (\ref{11})   and  in the equations throughout,  a
comma is used to denote  partial differentiation: For example
  $u_{i,i}$  denotes    $\frac{\partial u_i}{\partial x_i}$
and   $u_{i, t}$  denotes   $\frac{\partial u_i}{\partial t}$,
and we employ the convention of
 summing over repeated indices  from   $1$   to   $3$  .

\section{A priori bounds}

Multiplying $(\ref{11})_2$   by   $T$   and integrating over
$\Omega$,  we have
\begin{equation}
\frac{1}{2}\frac{d}{dt}\|T\|^2=-\|\nabla
T\|^2\label{21}\end{equation}

Multiplying  $(\ref{11})_3$ by $C$   and integrating over
$\Omega$,   then using arithmetic-geometric mean inequality,  we
obtain
\begin{equation}
\frac{1}{2}\frac{d}{dt}\|C \|^2+\frac{1}{2}\|\nabla
C\|^2\leq\frac{\rho^2}{2}\|\nabla T\|^2\label{22}
\end{equation}

Multiplying $  (\ref{11})_1$ by $u_i$ and integrating over $\Omega$,
 furthermore  using Cauchy-Schwarz,
  arithmetic-geometric mean,  then using Poincare's inequality,  we find
\begin{equation}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}\|u\|^2+\nu\|\nabla u
\|^2+\frac{a}{2}\|u\|^2
& \leq \frac{1}{a}[g^2\|T\|^2+h^2\|C \|^2] \\
& \leq \frac{1}{a\lambda_1}[g^2\|\nabla T\|^2+h^2\|
\nabla C\|^2]
\end{aligned}\label{23}
\end{equation}
where
\[
g^2=\max_\Omega g_ig_i; \quad h^2=\max_\Omega h_ih_i.
\]
and $\lambda_1$ is the first eigenvalue of the problem
\begin{gather*}
\Delta\phi+\lambda\phi=0 \quad \text{in }   \Omega\\
\phi=0 \quad   \text{on }  \partial\Omega
\end{gather*}

Multiplying  $(\ref{11})_1$ by $u_{i,t}$ and integrating over
$\Omega$ ,    furthermore  using Cauchy-Schwarz     and
arithmetic-geometric mean  inequality,  then using Poincare's
Poincare's inequality,  we find
\begin{equation}
\begin{aligned}
\frac{a}{2}\frac{d}{dt}\|u\|^2+\frac{\nu}{2}\frac{d}{dt}\|\nabla u\|^2
&\leq \frac{1}{2}[g^2\|T\|^2+h^2\|C \|^2] \\
&\leq \frac{1}{2\lambda_1}[g^2\|\nabla T\|^2+h^2\|\nabla C\|^2]
\end{aligned}\label{24}
\end{equation}

Multiplying   (\ref{21})   by $\Gamma_{11}$,  (\ref{22})   by
$\Gamma_{12}$ and   (\ref{23})   by $\Gamma_{13}$,  then adding all
results to   (\ref{24})  leads to
\begin{equation}
\frac{d}{dt}Q_1(t)+G(t)\leq 0 \label{25}
\end{equation}
where  $\Gamma_{1i},  (i=1,  2,  3)$ are positive constants at our
disposal,
\begin{equation}
Q_1(t)=\frac{\Gamma_{11}}{2}\|T\|^2+\frac{\Gamma_{12}}{2}\|C
\|^2+\frac{(\Gamma_{13}+a)}{2}\|u\|^2 +\frac{\nu}{2}\|\nabla
u \|^2\label{26}
\end{equation}
and
\begin{equation}
\begin{aligned}
G(t)&=[\frac{(2\Gamma_{11}-\rho^2\Gamma_{12})}{2}
 -\frac{1}{\lambda_1}g^2(\frac{1}{2}
 +\frac{\Gamma_{13})}{a}]\|\nabla T\|^2+\frac{a\Gamma_{13}}{2}\|u\|^2
 \\
&\quad +[\frac{\Gamma_{12}}{2}-\frac{1}{\lambda_11}h^2(\frac{1}{2}
+\frac{\Gamma_{13})}{a}]\|\nabla C \|^2
+\Gamma_{13}\nu\|\nabla u \|^2
\end{aligned}\label{27}
\end{equation}

We can select $\Gamma_{1i},  (i=1,  2,  3)$ to secure that all the
all the coefficients of   (\ref{27})   are   positive,  such as
\[
\Gamma_{13}=\frac{a}{2}; \quad
\Gamma_{12}=\frac{4h^2}{\lambda_1}; \quad
\Gamma_{11}>\frac{(2h^2\rho^2+g^2)}{\lambda_1}.
\]

Thus, with the help of Poincare's inequality, we can show that
\begin{equation}
\begin{aligned}
G(t)&\geq[\lambda_1\frac{(2\Gamma_{11}-\rho^2\Gamma_{12})}{2}
-g^2(\frac{1}{2}+\frac{\Gamma_{13})}{a}]\|
T\|^2+\Gamma_{13}\nu\|\nabla u \|^2 \\
&\quad +[\frac{2\lambda_1\Gamma_{12}}{2} -h^2(\frac{1}{2}
+\frac{\Gamma_{13}}{a}]\|C\|^2 +\Gamma_{13}\frac{a}{2}\|u\|^2
\end{aligned}\label{28}
\end{equation}

We can easily show that
\begin{equation}
\kappa_1 Q_1(t)\leq G(t) \label{29}
\end{equation}
where $\kappa_1$ is a positive constant represented by
\begin{align*}
\kappa_1
=\min \big\{&\frac{1}{\Gamma_{11}}[\lambda_1(2\Gamma_{11}-\rho^2\Gamma_{12})
  -g^2(1+\frac{2\Gamma_{13})}{a}], \\
&\frac{1}{\Gamma_{12}}[\Gamma_{12}\lambda_1
 -h^2(1+\frac{2\Gamma_{13})}{a}],
 \frac{a\Gamma_{13}}{\Gamma_{13}+a},2\Gamma_{13}\big\}
\end{align*}

Thus, from   (\ref{25}),  we can derive that
\begin{equation}
\frac{d}{dt}Q_1(t)+\kappa_1 Q_1(t) \leq 0\label{210}
\end{equation}

Furthermore, we obtain
\begin{equation}
Q_1(t)\leq Q_1(0)e^{-\kappa_1 t} \label{211}
\end{equation}

Therefore,  recalling the definition of $Q_1(t) $ and combining
(\ref{25}),  we can obtain
\begin{equation}
\begin{gathered}
\|T\|^2\leq M_1;  \quad
\|C\|^2\leq M_1; \quad
\|u\|^2\leq M_1;  \quad
\|\nabla u\|^2\leq M_1;  \quad
\int_0^t\|u\|^2d\eta\leq M_1 ;   \\
\int_0^t\|\nabla u \|^2d\eta\leq M_1; \quad
\int_0^t\|\nabla T\|^2d\eta\leq M_1; \quad
\int_0^t\|\nabla C\|^2d\eta\leq M_1.
\end{gathered} \label{212}
\end{equation}
where $M_1$ is a generic positive constant depending on the
coefficients of   (\ref{11}) and the initial data terms in
(\ref{13}).

It also follows  from inequality   (\ref{211})   that
$\|\nabla u \|^2$, $\|T\|^2$ and
$\|C\|^2$  decay exponentially as $t$   tends to  $\infty$.


\section{Continuous dependence on the   gravity coefficients}


Let $(u_i,  T,  C,  P)$ and $(u_i^*,  T^*,  C^*, P^*)$  be solutions
to  (\ref{11})   with the same boundary and initial data (\ref{12}),
(\ref{13}),  but with different gravity coefficients $(g_i,  h_i)$
and   $(g_i^*,  h_i^*)$, respectively.

 Namely,
 \begin{gather*}
     u_{i,t}=\nu \Delta u_i -au_i-p_{,i}+g_iT+h_iC\\
      T_{,t}+u_iT_{,i}=\Delta T\\
     C_{,t}+u_iC_{,i}=\Delta C+\rho\Delta T\quad \mbox{in }\Omega\times{t>0}\\
     u_{i,i}=0
\end{gather*}
and
  \begin{gather*}
     u_{i,t}^*=\nu \Delta u_i^* -au_i^*-p_{,i}^*+g_i^*T^*+h_i^*C^*\\
      T_{,t}^*+u_i^*T_{,i}^*=\Delta T^*\\
     C_{,t}^*+u_i^*C_{,i}^*=\Delta C^*+\rho\Delta T^*\quad \mbox{in}\quad\Omega\times{t>0}\\
     u_{i,i}^*=0\,.
\end{gather*}

Now set
\begin{equation}
\begin{gathered}
     w_i=u_i -u_i^*, \quad \Pi =p-p^*,S=T-T^*, \quad \Sigma=C-C^*\\
     \gamma_i=g_i -g_i^*,\quad \mu_i=h_i-h_i^*
\end{gathered}  \label{31}
\end{equation}

Clearly,   $(w_i,  \Pi,  S,  \Sigma)$  satisfies  the equations
\begin{equation}
  \begin{gathered}
   w_{i,  t}=\nu \Delta w_i -aw_i-\Pi_{,  i}+\gamma_iT +g_i^*S+\mu_iC+h_i^*\Sigma\\
   S_{,t}+w_iT_{,i}+u_i^*S_{,  i}=\Delta S\\
   \Sigma_{,t}+w_iC_{,  i}+u_i^*\Sigma_{,  i}=\Delta
   \Sigma+\rho\Delta S  \quad  \mbox{in }\Omega\times \{t>0\}\\
   w_{i,  i}=0
\end{gathered} \label{32}
\end{equation}
with the boundary-initial data
\begin{equation}
   \begin{gathered}
   w_i=S=\Sigma=0 \quad  \mbox{on }   \partial\Omega\times\{t>0\}\\
  w_i(x,  0)= S(x,  0)=\Sigma(x,  0)=0\quad  \mbox{in }   \Omega
\end{gathered}  \label{33}
\end{equation}

Multiplying $(\ref{32})_1$ by $w_i$ and integrating over $\Omega$,
then using Cauchy-Schwarz's inequality, and the arithmetic-geometric
mean inequality, we  obtain
\begin{equation}
\nu\|\nabla w\|^2+\frac{a}{2}\|
w\|^2+\frac{1}{2}\frac{d}{dt}\|w\|^2 \leq\frac{2}{a}[(g^*)^2\|S\|^2
+ (h^*)^2 \|\Sigma\|^2+ \mu^2\|C\|^2+ \gamma^2 \| T\|^2], \label{34}
\end{equation}
where
\[
\gamma^2=\max_\Omega \gamma_i\gamma_i;  \quad
\mu^2=\max_\Omega u_i; \quad
(g^*)^2=\max_\Omega g^*_ig^*_i; \quad
(h^*)^2=\max_\Omega h^*_ih^*_i.
\]

Multiplying $(\ref{32})_1$ by $w_{i,  t}$ and integrating over
$\Omega$,  then  using Cauchy-Schwarz inequality, and the
arithmetic-geometric mean inequality,  we have
\begin{equation}
\frac{1}{2}\nu\frac{d}{dt}\| \nabla
w\|^2+\frac{a}{2}\frac{d}{dt}\|w\|^2\leq (g^*)^2\|S\|^2 + (h^*)^2
\|\Sigma\|^2 +\mu^2\|C\|^2+ \gamma^2 \|T\|^2 \label{35}
\end{equation}

Multiplying $(\ref{32})_2$ by   $S$  and integrating over $\Omega$,
then  using  Cauchy-Schwarz,  the arithmetic-geometric mean
inequality,  the Sobolev inequality ,which holds for all $\varphi
\in C_0^1(\varphi)$,
\[
\|\varphi\|_4 \leq\Lambda \| \varphi_{,  i}\|
\]
where $\|\cdot\|_4$ is the norm on $L^4(\Omega)$.

we derive
\[
\frac{d}{dt}\|S \|^2 =-2 \|\nabla S  \|^2+2\langle T_{,  i},  w_iS\rangle;
\]
therefore
\begin{equation}
\begin{aligned}
\frac{d}{dt} \|S \|^2
&\leq -2 \|\nabla S  \|^2+  2\|\nabla T\|\cdot\|w\|_4\|S\|_4  \\
&\leq -2 \|\nabla S  \|^2+2
 \Lambda^\frac{1}{2}\|\nabla T\|\cdot\|\nabla w\|\cdot\|\nabla S \|   \\
&\leq -2\|\nabla S \|^2+2\alpha_{11}\|\nabla S\|^2
  +\frac{\Lambda}{2\alpha_{11}}\| \nabla w\|^2\|\cdot\|\nabla T\| ^2
\end{aligned} \label{36}
\end{equation}
where  $\alpha_{11}$ is a positive constant at our disposal.

Using  similar method as in (\ref{36}),   from equation
$(\ref{32})_3$,  we   find that
\begin{equation}
\frac{d}{dt}\|\Sigma \|^2\leq (-2+\alpha_{12}+\alpha_{13})\|
\nabla \Sigma\|^2+\frac{\Lambda}{\alpha_{13}} \|\nabla
w\|^2\cdot\|\nabla C \|^2+
 \frac{\rho^2}{\alpha_{12}}\|
\nabla S\|^2 \label{37}
\end{equation}
where $\alpha_{1i}$,  $i=2,  3$ are  positive constants at our
disposal.

Multiplying   (\ref{36})  by $\Gamma_{21}$ and adding (\ref{37})
leads to
\begin{equation}
\begin{aligned}
&\frac{d}{dt}(\| \Sigma\|^2+\Gamma_{21} \|S\|^2)\\
&\leq \frac{\Lambda}{2}\|\nabla w\|^2(\frac{\Gamma_{21}}{\alpha_{11}}
 \|\nabla T\|^2+\frac{2}{\alpha_{13}}\|\nabla C\|^2 ) \\
&\quad +2[(\alpha_{11}-1)+\frac{\rho^2}{2\alpha_{12}\Gamma_{21}}]
\Gamma_{21}\|\nabla S \|^2+(-2+\alpha_{12}+\alpha_{13})\| \nabla
\Sigma\|^2
\end{aligned} \label{38}
\end{equation}
where  $\Gamma_{21}$  is a   positive constant at our disposal.

In (\ref{38}),  we choose
\[\alpha_{11}=\frac{1}{4}; \quad
\alpha_{12}=\alpha_{13}=\frac{1}{2}; \quad
\Gamma_{21}=4\rho^2.
\]
it follows that
\begin{equation}
\frac{d}{dt}(\|\Sigma\|^2+4\rho^2 \|S\|^2)+\|  \nabla
\Sigma\|^2 +4\rho^2\|\nabla S  \|^2\leq
2\Lambda\|\nabla w\|^2(4\rho^2\|\nabla T\|
^2+\|\nabla C \|^2)\label{39}
\end{equation}

Multiplying   (\ref{34})  by $\Gamma_{22}$ and   (\ref{35})  by
$\Gamma_{23}$,  then adding all the results to  (\ref{39}),  then
using  Poincare's inequality,  we have
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\big[\frac{\Gamma_{23}}{2}\nu\|\nabla
w\|^2+\frac{(\Gamma_{22}+a\Gamma_{23})}{2}\|
w\|^2+(\|\Sigma\|^2+4\rho^2\|S\|^2)\big]\\
&+\big[\lambda_1-(\frac{2\Gamma_{22}}{a} +\Gamma_{23})(g^*)^2\big]
\|\Sigma\|^2 \\
&+\big[4\rho^2\lambda_1-(\frac{2\Gamma_{22}}{a} +\Gamma_{23})(h^*)^2\big]
\|S\|^2 + \Gamma_{22}\nu\|\nabla w\|^2
+\frac{a\Gamma_{22}}{2}\|w\|^2 \\
&\leq w\|^2\big[4\rho^2\|\nabla T\|^2 +\|C_{,  i}\|^2 \big]
+ (\frac{\Gamma_{22}}{a} +\Gamma_{23})\big[\mu^2\|C\|^2 +\gamma^2
\|T\|^2\big]
\end{aligned}\label{310}
\end{equation}
where $\Gamma_{2i}$,  $i=2,3$ are  positive constants at our
disposal.

We can choose $\Gamma_{22}$ and $\Gamma_{23}$ sufficient small to
make sure  that all   the coefficients in   (\ref{310})   are
positive,   such as
\[
\Gamma_{22}=\min \{\frac{a\lambda_1}{8(g^*)^2},
\frac{\rho^2a\lambda_1}{2(h^*)^2}\}; \quad
\Gamma_{23}=\min \{\frac{\lambda_1}{4(g^*)^2},
\frac{\rho^2\lambda_1}{(h^*)^2}\} .
\]

Let
\[
Q_2(t)=\{\frac{\Gamma_{23}}{2}\nu\|\nabla
w\|^2+\frac{(\Gamma_{22}+a\Gamma_{23})}{2}\|
w\|^2+\|\Sigma\|^2+4\rho^2\|S\|^2\}
\]
and
\begin{equation}
\begin{aligned}
\kappa_2=\min \big\{&\frac{2\Gamma_{22}\nu}{\Gamma_{23}\nu};
\frac{a\Gamma_{22}}{(\Gamma_{22}+a\Gamma_{23})};
(\lambda_1-(\frac{2\Gamma_{22}}{a}
+\Gamma_{23})(g^*)^2);   \\
&\frac{1}{4\rho^2}[4\rho^2\lambda_1-(\frac{2\Gamma_{22}}{a}
+\Gamma_{23})(h^*)^2]\big\}
\end{aligned}\label{311}
\end{equation}

It follows from  (\ref{310})  that
\begin{equation}
\frac{d}{dt}Q_2(t)+\kappa_2 Q_2(t)\leq (\frac{\Gamma_{22}}{a}
+\Gamma_{23})[\mu^2\|
C\|^2 + \gamma^2 \|T\|^2] \\
 +2\Lambda\|
\nabla w\|^2[4\rho^2\|\nabla T\|^2+\|\nabla C\|^2 ]\label{312}
\end{equation}

It is easy to show  that
\begin{equation}
2\Lambda\|\nabla w\|^2[4\rho^2\|\nabla
T\|^2+\|\nabla C\|^2 ]\leq Q_2(t)f_2(t) \label{313}
\end{equation}
where
\[
f_2(t)=\tau_2(\|\nabla T\|^2+\|\nabla C\|^2), \quad
\tau_2=\frac{4\Lambda}{\Gamma_{23}\nu}(4\rho^2+1) .
\]

Thus, combining   (\ref{312})  and  (\ref{313}),  we obtain
\begin{equation}
\frac{d}{dt}Q_2(t)+\kappa_2 Q_2(t)\leq
M_2(\mu^2+\gamma^2)+f_2(t)Q_2(t)\label{314}
\end{equation}
where $M_2=2(\frac{\Gamma_{22}}{a} +\Gamma_{23})M_1$.

 Now,  multiplying both sides of   (\ref{314})  by
$e^{\kappa_2t}$ and integrating from  0 and $t$,  we get
\begin{equation}
Q_2(t)\leq \frac{M_2}{\kappa_2}(\mu^2+\gamma^2)+\int_0^t
f_2(\eta)Q_2(\eta)d\eta\label{315}
\end{equation}

Hence,  applying Gronwall's lemma and using   (\ref{212}),  we
obtain
\begin{equation}
Q_2(t)\leq \frac{M_2}{\kappa_2}e^{\int_0^t
f_2(\eta)d\eta}\cdot(\mu^2+\gamma^2)\leq\frac{M_2}{\kappa_2}e^{2\tau_2M_1}\cdot(\mu^2+\gamma^2),
\forall t>0 \label{316}
\end{equation}

Consequently,  from inequality   (\ref{316}),  we can see that
$Q_2(t)\to0$,   as $g_i\to g_i^*$ and $h_i\to h_i^*$.   Recalling
the definition of $Q_2(t)$, so    $(w_i,  \Pi,  S,  \Sigma)\to 0$
and $(u_i, T,C)\to (u_i^*,  T^*,  C^*)$. So,  continuous dependence
on the gravity coefficients is  proved.


\section{Continuous dependence on   the Soret coefficient}


Let $(u_i,  T,  C,  P)$ and $(u_i^*,  T^*,  C^*,  P^*)$  be
solutions to (\ref{11} ),   with the same boundary and initial data
 (\ref{12}),   (\ref{13}),  but with different Soret
coefficient $\rho$ and $\rho^*$, respectively:
   \begin{gather*}
     u_{i,  t}=\nu \Delta u_i -au_i-p_{,  i}+g_iT+h_iC\\
      T_{,  t}+u_iT_{,  i}=\Delta T\\
     C_{,  t}+u_iC_{,  i}=\Delta C+\rho\Delta T\quad \mbox{in }
\Omega\times{t>0}\\
     u_{i,  i}=0
\end{gather*}
and
  \begin{gather*}
     u_{i,  t}^*=\nu \Delta u_i^* -au_i^*-p_{,  i}^*+g_iT^*+h_iC^*\\
      T_{,  t}^*+u_i^*T_{,  i}^*=\Delta T^*\\
     C_{,  t}^*+u_i^*C_{,  i}^*=\Delta C^*+\rho^*\Delta T^*\quad \mbox{in }
\Omega\times{t>0}\\
u_{i,  i}^*=0
\end{gather*}


Now set
\begin{equation}
w_i=u_i -u_i^*, \quad \Pi =p-p^*, \quad S=T-T^*, \quad
 \Sigma=C-C^* \label{41}
\end{equation}
Clearly,   $(w_i,  \Pi,  S,  \Sigma)$   satisfies  the equations
\begin{equation}
   \begin{gathered}
   w_{i,  t}=\nu \Delta w_i -aw_i-\Pi_{,  i}+ +g_iS++h_i\Sigma\\
   S_{,  t}+w_iT_{,  i}+u_i^*S_{,  i}=\Delta S\\
   \Sigma_{,  t}+w_iC_{,  i}+u_i^*\Sigma_{,  i}=\Delta
   \Sigma+(\rho-\rho^*)\Delta T+\rho^*\Delta S  \quad  \mbox{in }
\Omega\times \{t>0\}\\
   w_{i,  i}=0
\end{gathered} \label{42}
\end{equation}
with the boundary-initial data
\begin{equation}
  \begin{gathered}
   w_i=S=\Sigma=0 \quad  \mbox{on }   \partial\Omega\times\{t>0\}\\
   w_i(x,  0)=S(x,  0)=\Sigma(x,  0)=0\quad  \mbox{in }   \Omega
\end{gathered}  \label{43}
\end{equation}

Multiplying $ (\ref{42})_1$ by $w_i$ and integrating over $\Omega$,
furthermore using the Cauchy-Schwarz's inequality, and the
arithmetic-geometric mean inequality,   we have
\begin{equation}
\nu\|\nabla w\|^2+\frac{a}{2}\|
w\|^2+\frac{1}{2}\frac{d}{dt}\|w\|^2\leq \frac{1}{a}[g^2\|S\|^2 +
h^2 \|\Sigma\|^2]\label{44}
\end{equation}
where
 \[
g^2=\max_\Omega g_ig_i,  h^2=\max_\Omega
h_ih_i.
\]

Multiplying $(\ref{42})_1$ by $w_{i,  t}$ and integrating over
$\Omega$, then  using the Cauchy-Schwarz inequality, and the
arithmetic-geometric mean inequality,  we get
\begin{equation}
\frac{1}{2}\nu\frac{d}{dt}\|\nabla
w\|^2+\frac{a}{2}\frac{d}{dt}\|w\|^2 \leq \frac{1}{2}[g^2\|S\|^2+
h^2\| \Sigma\|^2]\label{45}
\end{equation}

Multiplying $(\ref{42})_2$ by   $S$    and integrating over
$\Omega$,  then  using  Cauchy-Schwarz,  the arithmetic-geometric
mean inequality,  and  using the Sobolev inequality, we can derive
\begin{equation}
\frac{d}{dt}\|S \|^2d\eta \leq -2\|\nabla S \|^2
+2\alpha_{21}\|\nabla S \|^2
+\frac{\Lambda}{2\alpha_{21}}\|\nabla w\|^2\|\nabla T
\|^2\label{46}
\end{equation}
for $\alpha_{21}$ is a  positive constant at our disposal.

Using  similar method as  in (\ref{46}),   from equation
$(\ref{42})_3$,  we can  find that
\begin{equation}
\begin{aligned}
\frac{d}{dt}\|\Sigma \|^2
& \leq (-2+\alpha_{22}+\alpha_{23}+\alpha_{24})\| \nabla \Sigma\|^2
+ \frac{\Lambda}{\alpha_{23}} \|\nabla w\|^2\|\nabla C \|^2 \\
&\quad +\frac{\rho^{*  2}}{\alpha_{22}}\|\nabla S\|^2
+\frac{(\rho-\rho^*)^2}{\alpha_{24}}\|\nabla T \|^2
\end{aligned}\label{47}
\end{equation}
for  $\alpha_{2i}$,  ($i=2,  3,  4$) are positive constants at our
disposal.

Multiplying   (\ref{46})   by $\Gamma_{31}$ and adding (\ref{47}),
leads to
\begin{equation}
\begin{aligned}
&\frac{d}{dt}( \|\Sigma\|^2+\Gamma_{31} \|S\|^2) \\
&\leq(-2+\alpha_{22}+\alpha_{23}+\alpha_{24})\|
 \nabla \Sigma\|^2
+2\big[(\alpha_{21}-1)+ \frac{\rho^{*  2}}{2\alpha_{22}\Gamma_{31}}\big]
\Gamma_{31}\|\nabla S  \|^2\\
&\quad +\frac{(\rho-\rho^*)^2}{\alpha_{24}}\|\nabla T\|^2
 + \frac{\Lambda}{2}\|
\nabla w\|^2\big[\frac{\Gamma_{31}}{\alpha_{21}} \|\nabla
T\|^2 +\frac{2}{\alpha_{23}}\|\nabla C\|^2 \big]
\end{aligned}\label{48}
\end{equation}
where  $\Gamma_{31}$ is a  positive constant at our disposal.

In (\ref{48}),  we choose $\alpha_{21}=1/2$, $ \alpha_{22}=1$,
$\alpha_{23}=\alpha_{24}=1/4$, It follows that
\begin{equation}
\begin{aligned}
&\frac{d}{dt}(\|\Sigma\|^2+\Gamma_{31} \|S\|^2)+\|
\nabla \Sigma\|^2 +(\Gamma_{31}-\rho^{*  2})\|\nabla S  \|^2\\
&\leq  \Lambda\|\nabla w\|^2(\Gamma_{31}\|
\nabla T\|^2+4\|\nabla C \|^2)
+4(\rho-\rho^*)^2\|\nabla T\|^2
\end{aligned}\label{49}
\end{equation}

Multiplying   (\ref{44})  by $\Gamma_{32}$ and   (\ref{45})  by
$\Gamma_{33}$,  furthermore  adding all the results to  (\ref{49}),
we get
\begin{equation}
\begin{aligned}
&\frac{d}{dt}[\frac{\Gamma_{33}}{2}\nu\|\nabla
w\|^2+\frac{(\Gamma_{32}+a\Gamma_{33})}{2}\| w\|^2 +(
\|\Sigma\|^2+\Gamma_4\|S\|^2)] \\
&+\Gamma_{32}\nu\| \nabla w\|^2 +\frac{a\Gamma_{32}}{2}\|
w\|^2+\| \nabla \Sigma\|^2  +(\Gamma_{31}-\rho^{*  2})\|\nabla S  \|^2 \\
& \leq(\frac{2\Gamma_{32}}{a}+\Gamma_{33})[g^2\|
\Sigma\|^2+h^2\|S\|^2]
 +2\Lambda\|\nabla w\|^2[\Gamma_{31}\|\nabla T\|^2 \\
&\quad +4\|\nabla C\|^2 ]+4(\rho-\rho^*)^2\|\nabla T \|^2
\end{aligned}\label{410}
\end{equation}
where   $\Gamma_{3i}$,  ($i=2,  3$) are positive constants at our
disposal.

We can choose $\Gamma_{31}$ to make sure that
 $\Gamma_{31}>\rho^{*2}$,  therefore,  by the  Poincare's inequality from
(\ref{410}),  we have
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\big[\frac{\Gamma_{33}}{2}\nu\|\nabla
w\|^2+\frac{(\Gamma_{32}+a\Gamma_{33})}{2}\|
w\|^2+(\|\Sigma\|^2+\Gamma_{31}\|S\|^2)\big] \\
&+\big[\lambda_1(\Gamma_{31}-\rho^{*  2})-(2\frac{\Gamma_{32}}{a}
+\Gamma_{33})h^2\big]\|S\|^2+\frac{a\Gamma_{32}}{2}\|w\|^2
 \\
&+ \Gamma_{32}\nu\| \nabla
w\|^2+\big[\lambda_1-(\frac{2\Gamma_{32}}{a}
+\Gamma_{33})g^2\big]\|\Sigma\|^2 \\
& \leq\Lambda\|\nabla w\|^2[\Gamma_{31}\|\nabla T\|^2 +4\|\nabla C\|^2
]+4(\rho-\rho^*)^2\|\nabla T\|^2
\end{aligned}\label{411}
\end{equation}

We can choose $\Gamma_{32}$ and $\Gamma_{33}$ sufficient small to
make sure that all the  coefficients in   (\ref{411})   are
positive,  such as we may choose
\[
\Gamma_{32}=\min \{\frac{a\lambda_1}{8g^2},
\frac{a\lambda_1(\Gamma_{31}-\rho^{*  2})}{8h^2}\}; \quad
\Gamma_{33}=\min \{\frac{\lambda_1}{4g^2},
\frac{\lambda_1(\Gamma_{31}-\rho^{*  2})}{4h^2}\} .
\]

Let
\[
Q_3(t)=\{\frac{\Gamma_{33}}{2}\nu\|\nabla
w\|^2+\frac{(\Gamma_{32}+a\Gamma_{33})}{2}\|
w\|^2+\|\Sigma\|^2+\Gamma_{31}\| S\|^2\}
\]
and
\begin{equation}
\begin{aligned}
\kappa_3=\min \{&\frac{2\Gamma_{32}}{\Gamma_{33}};
\frac{a\Gamma_{32}}{(\Gamma_{32}+a\Gamma_{33})};
(\lambda_1-(\frac{2\Gamma_{32}}{a}
+\Gamma_{33})g^2);   \\
&\frac{1}{\Gamma_{31}}[\lambda_1(\Gamma_{31}-\rho^{*
2})-(\frac{2\Gamma_{32}}{a} +\Gamma_{33})h^2]\}
\end{aligned}\label{412}
\end{equation}

Then,  it follows from  (\ref{411})  that
\begin{equation}
\frac{d}{dt}Q_3(t)+\kappa_3 Q_3(t)\leq \Lambda\|
w\|^2[\Gamma_{31}\|\nabla T\|^2 +4\| C_{,  i}\|^2 ]
+4(\rho-\rho^*)^2\|\nabla T\|^2\label{413}
\end{equation}

Also,  it can be easily shown that
\begin{equation}
\Lambda\|\nabla w\|^2[\Gamma_{31}\|\nabla
T\|^2+4\|\nabla C\|^2 ]\leq f_3(t)Q_3(t)\label{414}
\end{equation}
where
\[
f_3(t)=\tau_3(\|\nabla T\| ^2+\|\nabla C\|^2),
\tau_3=\frac{2\Lambda}{\Gamma_{33}\nu}(\Gamma_{31}+4) .
\]

Thus,  combing  (\ref{413})  and  (\ref{414}),  we obtain
\begin{equation}
\frac{d}{dt}Q_3(t)+\kappa_3 Q_3(t)\leq 4(\rho-\rho^*)^2\|\nabla
T\|^2 +f_3(t)Q_3(t)\label{415}\end{equation}

Now,  multiplying both sides of   (\ref{415})  by $e^{\kappa_3t}$
and integrating over  $[0,  t]$,  we obtain
\begin{equation}
Q_3(t)\leq \frac{M_3}{\kappa_3}(\rho-\rho^*)^2+\int_0^t
f_3(\eta)Q_3(\eta)d\eta\label{416}
\end{equation}
where $M_3=4M_1$.

Hence,  applying Gronwall's lemma and using   (\ref{212}),  we
obtain
\begin{equation}
Q_3(t)\leq \frac{M_3}{\kappa_3}e^{\int_0^t
f_3(\eta)d\eta}\cdot(\rho-\rho^*)^2\leq\frac{M_3}{\kappa_3}e^{2\tau_3M_1}
\cdot(\rho-\rho^*)^2, \quad
\forall t>0 \label{417}
\end{equation}

As a result,  from inequality   (\ref{417}),  we can  see that
$Q_3(t)\to0$,   as   $\rho\to \rho^*$.  Recalling the definition of
$Q_3(t)$,   so    $(w_i,  \Pi,  S, \Sigma)\to 0$   and   $(u_i,  T,
C)\to (u_i^*, T^*,  C^*)$.    Consequently,  continuous dependence
on the Soret coefficient is proved.

\subsection*{Acknowledgments}
The authors would like to express their gratitude to  Professor Yan Liu
for his help in writing parts of  this article.

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\end{document}