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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 75, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/75\hfil Reaction-diffusion system]
{Reaction-diffusion system of equations in non-stationary
medium and arbitrary non-smooth domains}

\author[S. A. Sanni\hfil EJDE-2010/75\hfilneg]
{Sikiru Adigun Sanni}

\address{ Sikiru Adigun Sanni \newline
Department of Mathematics,
Statistics  and Computer Science\\
University of Uyo,
Uyo, Akwa Ibom State, Nigeria}
\email{sikirusanni@yahoo.com}

\thanks{Submitted November 27, 2009. Published May 21, 2010.}
\subjclass[2000]{35B40, 35K57, 80A25}
\keywords{Irreversible reaction; reactant diffusivity; thermal
conductivity; \hfill\break\indent
a priori estimates; Banach's fixed point theorem}

\begin{abstract}
 A system of non-linear partial differential equations describing
 one-step irreversible reaction, reactant to product,
 in a non-stationary medium and non-smooth domain is considered.
 After obtaining the necessary a priori estimates, the existence
 of a unique local strong solution to the system is proved using
 a fixed point theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We consider the  semilinear parabolic system of partial
differential equations
\begin{gather}
\nabla . \bar{v}= 0\quad \text{in }\Omega_T \label{eqn1} \\
\frac{\partial \bar{v}}{\partial t}-\nu \Delta\bar{v}
=-\nabla.(\bar{v}\otimes\bar{v})-\frac{1}{\rho}\nabla p\quad\text{in }\Omega_T
 \label{eqn2}\\
\frac{\partial u}{\partial t}-k \Delta u
= -\nabla.(\bar{v} u)+Qwf(u)\quad\text{in }\Omega_T \label{eqn3}\\
\frac{\partial w}{\partial t}-d\Delta w
=-\nabla.(\bar{v} w)-wf(u)\quad\text{in }\Omega_T \label{eqn4}\\
\bar{v}=\bar{0},\ u=w=0\quad\text{on }
\partial\Omega\times [0,T)\label{eqn5}\\
\bar{v}(x,0)=\bar{v}_0(x),\quad u(x,0)=u_0(x),\quad
 w(x,0)=w_0(x)\label{eqn6}
\end{gather}
where $\bar{0}$ is the zero vector in $\mathbb{R}^3$,
$\otimes$ is the matrix multiplication defined by the tensor
$\bar{v}\otimes\bar{v}:=v_iv_j$ ($i,j=1,2,3$) and
 $\Omega_T:=\Omega\times[0,T)$. Notice then that
$\nabla.(\bar{v}\otimes\bar{v})
=\frac{\partial}{\partial x_i}(\bar{v}_i\bar{v}_j)
=\frac{\partial}{\partial x_j}(\bar{v}_i\bar{v}_j) =v_i\frac{\partial
v_j}{\partial x_i}=\bar{v}.\nabla\bar{v}$ (using \eqref{eqn1}).

In applications, the system models a single-step irreversible reaction,
reactant $\to$ product in non-stationary incompressible medium.
$\bar{v}(x,t)$ is the velocity of the medium; $\nu$ and $\rho$ are the
kinematic viscosity and the density of the medium respectively.
$u(x,t)$ is the temperature in the reaction vessel, $w(x,t)$
is the mass fraction of the reactant, $1-w(x,t)$ is the mass
fraction of the product, $k$ the positive thermal conductivity
and $d$ the reactant diffusivity. $Qwf(u)$ and $-wf(u)$ are the
reaction kinetics, determined by a positive, uniformly bounded
and differentiable function $f(u)$.
Furthermore, $f'(u)$ is assumed to be Lipschitz continuous.
It is assumed that $\Omega$ is an open and bounded arbitrary
non-smooth domain in $\mathbb{R}^3$. Theoretically, the reactant
decomposes at a rate which is proportional to $w(x,t)f(u)$,
where $f(u)$ is the approximate number of molecules that have
sufficient energy for the reaction to begin. In this paper,
we shall assume that
\begin{gather}
0\leq f(u)\leq B \label{eqn7}\\
|f'(u)|\leq B',\quad
|f'(u)-f'(\tilde{u})|\leq L|u-\tilde{u}|\label{eqn8}
\end{gather}

For further information on chemical kinetics and combustion,
the reader is referred to Buckmaster\cite{Buckmaster}, Buckmaster
and Ludford \cite{BL}, and Frank-Kamenetskii \cite{FK}.

Several combustion models assumed some smoothness on the boundary
vis-a-vis stationary media. Authors of these models include Avrin
\cite{Avrin1, Avrin2}, Daddiouaissa \cite{Daddiouaissa}, De
Oliviera et al \cite{DPP}, Fitzgibbon and Martin \cite{FM}, Henry
\cite{Henry}, Konach \cite{Kouach}, Sanni \cite{Sanni}, Sattinger
\cite{Sattinger}, and some literature cited in them.

In this paper, we  establish the existence of a unique
local-in-time strong solution to the system
\eqref{eqn1}-\eqref{eqn6}, in arbitrary non-smooth domains.
Clearly, the inclusion of the Navier-Stokes equations in the
system implies that the medium is non-stationary.

Using Leray projector \cite{Temam}, the problem
\eqref{eqn1}-\eqref{eqn6} can be reduced to that of finding only
$(\bar{v},u,w)$ by a variational formulation. We are thus
motivated to define:

\begin{definition} \rm
We call a solution $(\bar{v},u,w)$ of the system
\eqref{eqn1}-\eqref{eqn6} a strong solution, provided
$(\bar{v},u,w)\in X^3$, where $X$ is defined by
\begin{equation}
X:={L^\infty[0 ,T; H^1_0(\Omega)]}\cap{H^1[0 ,T; H^1_0(\Omega)]}\cap
W^{1,\infty}[0,T;L^2(\Omega)]\label{eqn8.1}
\end{equation}
\end{definition}

\section{A priori estimates}

We will need the following Sobolev embedding theorem,
stated and proved in \cite[pp. 265-266]{Evans}.

\begin{theorem}\label{thm1}
Assume that $\Omega\subset \mathbb{R}^n$ is  open and bounded.
Suppose $U\in W^{1,p}_0(\Omega)$ for some $1\leq p < n$.
Then we have the estimate
\begin{equation}
\|U\|_{L^q(\Omega)}\leq C\|\nabla U\|_{L^p(\Omega)}\label{eqn8.2}
\end{equation}
for each $q\in [1,p*]$, the constant $C$ depending only on $p,q,n$
and $\Omega$, where $p*:=\frac{np}{n-p}$ is the Sobolev conjugate.
\end{theorem}

Notice that the hypothesis of Theorem \ref{thm1} requires no
smoothness assumption on the boundary $\Omega$.

We now set out to obtain a priori estimates required to prove the
existence of a unique local strong solution to the system
\eqref{eqn1}-\eqref{eqn6}. We first state and prove the following
Lemmas.

\begin{lemma}\label{lem1}
Let $u\in H^1(\Omega)$ and $v, w, p \in {H^1_0(\Omega)}$. Then
\begin{gather}
\int_\Omega uwp\,dx \leq \epsilon
\|u\|_{{L^2(\Omega)}}^2+C(\Omega)\epsilon^{-1}
\|w\|_{{H^1_0(\Omega)}}^2\|p\|_{{H^1_0(\Omega)}}^2\label{eqn9}
\\
\int_\Omega uwp\,dx \leq \epsilon\left(\|u\|_{{L^2(\Omega)}}^2
+\|p\|_{{H^1_0(\Omega)}}^2\right)
+C(\Omega)\epsilon^{-3}\|w\|_{{H^1_0(\Omega)}}^4\|p\|_{{L^2(\Omega)}}^2\label{eqn10}
\\
\int_\Omega uwp\,dx \leq \epsilon\|u\|_{{L^2(\Omega)}}^2
 +C(\Omega)\epsilon^{-1}\|w\|_{{H^1_0(\Omega)}}^2
 \Big(T^{-1/2}\|p\|_{L^2(\Omega)}^2+T^{1/2}\|p\|_{H^1_0(\Omega)}^2\Big)
 \label{eqn11}\\
\begin{aligned}
&\int_\Omega u\left(pw-\tilde{p} \tilde{w}\right)dx\\
&\leq \epsilon\|u\|_{{L^2(\Omega)}}^2
+ C(\Omega)\epsilon^{-1}\Big[\|p\|_{{H^1_0(\Omega)}}^2
\|w-\tilde{w}\|_{{H^1_0(\Omega)}}^2
+\|p-\tilde{p}\|_{{H^1_0(\Omega)}}^2\|\tilde{w}\|_{{H^1_0(\Omega)}}^2\Big]
\end{aligned} \label{eqn12}
\\
\begin{aligned}
&\int_\Omega u\left(pw-\tilde{p} \tilde{w}\right)dx\\
&\leq \epsilon\|u\|_{{L^2(\Omega)}}^2 + C(\Omega)\epsilon^{-1}
 \Big[\|p\|_{{H^1_0(\Omega)}}^2\Big(T^{-1/2}\|w-\tilde{w}\|_{{L^2(\Omega)}}^2\\
&\quad +T^{1/2}\|w-\tilde{w}\|_{{H^1_0(\Omega)}}^2\Big)
 +\|p-\tilde{p}\|_{{H^1_0(\Omega)}}^2\Big(T^{-1/2}\|\tilde{w}\|_{{L^2(\Omega)}}^2
+T^{1/2}\|\tilde{w}\|_{{H^1_0(\Omega)}}^2\Big)\Big]
\end{aligned} \label{eqn13}
\\
\begin{aligned}
\int_\Omega uvwp\,dx&\leq \epsilon \|\nabla u\|_{L^2(\Omega)}^2
+ C(\Omega)\epsilon^{-1}\|v\|_{H^1_0(\Omega)}^4\|u\|_{{L^2(\Omega)}}^2 \\
&\quad+ C(\Omega) \Big(T^{-1/2}\|p\|_{L^2(\Omega)}^2+T^{1/2}
\|p\|_{{H^1_0(\Omega)}}^2\Big)
 \|w\|_{H^1_0(\Omega)}^2
\end{aligned} \label{eqn14}
\\
\begin{aligned}
&\int_\Omega u(vwp-\tilde{\bar{v}}\tilde{w}\tilde{p})dx \\
&\leq  C(\Omega)\epsilon^{-1}\Big(\|v\|_{H^1_0(\Omega)}^4
+\|\tilde{w}\|_{H^1_0(\Omega)}^4\Big)\|u\|_{{L^2(\Omega)}}^2 \\
&\quad+\epsilon \|\nabla u\|_{L^2(\Omega)}^2
 +C(\Omega)\Big[\|w\|_{H^1_0(\Omega)}^2\Big(T^{-1/2}
 \|p-\tilde{p}\|_{L^2(\Omega)}^2
 +T^{1/2}\|p-\tilde{p}\|_{{H^1_0(\Omega)}}^2\Big) \\
&\quad+\Big(T^{-1/2}\|\tilde{p}\|_{L^2(\Omega)}^2+T^{1/2}
\|\tilde{p}\|_{{H^1_0(\Omega)}}^2\Big)
\Big(\|w-\tilde{w}\|_{H^1_0(\Omega)}^2
+\|v-\tilde{\bar{v}}\|_{H^1_0(\Omega)}^2\Big)\Big]
\end{aligned}\label{eqn15}
\end{gather}
\end{lemma}

\begin{proof}
 1. Proof of \eqref{eqn9}. By H\"older's inequality,
\begin{equation}
\begin{aligned}
\int_\Omega uwp\,dx
&\leq \|u\|_{L^2(\Omega)}\|w\|_{L^4(\Omega)}\|p\|_{L^4(\Omega)} \\
&\leq C(\Omega)\|u\|_{L^2(\Omega)}\|w\|_{H^1_0(\Omega)}\|p\|_{H^1_0(\Omega)}
\end{aligned}  \label{eqn16}
\end{equation}
by Sobolev embedding theorem.
Then \eqref{eqn9} follows easily from \eqref{eqn16} by Cauchy's
inequality with $\epsilon$.

2. Proof of \eqref{eqn10} and \eqref{eqn11}.
\begin{equation}
\begin{aligned}
\int_\Omega uwp\,dx
&\leq  \epsilon\|u\|_{L^2(\Omega)}^2+ \frac{1}{4\epsilon}
 \int_\Omega p^2w^2dx\quad \text{(by Cauchy's inequality with $\epsilon$)} \\
&\leq  \epsilon\|u\|_{L^2(\Omega)}^2+\frac{1}{4\epsilon}
\|p\|_{L^2(\Omega)}\|p\|_{L^6(\Omega)}\|w\|_{L^6(\Omega)}^2\quad
\text{(by H\"older's inequality)} \\
&\leq  \epsilon\|u\|_{L^2(\Omega)}^2+C(\Omega)(4\epsilon)^{-1}\|p\|_{L^2(\Omega)}
\|p\|_{H^1_0(\Omega)}\|w\|_{H^1_0(\Omega)}^2,
\end{aligned}\label{eqn17}
\end{equation}
by Sobolev embedding theorem. Then \eqref{eqn10} and \eqref{eqn11}
follow by applying Cauchy's inequality with $\epsilon^2$ and
$T^{1/2}$, respectively, to the appropriate factors of the second term
on the right side of \eqref{eqn17}.

3. Proof of \eqref{eqn12} and \eqref{eqn13}.
\begin{equation}
\int_\Omega u(pw-\tilde{p}\tilde{w})dx=\int_\Omega up(w-\tilde{w})dx
+ \int_\Omega u\tilde{w}(p-\tilde{p})dx.\label{eqn18}
\end{equation}
Then \eqref{eqn12} and \eqref{eqn13} follows by applying \eqref{eqn9}
and \eqref{eqn11} to \eqref{eqn18} respectively.

4. Proof of \eqref{eqn14}.
By Young's inequality and then  by H\"older's inequality,
\begin{equation}
\begin{aligned}
\int_\Omega uvwp\,dx
&\leq  \frac{1}{2}\int_\Omega u^2v^2dx
+ \frac{1}{2}\int_\Omega w^2p^2dx \\
&\leq \frac{1}{2} \|u\|_{L^2(\Omega)}\|u\|_{L^6(\Omega)}\|v\|_{L^6(\Omega)}^2
 +\|w\|_{L^6(\Omega)}^2\|p\|_{L^2(\Omega)}\|p\|_{L^6(\Omega)} \\
&\leq  \|u\|_{L^2(\Omega)}\|u\|_{H^1_0(\Omega)}
\|v\|_{H^1_0(\Omega)}^2+\|w\|_{H^1_0(\Omega)}^2\|p\|_{L^2(\Omega)}
\|p\|_{H^1_0(\Omega)},
\end{aligned} \label{eqn19}
\end{equation}
by By Sobolev embedding theorem. \eqref{eqn14} follows by applying
Cauchy's inequalities with $\epsilon$ and $T^{1/2}$ to the first and
second terms on the right side of \eqref{eqn19} respectively.

5. Proof of \eqref{eqn15}.
\begin{equation}
\begin{aligned}
&\int_\Omega u(vwp-\tilde{\bar{v}}\tilde{w}\tilde{p})dx\\
&=\int_\Omega uvw(p-\tilde{p})dx + \int_\Omega uv\tilde{p}(w-\tilde{w})dx
+ \int_\Omega u\tilde{p}\tilde{w}(v-\tilde{\bar{v}})dx\,.
\end{aligned} \label{eqn20}
\end{equation}
Then \eqref{eqn15} follows by applying \eqref{eqn14} to the each term
on the right side of \eqref{eqn20}.
This concludes the proof of Lemma \ref{lem1}
\end{proof}

\begin{lemma}\label{lem2}
Let \eqref{eqn1}-\eqref{eqn4} hold. Suppose $\bar{v}_0,u_0,w_0\in
{H^1_0(\Omega)}\cap{H^2(\Omega)}$, then
\begin{equation}
\begin{aligned}
&\|\partial_t\bar{v}_0\|_{L^2(\Omega)}^2 + \|\partial_tu_0\|_{L^2(\Omega)}^2 
+ \|\partial_tw_0\|_{L^2(\Omega)}^2 \\
&\leq C(\|\nabla \bar{v}_0\|_H^1(\Omega)^2
 + \|\nabla u_0\|_H^1(\Omega)^2+\|\nabla w_0\|_H^1(\Omega)^2)
 (1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2),
\end{aligned}\label{eqn21}
\end{equation}
where $C=C(\nu,k,d,B,\rho,\Omega,Q)$
\end{lemma}

\begin{proof}
Taking \eqref{eqn1} and \eqref{eqn3} on $t=0$ and multiplying
the corresponding equation to \eqref{eqn3} by $\partial_tu_0$, we estimate
\begin{equation}
\begin{aligned}
&\int_\Omega |\partial_tu_0|^2dx\\
&=-\int_\Omega\partial_tu_0.\bar{v}_0.\nabla u_0dx+k\int_\Omega 
 \partial_tu_0\Delta u_0dx
+Q\int_\Omega\partial_tu_0 w_0f(u_0)dx \\
&\leq 2\epsilon\int_\Omega|\partial_tu_0|^2dx+\frac{1}{4\epsilon}\Big(QB\int_0|w_0|^2dx
 +k\int_\Omega|\Delta u_0|^2dx\Big) \\
&\quad\text{(Integrating by parts, using Cauchy's inequality with
$\epsilon$ and \eqref{eqn7})} \\
&\leq2\epsilon\int_\Omega|\partial_t\bar{v}_0|^2dx
 +\frac{C(Q,B,k,\Omega)}{\epsilon}\\
&\quad\times \Big(\|\bar{v}_0\|_{H^1_0(\Omega)}^2\|\nabla u_0\|_H^1(\Omega)^2
+\|w_0\|_{H^1_0(\Omega)}^2+
 \|\nabla^2 u_0\|_{L^2(\Omega)}^2\Big),
\end{aligned} \label{eqn22}
\end{equation}
by H\"older and Poincare's inequalities and using that
$\|\Delta \bar{v}_0\|_{L^2(\Omega)}\leq \|\nabla^2\bar{v}_0\|_{L^2(\Omega)}$. 
Choosing
$\epsilon>0$ sufficiently small and simplifying, we deduce
\begin{equation}
\|\partial_tu_0\|_{L^2(\Omega)}^2\leq C(Q,B,k,\Omega)
\left[(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)\|\nabla  u_0\|_H^1(\Omega)^2 
+ \|w_0\|_{H^1_0(\Omega)}^2\right]
\label{eqn23}
\end{equation}
Evaluating \eqref{eqn1}, \eqref{eqn2} and \eqref{eqn4} at $t=0$, we
 obtain analogous estimates to \eqref{eqn23}, viz:
\begin{gather}
\|\partial_t\bar{v}_0\|_{L^2(\Omega)}^2 \leq  C(\nu,\Omega)
(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)
 \|\nabla  \bar{v}_0\|_H^1(\Omega)^2\label{eqn24}\\
\|\partial_tw_0\|_{L^2(\Omega)}^2 \leq  C(d,B,\Omega)
\left[(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)\|
\nabla  w_0\|_H^1(\Omega)^2 + \|w_0\|_{H^1_0(\Omega)}^2\right]\label{eqn25}
\end{gather}
Combining \eqref{eqn23}, \eqref{eqn24} and \eqref{eqn25}, we
deduce \eqref{eqn21}.
\end{proof}

\begin{theorem}\label{thm2}
Let $\bar{v}_0, u_0, w_0 \in {H^1_0(\Omega)}\cap{H^2(\Omega)}$. 
Suppose $(\bar{v}, u, w)$ is a
strong solution of the system \eqref{eqn1}-\eqref{eqn6}. Then
we have the estimate
\begin{equation}
\begin{aligned}
&\sup_{[0,T]}\Big(\|\partial_t\bar{v}\|_{L^2(\Omega)}^2
+\|\bar{v}\|_{H^1_0(\Omega)}^2+\|\partial_tu\|_{L^2(\Omega)}^2
+\|u\|_{H^1_0(\Omega)}^2+\|\partial_tw\|_{L^2(\Omega)}^2
+ \|w\|_{H^1_0(\Omega)}^2\Big)\\
&+ \|\nabla \left(\partial_t\bar{v}\right)\|_{L^2[0 ,T; L^2(\Omega)]}^2
+\|\nabla (\partial_tu)\|_{L^2[0 ,T; L^2(\Omega)]}^2
+ \|\nabla \left(\partial_tw\right)\|_{L^2[0 ,T; L^2(\Omega)]}^2 \\
&\leq \frac{CT[(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)(G(\bar{v}_0,u_0, w_0)+1)]^3}
{\{1-2CT[(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)(G(\bar{v}_0,u_0, w_0)+1)]^2\}^{3/2}} \\
&=:\Sigma=\text{constant},
\end{aligned}\label{eqn26}
\end{equation}
for
\begin{equation}
T<\{2C[(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)(\|\nabla\bar{v}_0\|_H^1(\Omega)^2
+\|\nabla u_0\|_H^1(\Omega)^2+\|\nabla w_0\|_H^1(\Omega)^2)+ 1]^2\}^{-1},\label{eqn27}
\end{equation}
where
$$
G(\bar{v}_0,u_0, w_0)=\|\nabla\bar{v}_0\|_H^1(\Omega)^2+\|\nabla
u_0\|_H^1(\Omega)^2+\|\nabla w_0\|_H^1(\Omega)^2
$$
 and $C=C(\nu,k,d,Q,B,B',\Omega)$.
\end{theorem}

We will use \eqref{eqn3} and the corresponding conditions in
\eqref{eqn5} and \eqref{eqn6} to obtain estimates for $u$; and
thereafter, for brevity, state analogous estimates for $\bar{v}$ and
$w$.

\begin{proof}
1. Multiplying \eqref{eqn3} by $\partial_tu$, integrating the ensuing
equation by parts over $\Omega$ and using \eqref{eqn1} and
\eqref{eqn5}, we deduce
\begin{equation}
\begin{aligned}
&\int_\Omega|\partial_tu|^2dx + \frac{k}{2}\frac{d}{dt}
\left(\int_\Omega|\nabla u|^2dx\right)\\
&=\int_\Omega\nabla(\partial_tu).(\bar{v} u)dx+Q\int_\Omega\partial_tu wf(u)dx \\
&\leq \epsilon\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2
+C(\Omega)\Big(\frac{1}{\epsilon}\|\bar{v}\|_H^1(\Omega)^2\|u\|_H^1
(\Omega)^2+Q^2B^2\|w\|_{H^1_0(\Omega)}^2\Big)\\
&\quad + \|\partial_tu\|_{L^2(\Omega)}^2,
\end{aligned} \label{eqn28}
\end{equation}
using \eqref{eqn9}) of Lemma \ref{lem1}.
Simplifying, \eqref{eqn28} yields
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\big(\frac{k}{2}\|u\|_{H^1_0(\Omega)}^2\big) \\
&\leq\epsilon\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2+C(\Omega)
\Big(\frac{1}{\epsilon}\|\bar{v}\|_H^1(\Omega)^2
\|u\|_H^1(\Omega)^2+Q^2B^2\|w\|_{H^1_0(\Omega)}^2\Big)
\end{aligned}\label{eqn29}
\end{equation}

2. Differentiating \eqref{eqn3} with respect to $t$ yields
\begin{equation}
\frac{\partial}{\partial t}(\partial_tu)-k\Delta(\partial_tu)
=-\partial_t\bar{v}.\nabla u+\bar{v}.\nabla(\partial_tu)+Q\partial_tw.f(u)
+Qw\partial_tu f'(u)\label{eqn30}
\end{equation}
Multiply  by $\partial_tu$ and integrating by
parts over $\Omega$ and use \eqref{eqn1},  \eqref{eqn5} to deduce:
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(\|\partial_tu\|_{L^2(\Omega)}^2)
+k\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2\\
&=\int_\Omega\nabla(\partial_tu).\partial_t\bar{v}.udx
+ \int_\Omega\nabla(\partial_tu).\bar{v}.\partial_tu dx\\
&\quad +Q\int_\Omega\partial_tu\partial_tu wf'(u)dx
+Q\int_\Omega\partial_tu\partial_tw f(u)dx \\
&\leq \epsilon\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2+\epsilon
 \|\nabla(\partial_t\bar{v})\|_{L^2(\Omega)}^2
 +C(\Omega)\epsilon^{-3}\|\partial_t\bar{v}\|_{L^2(\Omega)}^2\|u\|_{H^1_0(\Omega)}^4\\
&\quad \epsilon\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2+\epsilon\|
  \nabla(\partial_tu)\|_{L^2(\Omega)}^2
  +C(\Omega)\epsilon^{-3}\|\partial_tu\|_{L^2(\Omega)}^2\|v\|_{H^1_0(\Omega)}^4\\
&\quad + QB'[\epsilon\|\partial_tu\|_{L^2(\Omega)}^2
+\epsilon\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2
+C(\Omega)\epsilon^{-3}\|\partial_tu\|_{L^2(\Omega)}^2\|w\|_{H^1_0(\Omega)}^4]\\
&\quad + QB(\|\partial_tu\|_{L^2(\Omega)}^2+\|\partial_tw\|_{L^2(\Omega)}^2),
\end{aligned} \label{eqn31}
\end{equation}
using \eqref{eqn10} of Lemma \ref{lem1} and Young's
inequality.

3. Combining \eqref{eqn29} and \eqref{eqn31} we deduce
\begin{equation}
\begin{aligned}
&\frac{d}{dt}(\|\partial_tu\|_{L^2(\Omega)}^2
+k\|u\|_{H^1_0(\Omega)}^2)+2k\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2\\
&\leq C\Big[\epsilon\|\nabla (\partial_tu)\|_{L^2(\Omega)}^2
+\epsilon\|\nabla(\partial_t\bar{v})\|_{L^2(\Omega)}^2 
+\epsilon\|\partial_tu\|_{L^2(\Omega)}^2\\
&\quad +\big(1+\epsilon^{-1} +\epsilon^{-3}\big)
\Big(1+\|\partial_t\bar{v}\|_{L^2(\Omega)}^2\|\bar{v}\|_{H^1_0(\Omega)}^2
 +\|\partial_tu\|_{L^2(\Omega)}^2\\
&\quad  +\| u\|_{H^1_0(\Omega)}^2+\|\partial_tw\|_{L^2(\Omega)}^2
+\| w\|_{H^1_0(\Omega)}^2\Big)^3\Big]
\end{aligned} \label{eqn32}
\end{equation}
where $C=C(Q,B,B',\Omega)$.

4. Following steps 1-3 in respect of \eqref{eqn1}, \eqref{eqn2},
\eqref{eqn4} and the corresponding conditions in \eqref{eqn5}, we
obtain analogous estimates to \eqref{eqn32}:
\begin{gather}
\begin{aligned}
&\frac{d}{dt}(\|\partial_t\bar{v}\|_{L^2(\Omega)}^2
+\nu\|\bar{v}\|_{H^1_0(\Omega)}^2)+2\nu\|\nabla(\partial_t\bar{v})\|_{L^2(\Omega)}^2\\
&\leq C(\Omega)\Big[\epsilon\|\nabla (\partial_t\bar{v})\|_{L^2(\Omega)}^2
+(\epsilon^{-1}+\epsilon^{-3})\Big(\|\partial_t\bar{v}\|_{L^2(\Omega)}^2
 +\| \bar{v}\|_{H^1_0(\Omega)}^2\big)\Big]\,,
\end{aligned} \label{eqn33}
\\
\begin{aligned}
&\frac{d}{dt}(\|\partial_tw\|_{L^2(\Omega)}^2+d\|w\|_{H^1_0(\Omega)}^2)
 +2d\|\nabla(\partial_tw)\|_{L^2(\Omega)}^2\\
&\leq C\Big[\epsilon\|\nabla (\partial_tw)\|_{L^2(\Omega)}^2
+\epsilon\|\nabla(\partial_t\bar{v})\|_{L^2(\Omega)}^2
+\epsilon\|\partial_tu\|_{L^2(\Omega)}^2\\
&\quad +(1+\epsilon^{-1}+\epsilon^{-3})
 \Big(1+\|\partial_t\bar{v}\|_{L^2(\Omega)}^2
 + \|\bar{v}\|_{H^1_0(\Omega)}^2+\|\partial_tu\|_{L^2(\Omega)}^2\\
&\quad +\| u\|_{H^1_0(\Omega)}^2+\|\partial_tw\|_{L^2(\Omega)}^2
+\| w\|_{H^1_0(\Omega)}^2\Big)^3\Big],
\end{aligned}\label{eqn34}
\end{gather}
where $C=C(B,B',\Omega)$.


5. Combining \eqref{eqn32}-\eqref{eqn34}, choosing $\epsilon>0$
sufficiently small and simplifying, we deduce
\begin{align*}
&\frac{d}{dt}\Big(\|\partial_t\bar{v}\|_{L^2(\Omega)}^2
+\|\bar{v}\|_{H^1_0(\Omega)}^2+\|\partial_tu\|_{L^2(\Omega)}^2+\|u\|_{H^1_0(\Omega)}^2
 +\|\partial_tw\|_{L^2(\Omega)}^2\\
&+ \|w\|_{H^1_0(\Omega)}^2\Big)+\|\nabla(\partial_t\bar{v})\|_{L^2(\Omega)}^2
+\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2
 +\|\nabla(\partial_tw)\|_{L^2(\Omega)}^2 \\
&\leq C(\nu,k,d,Q,B,B',\Omega)
 \Big(1+\|\partial_t\bar{v}\|_{L^2(\Omega)}^2+\|\bar{v}\|_{H^1_0(\Omega)}^2
 +\|\partial_tu\|_{L^2(\Omega)}^2\\
&\quad + \| u\|_{H^1_0(\Omega)}^2+\|\partial_tw\|_{L^2(\Omega)}^2
+\| w\|_{H^1_0(\Omega)}^2\Big)^3
\end{align*} %\label{eqn35}
Solving the above equation, maximizing the left and right sides of the
ensuing inequalities and using Lemma \ref{lem2} concludes the
proof of the Theorem \ref{thm1}.
\end{proof}

\section{Existence of a Solution}

We  prove the existence of a unique local strong
solution to the system \eqref{eqn1}-\eqref{eqn6}, in a subset
$K$ of the space $X^3$ equipped with the norm
\begin{equation}
\begin{aligned}
&\|(\eta,\xi,\zeta)\|_{X^3}\\
&\leq \Big[\|\eta\|_{L^\infty[0 ,T; H^1_0(\Omega)]}^2
+\|\partial_t\eta\|_{L^\infty[0 ,T; L^2(\Omega)]}^2
+\|\xi\|_{L^\infty[0 ,T; H^1_0(\Omega)]}^2 \\
&\quad +\|\partial_t\xi\|_{L^\infty[0 ,T; L^2(\Omega)]}^2
+\|\zeta\|_{L^\infty[0 ,T; H^1_0(\Omega)]}^2
+\|\partial_t\zeta\|_{L^\infty[0 ,T; L^2(\Omega)]}^2\\
&\quad +\|\nabla(\partial_t\eta)\|_{L^2[0 ,T; L^2(\Omega)]}^2
+\|\nabla(\partial_t\xi)\|_{L^2[0 ,T; L^2(\Omega)]}^2
 +\|\nabla(\partial_t\zeta)\|_{L^2[0 ,T; L^2(\Omega)]}^2\Big]^{\frac{1}{2}},
\end{aligned}\label{eqn36}
\end{equation}
where $X$ is defined by \eqref{eqn8.1}.

\begin{theorem}\label{thm3}
Let $\bar{v}_0, u_0$ and $w_0 \in {H^1_0(\Omega)}\cap{H^2(\Omega)}$. 
Then there exists a
unique local strong solution to the system
\eqref{eqn1}-\eqref{eqn6}.
\end{theorem}

\begin{proof}
1. The fixed point arguments for the system \eqref{eqn1}-\eqref{eqn6}
are
\begin{gather}
\nabla . \bar{Q}=0\quad\text{in }\Omega_T \label{eqn37}\\
\frac{\partial \bar{Q}}{\partial t}-\nu \Delta\bar{Q}
 =-\nabla.(\bar{v}\otimes\bar{v})-\frac{1}{\rho}\nabla Y\quad\text{in }\Omega_T \label{eqn38}\\
\frac{\partial R}{\partial t}-k \Delta R
 =-\nabla.(\bar{v} u)+Qwf(u)\quad\text{in }\Omega_T\label{eqn39}\\
\frac{\partial S}{\partial t}-d\Delta S
 =-\nabla.(\bar{v} w)-wf(u)\quad\text{in }\Omega_T\label{eqn40}\\
\bar{Q}=\bar{0},\quad R=S=0\quad\text{on }\partial\Omega\times [0,T)
\label{eqn41}\\
\bar{Q}(x,0)=\bar{v_0}(x),\quad R(x,0)=u_0(x),\quad
 S(x,0)=w_0(x),\label{eqn42}
\end{gather}
where $Y$ is the pressure distribution corresponding to the solution
$(\bar{Q},R,S)$.

2. We next define a mapping
\begin{equation}
\tau: X^3\to X^3\label{eqn43}
\end{equation}
by setting $\tau[(\bar{v},u,w)]=(\bar{Q},R,S)$, whenever $(\bar{Q},R,S)$ is
derived from $(\bar{v},u,w)$ via \eqref{eqn37}-\eqref{eqn42}. We will
prove that for sufficiently small $T>0$, $\tau$ is a
contraction mapping. Choose 
$(\bar{v},u,w), (\tilde{\bar{v}},\tilde{u},\tilde{w})\in X^3$ and
define
$$
\tau[(\bar{v},u,w)]=(Q,R,S),\quad
\tau[(\tilde{\bar{v}},\tilde{u},\tilde{w})]=(\tilde{\bar{Q}},\tilde{R},\tilde{S}).
$$
 Thus, for two solutions
$(Q,R,S)$, and $(\tilde{\bar{Q}},\tilde{R},\tilde{S})$ of the system
\eqref{eqn37}-\eqref{eqn42}, we have
\begin{gather}
\nabla . (\bar{Q}-\tilde{\bar{Q}})=0\quad\text{in }\Omega_T \label{eqn44}\\
\frac{\partial }{\partial t}(\bar{Q}-\tilde{\bar{Q}})
-\nu \Delta(\bar{Q}-\tilde{\bar{Q}})
 =-\nabla.(\bar{v}\otimes\bar{v}-\tilde{\bar{v}}\otimes\tilde{\bar{v}})-\frac{1}{\rho}
 \nabla(Y-\tilde{Y})\quad\text{in }\Omega_T \label{eqn45}\\
\frac{\partial}{\partial t}(R-\tilde{R})-k \Delta (R-\tilde{R})
 =-\nabla.(\bar{v} u-\tilde{\bar{v}} \tilde{u})+Q(wf(u)
 -\tilde{w} f(\tilde{u}))\quad\text{in }\Omega_T\label{eqn46}\\
\frac{\partial}{\partial t}(S-\tilde{S})-d\Delta (S-\tilde{S})
 =-\nabla.(\bar{v} w-\tilde{\bar{v}}\tilde{w})-(wf(u)
 -\tilde{w} f(\tilde{u}))\quad\text{in }\Omega_T\label{eqn47}\\
\bar{Q}-\tilde{\bar{Q}}  =\bar{0},\quad
R-\tilde{R}=S-\tilde{S} =0\quad\text{on }\partial\Omega\times [0,T)\label{eqn48}\\
(\bar{Q}-\tilde{\bar{Q}})(x,0)= \bar{0},\quad (R-\tilde{R})(x,0)=0,\quad
 (S-\tilde{S})(x,0)=0\label{eqn49}
\end{gather}

3. Multiplying \eqref{eqn46} by $\partial_t(R-\tilde{R})$, integrating the
ensuing equation by parts over $\Omega$, using \eqref{eqn48} and
applying \eqref{eqn12} of Lemma \ref{lem1}, we deduce
\begin{equation}
\begin{aligned}
&\|\partial_t(R-\tilde{R})\|_{L^2(\Omega)}^2
+\frac{k}{2}\frac{d}{dt}\left(\|\nabla(\partial_t(R-\tilde{R}))\|_{L^2(\Omega)}^2\right)\\
&= \int_\Omega\nabla\left(\partial_t(R-\tilde{R})\right).(\bar{v} u
-\tilde{\bar{v}}\tilde{u})dx+Q\int_\Omega\partial_t(R-\tilde{R})(wf(u)
 -\tilde{w} f(\tilde{u}))dx \\
&\leq \epsilon\|\nabla(\partial_t(R-\tilde{R}))\|_{L^2(\Omega)}^2+\epsilon\|
 \partial_t(R-\tilde{R})\|_{L^2(\Omega)}^2\\
&\quad + C(\Omega)\epsilon^{-1} \Big(\|\bar{v}\|_{H^1_0(\Omega)}^2
\|u-\tilde{u}\|_{H^1_0(\Omega)}^2
 +\|\tilde{u}\|_{H^1_0(\Omega)}^2\|v-\tilde{\bar{v}}\|_{H^1_0(\Omega)}^2\Big)\\
&\quad + C(Q,B,B',\Omega)\epsilon^{-1}\Big(\|w\|_{H^1_0(\Omega)}^2
\|u-\tilde{u}\|_{H^1_0(\Omega)}^2
 +\|w-\tilde{w}\|_{H^1_0(\Omega)}^2\Big),
\end{aligned} \label{eqn50}
\end{equation}
where we have used some bounds in \eqref{eqn7} and \eqref{eqn8}.

4. Further, we differentiate \eqref{eqn46} with respect $t$ to get
\begin{equation}
\begin{aligned}
&\frac{\partial}{\partial t}(\partial_t(R-\tilde{R}))
-k\Delta(\partial_t(R-\tilde{R}))\\
&=-\nabla.\left(\partial_t\bar{v} u-\partial_t\tilde{\bar{v}}\tilde{u}
+\bar{v}\partial_tu-\tilde{\bar{v}}\partial_t\tilde{u}\right)
Q\big(\partial_tw f(u)\\
&\quad -\partial_t\tilde{w} f(\tilde{u})+w\partial_tu f'(u)
-\tilde{w}\partial_t\tilde{u} f'(\tilde{u})\big)
\end{aligned} \label{eqn51}
\end{equation}
Multiplying \eqref{eqn51} by $\partial_t(R-\tilde{R})$, integrating by parts
over $\Omega$, and applying Young's inequality with $\epsilon$,
\eqref{eqn13} and \eqref{eqn15} as appropriate, we deduce
%\begin{equation}
\begin{align*}
&\frac{1}{2}\frac{d}{dt}(\|\partial_t(R-\tilde{R})\|_{L^2(\Omega)}^2)
+k\|\nabla(\partial_t(R-\tilde{R}))\|_{L^2(\Omega)}^2\\
&=\int_\Omega(\partial_t\bar{v} u-\partial_t\tilde{\bar{v}}\tilde{u}
+ \bar{v}\partial_tu-\tilde{\bar{v}}\partial_t\tilde{u}).
\nabla(\partial_t(R-\tilde{R}))dx\\
&\quad +Q\int_\Omega\partial_t(R-\tilde{R}).(\partial_tw f(u)-\partial_t\tilde{w} 
f(\tilde{u})+ w\partial_tu f'(u)-\tilde{w}\partial_t\tilde{u} f'(\tilde{u}))dx \\
&\leq 3\epsilon\|\nabla(\partial_t(R-\tilde{R}))\|_{L^2(\Omega)}^2+\Big(2\epsilon
 +C(\Omega)\epsilon^{-1}\|w\|_{H^1_0(\Omega)}^4\Big)
 \|\partial_t(R-\tilde{R})\|_{L^2(\Omega)}^2\\
&\quad + C(\Omega,B,B',Q,L)\epsilon^{-1}
 \Big\{\Big[T^{-1/2}\Big(\|\partial_t\tilde{\bar{v}}\|_{L^2(\Omega)}^2
 +\|\partial_t\tilde{u}\|_{L^2(\Omega)}^2+\|\partial_tw\|_{L^2(\Omega)}^2\\
&\quad+ \|\partial_tu\|_{L^2(\Omega)}^2\Big)+T^{1/2}
\Big(\|\nabla(\partial_t\tilde{\bar{v}})\|_{L^2(\Omega)}^2
 +\|\nabla(\partial_t\tilde{u})\|_{L^2(\Omega)}^2
 +\|\nabla(\partial_tw)\|_{L^2(\Omega)}^2\\
&\quad +\|\nabla(\partial_tu)\|_{L^2(\Omega)}^2\Big)\Big]
 \Big(\|u-\tilde{u}\|_{H^1_0(\Omega)}^2
 +\|\bar{v}-\tilde{\bar{v}}\|_{H^1_0(\Omega)}^2
 +\|w-\tilde{w}\|_{H^1_0(\Omega)}^2\Big)\\
&\quad + \Big[T^{-1/2}\Big(\|\partial_t(\bar{v}-\tilde{\bar{v}})
\|_{L^2(\Omega)}^2
 +\|\partial_t(u-\tilde{u})\|_{L^2(\Omega)}^2
 +\|\partial_t(w-\tilde{w})\|_{L^2(\Omega)}^2\Big)\\
&\quad + T^{1/2}\Big(\|\nabla(\partial_t(\bar{v}
-\tilde{\bar{v}}))\|_{L^2(\Omega)}^2
 +\|\nabla(\partial_t(u-\tilde{u}))\|_{L^2(\Omega)}^2\\
&\quad +\|\nabla(\partial_t(w-\tilde{w}))\|_{L^2(\Omega)}^2\Big)\Big]
\Big(1+\|u\|_{H^1_0(\Omega)}^2+\|\bar{v}\|_{H^1_0(\Omega)}^2
+\|\tilde{w}\|_{H^1_0(\Omega)}^2\Big)\Big\}.
\end{align*}% \label{eqn52}

5. Combining the above inequality with \eqref{eqn50}, Choosing $\epsilon
> 0$ sufficiently small, and simplifying, we deduce
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\|\partial_t(R-\tilde{R})\|_{L^2(\Omega)}^2
+\|R-\tilde{R}\|_{H^1_0(\Omega)}^2\Big)
 +\|\nabla(\partial_t(R-\tilde{R}))\|_{L^2(\Omega)}^2 \\
&\leq C\Big\{(1+\|w\|_{H^1_0(\Omega)}^2)^2
\Big(\|\partial_t(R-\tilde{R})\|_{L^2(\Omega)}^2
+\|R-\tilde{R}\|_{H^1_0(\Omega)}^2\Big)+\Big[1+\|\bar{v}\|_{H^1_0(\Omega)}^2 \\
&\quad+\|\tilde{u}\|_{H^1_0(\Omega)}^2+\|w\|_{H^1_0(\Omega)}^2+T^{-1/2}
\Big(\|\partial_t\tilde{\bar{v}}\|_{L^2(\Omega)}^2
 +\|\partial_t\tilde{u}\|_{L^2(\Omega)}^2+\|\partial_tw\|_{L^2(\Omega)}^2\\
&\quad + \|\partial_tu\|_{L^2(\Omega)}^2\Big)+T^{1/2}
\Big(\|\nabla(\partial_t\tilde{\bar{v}})\|_{L^2(\Omega)}^2
 +\|\nabla(\partial_t\tilde{u})\|_{L^2(\Omega)}^2
 +\|\nabla(\partial_tw))\|_{L^2(\Omega)}^2\\
&\quad + \|\nabla(\partial_tu)\|_{L^2(\Omega)}^2\Big)\Big]
\Big(\|u-\tilde{u}\|_{H^1_0(\Omega)}^2
+\|\bar{v}-\tilde{\bar{v}}\|_{H^1_0(\Omega)}^2+\|w-\tilde{w}\|_{H^1_0(\Omega)}^2\Big)\\
&\quad + \Big[T^{-1/2}\Big(\|\partial_t(\bar{v}-\tilde{\bar{v}})\|_{L^2(\Omega)}^2
 +\|\partial_t(u-\tilde{u})\|_{L^2(\Omega)}^2
 +\|\partial_t(w-\tilde{w})\|_{L^2(\Omega)}^2\Big)\\
&\quad + T^{1/2}\Big(\|\nabla(\partial_t(\bar{v}-\tilde{\bar{v}}))\|_{L^2(\Omega)}^2
 +\|\nabla(\partial_t(u-\tilde{u}))\|_{L^2(\Omega)}^2\\
&\quad +\|\nabla(\partial_t(w-\tilde{w}))\|_{L^2(\Omega)}^2\Big)\Big]
\Big(1+\|u\|_{H^1_0(\Omega)}^2+\|\bar{v}\|_{H^1_0(\Omega)}^2
+\|\tilde{w}\|_{H^1_0(\Omega)}^2\Big)\Big\},
\end{aligned} \label{eqn53}
\end{equation}
where $C=C(k,\Omega,B,B',Q,L)$.

6. There exist analogous estimates to \eqref{eqn53} for $\bar{Q}-\tilde{\bar{Q}}$
and $S-\tilde{S}$, which for brevity, we do not render here. If we
combine these estimates with \eqref{eqn53}, we deduce, after an
application of the differential form of the Gronwall's inequality,
the estimates:
\begin{equation}
\begin{aligned}
&\|(\bar{Q},R,S)-(\tilde{\bar{Q}},\tilde{R},\tilde{S})\|_{X^3}\\
&=\|\tau[(\bar{v},u,w)]-\tau[(\tilde{\bar{v}},\tilde{u},\tilde{w})]\|_{X^3}  \\
&\leq C\left(T+T^{1/2}\right)^{1/2}\exp\Big[2^{-1}
 T(1+\|w\|_{H^1_0(\Omega)}^2)\Big]\\
&\quad\times \Big(1+\|(\bar{v},u,w)\|_{X^3}^2
+ \|(\tilde{\bar{v}},\tilde{u},\tilde{w})\|_{X^3}^2\Big)^{1/2}\|(\bar{v},u,w)
 -(\tilde{\bar{v}},\tilde{u},\tilde{w})\|_{X^3}
\end{aligned} \label{eqn54}
\end{equation}
where $C=C(Q,B,B',\Omega,k,d,\nu,L)$.

7. Notice that the bound in \eqref{eqn54} is not uniform. Thus we
need to prove the existence of a unique solution in a subset of
$X^3$. Define a convex set
\begin{equation}
K:=\{(\bar{v},u,w)|(\bar{v},u,w)-(\bar{v}_0,u_0,w_0)\in X_0^3\text{ and }
\|(\bar{v},u,w)\|_{X^3}\leq 2\sqrt{\Sigma}\},\label{eqn54.2}
\end{equation}
where $X_0^3$ is the set where the initial and the boundary values
are zero; and $\Sigma=$ constant is the bound in \eqref{eqn26}. We
will show that, if $T>0$ is sufficiently small, then
\begin{equation}
\tau[K]\subseteq K,\quad
 \|\tau[(\bar{v},u,w)]-\tau[(\tilde{\bar{v}},\tilde{u},\tilde{w})]\|_{X^3}
 \leq \gamma\|(\bar{v},u,w)-(\tilde{\bar{v}},\tilde{u},\tilde{w})\|_{X^3}\label{eqn55}
\end{equation}
for all $(\bar{v},u,w),(\tilde{\bar{v}},\tilde{u},\tilde{w})\in K$ and 
some $\gamma<1$.
Using \eqref{eqn26} and \eqref{eqn42}, we have
\begin{equation}
\|\tau[(\bar{v}_0,u_0,w_0)]\|_{X^3}
=\|(\bar{Q}(x,0),R(x,0),S(x,0))\|_{X^3}
=\|(\bar{v}_0,u_0,w_0)\|_{X^3}\leq \sqrt{\Sigma}\label{eqn56}
\end{equation}
Therefore, for $(\bar{v},u,w)\in K$, using \eqref{eqn54} and \eqref{eqn56},
\begin{equation}
\begin{aligned}
&\|\tau[(\bar{v},u,w)]\|_{X^3}\\
&\leq \|\tau[(\bar{v}_0,u_0,w_0)]\|_{X^3}
 +\|\tau[(\bar{v},u,w)]-\tau[(\bar{v}_0,u_0,w_0)]\|_{X^3} \\
&\leq \sqrt{\Sigma}+  C\left(T+T^{1/2}\right)^{1/2}
\exp\Big[2^{-1}T(1+\|w\|_{H^1_0(\Omega)}^2)\Big]\\
&\quad \times \Big(1+\|(\bar{v},u,w)\|_{X^3}^2+
\|(\bar{v}_0,u_0,w_0)\|_{X^3}^2\Big)^{1/2}\|
(\bar{v},u,w)-(\bar{v}_0,u_0,w_0)\|_{X^3}\\
&\leq \sqrt{\Sigma} + C\left(T+T^{1/2}\right)^{1/2}\exp
\big[2^{-1}T(1+4\Sigma)\big](1+5\Sigma)^{1/2}(4\sqrt{\Sigma}) \\
&\leq 2\sqrt{\Sigma},
\end{aligned}\label{eqn57}
\end{equation}
for $T>0$ sufficiently small such that
\begin{equation}
4C\big(T+T^{1/2}\big)^{1/2}\exp[2^{-1}T(1+4\Sigma)]
(1+5\Sigma)^{1/2}\leq 1\label{eqn58}
\end{equation}
Thus $\tau[(\bar{v},u,w)]\in K$, and hence $\tau(K)\subseteq K$
for $T>0$ sufficiently small. Furthermore, if $T$ is
chosen sufficiently small such that
\begin{equation}
C\big(T+T^{1/2}\big)^{1/2}\exp[2^{-1}T(1+4\Sigma)](1+5\Sigma)^{1/2}
=\gamma<1,\label{eqn59}
\end{equation}
then,  \eqref{eqn54} implies
\begin{equation}
\|\tau[(\bar{v},u,w)]-\tau[(\tilde{\bar{v}},\tilde{u},\tilde{w})]\|_{X^3}
< \gamma\|(\bar{v},u,w)-(\tilde{\bar{v}},\tilde{u},\tilde{w})\|_{X^3}\label{eqn60}
\end{equation}
for all $(\bar{v},u,w),\ (\tilde{\bar{v}},\tilde{u},\tilde{w})\in K$. 
Thus, the mapping
$\tau$ is a strict contraction for sufficiently small $T>0$.

8. Given $( \bar{v}_k,u_k,w_k)\ (k=0,1,2,\dots)$, inductively define
$$
(Q,R,S):=(\bar{v}_{k+1},u_{k+1},w_{k+1})\in K
$$
to be the unique weak solution of the linear initial boundary
value problem
\begin{gather}
\nabla . \bar{v}_{k+1} = 0\quad\text{in }\Omega_T \label{eqn61}\\
\frac{\partial \bar{v}_{k+1}}{\partial t}-\nu \Delta\bar{v}_{k+1}
 =-\nabla.( \bar{v}_k\otimes \bar{v}_k)-\frac{1}{\rho}\nabla p_{k+1}\quad\text{in }
 \Omega_T \label{eqn62}\\
\frac{\partial u_{k+1}}{\partial t}-k \Delta u_{k+1}
 =-\nabla.( \bar{v}_ku_k)+Qw_k f(u_k)\quad\text{in }\Omega_T\label{eqn63}\\
\frac{\partial w_{k+1}}{\partial t}-d\Delta w_{k+1}
= -\nabla.( \bar{v}_k w_k)-w_k f(u_k)\quad\text{in }\Omega_T\label{eqn64}\\
\bar{v}_{k+1}=\bar{0},\quad u_{k+1}=w_{k+1}=0\quad\text{on }\partial\Omega\times [0,T)
 \label{eqn65}\\
\bar{v}_{k+1}(x,0)=\bar{v}_0(x),\quad u_{k+1}(x,0)=u_0(x),\quad
 w_{k+1}(x,0)=w_0(x),\label{eqn66}
\end{gather}
where $Y:=p_{k+1}$ is the pressure distribution corresponding
to  $(\bar{v}_{k+1},u_{k+1},w_{k+1})$.

By the definition of the mapping $\tau$, we have (for
$k=0,1,2,\dots$), using \eqref{eqn61}-\eqref{eqn66} that
\begin{equation}
(\bar{v}_{k+1},u_{k+1},w_{k+1})=\tau[( \bar{v}_k,u_k,w_k)].\label{eqn67}
\end{equation}
Consider the series
\begin{equation}
(\bar{v}_1,u_1,w_1) + \sum_{r\geq 2}[(\bar{v}_r,u_r,w_r)
-(\bar{v}_{r-1},u_{r-1},w_{r-1})]\label{eqn68}
\end{equation}
The partial sum of the first $k+1$ terms of the series
\eqref{eqn68} is
\begin{equation}
(\bar{v}_1,u_1,w_1) + \sum_{r= 2}^{k+1}[(\bar{v}_r,u_r,w_r)
-(\bar{v}_{r-1},u_{r-1},w_{r-1})]=(\bar{v}_{k+1},u_{k+1},w_{k+1})\label{eqn69}
\end{equation}
Now,  using \eqref{eqn60}, we have
\begin{gather}
\begin{aligned}
\|(\bar{v}_2,u_2,w_2)-(\bar{v}_1,u_1,w_1)\|_{X^3}
&=\|\tau[(\bar{v}_1,u_1,w_1)]-\tau[(\bar{v}_0,u_0,w_0)]\|_{X^3} \\
&< \gamma\|(\bar{v}_1,u_1,w_1)-(\bar{v}_0,u_0,w_0)\|_{X^3}
\end{aligned} \label{eqn70}
\\
\begin{aligned}
\|(\bar{v}_3,u_3,w_3)-(\bar{v}_2,u_2,w_2)\|_{X^3}
&= \|\tau[(\bar{v}_2,u_2,w_2)]-\tau[(\bar{v}_1,u_1,w_1)]\|_{X^3} \\
&<  \gamma^2\|(\bar{v}_1,u_1,w_1)-(\bar{v}_0,u_0,w_0)\|_{X^3}
\end{aligned} \label{eqn71}
\end{gather}
By induction,
\begin{equation}
\begin{aligned}
\|(\bar{v}_{k+1},u_{k+1},w_{k+1})-( \bar{v}_k,u_k,w_k)\|_{X^3}
&< \gamma^k\|(\bar{v}_1,u_1,w_1)-(\bar{v}_0,u_0,w_0)\|_{X^3} \\
&< 4\gamma^k\sqrt{\Sigma},
\end{aligned} \label{eqn72}
\end{equation}
since $(\bar{v}_1,u_1,w_1),\ (\bar{v}_0,u_0,w_0)$ are in $K$, defined by
\eqref{eqn54.2}. Hence the series \eqref{eqn68} is absolutely
convergent, since using \eqref{eqn72}, the series
\begin{equation}
\sum_{k=0}4\gamma^k\sqrt{\Sigma},
\end{equation}
which converges, dominates
\begin{equation}
\|(\bar{v}_1,u_1,w_1)\|_{X^3} + \sum_{r\geq 2}\|(\bar{v}_r,u_r,w_r)
-(\bar{v}_{r-1},u_{r-1},w_{r-1})\|_{X^3}.
\end{equation}
 This implies that the series \eqref{eqn68} is
convergent. Define
$$
\lim_{k\to \infty}(\bar{v}_{k+1},u_{k+1},w_{k+1}):=(\bar{v},u,w).
$$
Thus
$(\bar{v}_{k+1},u_{k+1},w_{k+1})\to (\bar{v},u,w)$ uniformly in $K$.
Thus
\begin{equation}
\lim_{k\to \infty}(\bar{v}_{k+1},u_{k+1},w_{k+1})=(\bar{v},u,w)
=\lim_{k\to \infty}\tau[( \bar{v}_k,u_k,w_k)]=\tau[(\bar{v},u,w)]\label{eqn73}
\end{equation}
By \eqref{eqn73}, $(\bar{v},u,w)\in K$ is the unique fixed point of
$\tau$.

9. As in \cite{Temam}, define
\begin{equation}
V:=\text{The  closure  of $
 \{\bar{\zeta}\in C_c^\infty(\Omega): \nabla.\bar{\zeta}=0\}$
 in }{H^1_0(\Omega)}.
\end{equation}
 In view of the previous steps of this section, we are motivated
to give the following definition.

\begin{definition} \rm
The weak formulation of  \eqref{eqn1}-\eqref{eqn6} is:
For given $(\bar{v}_0,u_0,w_0)\in [{H^1_0(\Omega)}\cap {H^2(\Omega)}]^3$, 
find $(\bar{v},u,w)\in
K$ satisfying
\begin{gather}
\int_\Omega\partial_t\bar{v}.\bar{\zeta}dx+\nu\int_\Omega \nabla \bar{v}:\nabla\bar{\zeta}dx
= -\int_\Omega\nabla.(\bar{v}\otimes\bar{v}).\bar{\zeta}dx\label{eqn73.01}\\
 \int_\Omega \partial_tu\xi dx + k\int_\Omega \nabla u.\nabla\xi dx
 = -\int_\Omega \nabla.(\bar{v} u)\xi dx + Q\int_\Omega wf(u)\xi dx\label{eqn73.03}\\
 \int_\Omega \partial_tw\xi dx + d\int_\Omega \nabla w.\nabla\xi dx
 = -\int_\Omega \nabla.(\bar{v} w)\xi dx - \int_\Omega wf(u)\xi dx  \label{eqn73.05}\\
 \bar{v}(x,0)=\bar{v}_0(x),\quad  u(x,0) = u_0(x),\quad  w(x,0)=w_0(x),  \label{eqn73.07}
\end{gather}
for each $\zeta\in V$ and each $\xi\in {H^1_0(\Omega)}$.
\end{definition}

10. Before verifying that $(\bar{v},u,w)$ is weak solution of
\eqref{eqn1}-\eqref{eqn6}, we first prove the following
Lemma.

\begin{lemma}\label{lem3}
If $( \bar{v}_k,u_k,w_k)\in K$, $\xi\in {H^1_0(\Omega)}$ and 
$\bar{\zeta}\in V$, then
\begin{gather}
\int_\Omega \nabla.( \bar{v}_k\otimes \bar{v}_k).\bar{\zeta} dx\to
 \int_\Omega \nabla.(\bar{v}\otimes\bar{v}).\bar{\zeta} dx\label{eqn73.2}\\
\int_\Omega \nabla.( \bar{v}_k u_k)\xi dx \to
 \int_\Omega \nabla.(\bar{v} u)\xi dx\label{eqn73.4}\\
\int_\Omega \nabla.( \bar{v}_k w_k)\xi dx \to
 \int_\Omega \nabla.(\bar{v} w)\xi dx\label{eqn73.6}\\
 f(u_k)\to f(u)\quad\text{in } {L^2(\Omega)}\label{eqn73.8}\\
 \int_\Omega w_k f(u_k)\xi dx\to\int_\Omega wf(u)\xi dx\label{eqn73.10}
\end{gather}
\end{lemma}

\begin{proof} (i). Proof of \eqref{eqn73.2}.
Integrating by parts, we have
\begin{equation}
|\int_\Omega \nabla.( \bar{v}_k\otimes \bar{v}_k).\bar{\zeta} dx|
=|\int_\Omega  \bar{v}_k\otimes \bar{v}_k:\nabla\bar{\zeta}dx|
\leq \|\zeta\|_{H^1_0(\Omega)}\|u_k\|_{H^1_0(\Omega)}^2,\label{eqn73.11}
\end{equation}
by using \eqref{eqn16} of Lemma \ref{lem1}.
Equation \eqref{eqn73.2} follows by taking limits on both sides
of \eqref{eqn73.11}.
Further, the proofs of \eqref{eqn73.4} and \eqref{eqn73.6} follow
by similar calculations.

(ii). Proof of \eqref{eqn73.8}. We have
\begin{gather}
\begin{aligned}
\int_\Omega |f(u_k)|^2dx
=\int_\Omega \Big|\int_0^{u_k} f'(r)dr+f(0)\Big|^2dx 
\leq \int_\Omega \big|B'|u_k| + f(0)\big|^2dx\\
\leq C_1(B',f(0))\int_\Omega(|u_k|^2+2|u_k|+1|)dx
\leq C_2(B',f(0),\Omega)(\|u_k\|_{L^2(\Omega)}+1)^2
\end{aligned}\label{eqn73.15}
\end{gather}
where  we have  used the first inequality in \eqref{eqn8} and the estimate
$\int_\Omega|u_k|dx\leq |\Omega|^\frac{1}{2}\|u_k\|_{L^2(\Omega)})$.
Then \eqref{eqn73.8} follows by taking limits on both sides of
\eqref{eqn73.15}.

(iii). Proof of \eqref{eqn73.10}. We estimate
\begin{equation}
 |\int_\Omega w_k f(u_k)\xi dx|
 \leq \|w_k\|_{H^1_0(\Omega)}\|f(u_k)\|_{L^2(\Omega)}\|\xi\|_{H^1_0(\Omega)},\label{eqn73.20}
\end{equation}
using \eqref{eqn16} of Lemma \ref{lem1}. Hence, \eqref{eqn73.10}
follows by taking limits in \eqref{eqn73.20}.

11. We now verify that $(\bar{v},u,w)\in K$ is a weak solution of \eqref{eqn1}.
Fix $\zeta\in V$ and $\xi\in {H^1_0(\Omega)}$. Using
\eqref{eqn61}-\eqref{eqn66}, we have
\begin{gather}
\int_\Omega\partial_t\bar{v}_{k+1}.\bar{\zeta}dx
+\nu\int_\Omega \nabla \bar{v}_{k+1}:\nabla\bar{\zeta}dx
=-\int_\Omega\nabla.( \bar{v}_k\otimes \bar{v}_k).\bar{\zeta}dx\label{eqn73.23}
\\
\begin{aligned}
&\int_\Omega \partial_tu_{k+1}\xi dx + k\int_\Omega \nabla u_{k+1}.\nabla\xi dx\\
&= -\int_\Omega \nabla.( \bar{v}_k u_k)\xi dx +  Q\int_\Omega w_k f(u_k)\xi dx
\end{aligned} \label{eqn73.25}
\\
\begin{aligned}
& \int_\Omega \partial_tw_{k+1}\xi dx + d\int_\Omega \nabla w_{k+1}.\nabla\xi dx \\
&= -\int_\Omega \nabla.( \bar{v}_k w_k)\xi dx
  - \int_\Omega w_k f(u_k)\xi dx
\end{aligned}\label{eqn73.27}\\
\bar{v}_{k+1}(x,0)=\bar{v}_0(x),\quad  u_{k+1}(x,0) = u_0(x),\quad
  w_{k+1}(x,0)=w_0(x)\,.  \label{eqn73.29}
\end{gather}
Letting $k\to \infty$ in \eqref{eqn73.23}-\eqref{eqn73.29}
and using Lemma \ref{lem2} to handle the nonlinear terms yield
\eqref{eqn73.01}-\eqref{eqn73.07} as desired.
\end{proof}

12. We next demonstrate how to obtain the pressure $Y=p_{k+1}$.
First, we obtain the boundary condition on pressure by taking
\eqref{eqn2} on the boundary and using \eqref{eqn5} to deduce
\begin{equation}
\frac{1}{\rho}\nabla p=\nu\Delta \bar{v}\quad\text{on }\partial\Omega\label{eqn73.32}
\end{equation}
Following the steps in \cite{Mccomb}, we express \eqref{eqn73.32}
in terms of the standard normal derivatives as
\begin{equation}
\frac{1}{\rho}\frac{\partial p}{\partial n}
=\nu\hat{n}.\frac{\partial^2 \bar{v}}{\partial n^2}\label{eqn73.34}
\end{equation}
where $\hat{n}(x)$ is the inward normal at $x$ on $\partial\Omega$.

Taking the divergence of \eqref{eqn62} yields the equation
satisfied by $Y=p_{k+1}$ as
\begin{equation}
\Delta p_{k+1} = \rho \nabla.[\nabla .( \bar{v}_k\otimes  \bar{v}_k)],\label{eqn74}
\end{equation}
which is a form of Poisson's equation. Further, in  sympathy with
the boundary condition \eqref{eqn73.34}, we impose the the
boundary condition on the pressure  $p_{k+1}$ as
\begin{equation}
\frac{1}{\rho}\frac{\partial p_{k+1}}{\partial n}
=\nu\hat{n}.\frac{\partial^2 \bar{v}}{\partial n^2}\label{eqn75}
\end{equation}
Hence, the formal solution of \eqref{eqn74} subject to the
condition \eqref{eqn75} is
\begin{equation}
\begin{aligned}
p_{k+1}(x,t)&=-\rho\int_\Omega G(x,y)\nabla.[\nabla.( \bar{v}_k(y,t)\otimes  
\bar{v}_k(y,t))]dy \\
&\quad +\rho\nu\int_{\partial \Omega}G(x,y)\hat{n}.
\frac{\partial^2 \bar{v}(y,t)}{\partial n^2}dS(y)\label{eqn76}
\end{aligned}
\end{equation}
where $G(x,y)$ is the Green's function satisfying the Laplace's
equation in the form
\begin{equation}
\Delta G(x,y)=\delta(x-y)
\end{equation}
with the condition
\begin{equation}
\frac{\partial G(x,y)}{\partial n}=0\quad (x\text{ on }\partial\Omega).
\end{equation}
where $\delta$ is the Dirac delta function.

For $n=3$, $G(x,y)=\frac{1}{|x-y|}$, $x,y\in\Omega$, we define
\begin{gather}
\begin{aligned}
p_{k+1}^\epsilon(x,t)
&:=-\rho\int_\Omega \frac{1}{|x-y|
 +\epsilon}\nabla.[\nabla.( \bar{v}_k(y,t)\otimes  \bar{v}_k(y,t))]dy \\
&\quad +\rho\nu\int_{\partial \Omega}\frac{1}{|x-y|+\epsilon}\hat{n}.
\frac{\partial^2 \bar{v}(y,t)}{\partial n^2}dS(y),
\end{aligned} \label{eqn77}\\
\begin{aligned}
p^\epsilon(x,t)
&:=-\rho\int_\Omega \frac{1}{|x-y|+\epsilon}\nabla.[\nabla.(\bar{v}(y,t)
 \otimes \bar{v}(y,t))]dy \\
&\quad +\rho\nu\int_{\partial \Omega}\frac{1}{|x-y|+\epsilon}\hat{n}.
\frac{\partial^2 \bar{v}(y,t)}{\partial n^2}dS(y)
\end{aligned} \label{eqn78}
\\
\begin{aligned}
p(x,t)&:=-\rho\int_\Omega \frac{1}{|x-y|}\nabla.[\nabla.(\bar{v}(y,t)\otimes
\bar{v}(y,t))]dy \\
&\quad +\rho\nu\int_{\partial \Omega}\frac{1}{|x-y|}\hat{n}.
\frac{\partial^2 \bar{v}(y,t)}{\partial n^2}dS(y)
\end{aligned}\label{eqn79}
\end{gather}
where $\epsilon>0$. Notice that
\[
\lim_{\epsilon\to\ 0}p_{k+1}^\epsilon(x,t)=p_{k+1}(x,t), \quad
\lim_{\epsilon\to\ 0}p^\epsilon(x,t)= p(x,t)
\]
Hence, integrating twice by parts, using \eqref{eqn1} and
\eqref{eqn5}, we have
\begin{equation}
\begin{aligned}
&|p_{k+1}^\epsilon(x,t)-p^\epsilon(x,t)|\\
&= |\rho\int_\Omega \frac{1}{|x-y|+\epsilon}\nabla.\{\nabla.
[ \bar{v}_k(y,t)\otimes  \bar{v}_k(y,t)-\bar{v}(y,t)\otimes \bar{v}(y,t)]\}dy|\\
&= |\rho\int_\Omega \nabla\big\{\nabla[\frac{1}{|x-y|+\epsilon}]\big\}:
[ \bar{v}_k(y,t)\otimes  \bar{v}_k(y,t)-\bar{v}(y,t)\otimes \bar{v}(y,t)]dy|
\end{aligned} \label{eqn80}
\end{equation}
which tends to $0$ as  $k\to\ \infty$.
Therefore
\begin{equation}
\lim_{k\to \infty}p_{k+1}^\epsilon(x,t)=p^\epsilon(x,t),\label{eqn81}
\end{equation}
 From whence sending $\epsilon$ to $0$, we obtain
\begin{equation}
\lim_{k\to\ \infty}p_{k+1}(x,t)=p(x,t),\label{eqn82}
\end{equation}
where, $p(x,t)$ given by \eqref{eqn79}, is the pressure
corresponding to the solution $(\bar{v},u,w)$. Indeed, \eqref{eqn79} is
the formal solution for the pressure $p(x,t)$ satisfying
\begin{equation}
\Delta p = \rho \nabla.[\nabla .(\bar{v}\otimes \bar{v})],\label{eqn83}
\end{equation}
in terms of $G(x,y)$, as obtained in \cite{Mccomb}.
\end{proof}

\section{Regularity}

The Analysis so far carried out requires no smoothness assumption
on the boundary. However, for smooth solution up to the boundary,
one requires the boundary $\partial\Omega$ to be $C^\infty$.
The lengthy proofs of the associated regularity theorems are
currently being established by an analysis of certain difference
quotients in another paper.


\subsection*{Acknowledgments}
The author would like to thank the anonymous referee whose
thoughtful comments improved the original version of this manuscript.

\begin{thebibliography}{20}
\bibitem{Avrin1} {Avrin, J. D.};
\emph{Asymptotic behavior of one-step combustion models with multiple
reactants on bounded domains}, SIAM J. Math. Anal., 24 (1993), 290-298.

\bibitem{Avrin2} {Avrin, J. D.};
\emph{Qualitative theory of the Cauchy problem for a one-step reaction
model on bounded domains}, SIAM J. Math. Anal., 22 (1991), 379-391.

\bibitem{Buckmaster} {Buckmaster J.};
\emph{The Mathematics of Combustion}, SIAM, Philadelphia (1985).

\bibitem{BL} {Buckmaster, J.; Ludford, G.};
\emph{Theory of Laminar Flames}, Cambridge University Press,
Cambridge (1982).

\bibitem{Daddiouaissa} {Daddiouaissa, E.};
\emph{Existence of global solutions for a system of equations
having a triangular matrix}, Electron. Journal of Diff. Equations,
2009(2009), No. 09, 1-7.

\bibitem{DPP}  {De Oliveira, L. F.; Pereira, A. L.; Pereira, M. C.};
 \emph{Continuity of attractors for a reaction-diffusion problem
with respect to variations of the domain}, Electron. Journal of Diff.
 Equations, 2005(2005), No. 100, 1-18.

\bibitem{Evans} {Evans, L. C.};
\emph{Partial Differential Equations}, American Mathematical Society,
Providence, Rhode Island,(1998).

\bibitem{FM} {Fitzgibbon, W. E.; Martin, C. B.};
\emph{The longtime behavior of solutions to a quasilinear combustion
model}, Nonlinear Analysis, Theory, Methods \& Applications,
19(1992), No. 10, 947-961.

\bibitem{FK} {Frank-Kamenetski, D.};
\emph{Diffusion and Heat Transfer in Chemical Kinetics}.
Plenum Press, New York (1969).

\bibitem{Henry} {Henry, D.};
\emph{Geometric Theory of Semilinear Parabolic Equations},
Lecture Notes in Math. 840, Springer Verlag, Berlin, New York, 1981.

\bibitem{Kouach} {Kouach, S.}, 
\emph{Existence of global solutions to reaction-diffusion systems
 with nonhomogeneous boundary conditions via Lyapunov functional}, 
 Electron. Journal of Diff. Equations, 2002(2002), No. 88, 1-13.

\bibitem{Mccomb} {McCOMB, W. D.};
 \emph{The Physics of Fluid Turbulence}, Clarendon Press, Oxford,
(1990).

\bibitem{Sanni} {Sanni, S.~A.};
 \emph{A Combustion model with unbounded thermal conductivity
and reactant diffusivity in non-smooth domains},
Electron. Journal of Diff. Equations, 2009(2009), No. 60, 1-14.

\bibitem{Sattinger} {Sattinger, D.};
 \emph{A nonlinear parabolic system in the theory of combustion},
Quart. Appl. math., 33(1975), 47-61.

\bibitem{Temam} {Temam, R.};
\emph{Navier-Stokes Equations}, North-Holland Publishing Company,
Oxford, (1977).
\end{thebibliography}

\end{document}