\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 $$X:={L^\infty[0 ,T; H^1_0(\Omega)]}\cap{H^1[0 ,T; H^1_0(\Omega)]}\cap W^{1,\infty}[0,T;L^2(\Omega)]\label{eqn8.1}$$ \end{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 $$\|U\|_{L^q(\Omega)}\leq C\|\nabla U\|_{L^p(\Omega)}\label{eqn8.2}$$ 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{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} 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{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} 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}. $$\int_\Omega u(pw-\tilde{p}\tilde{w})dx=\int_\Omega up(w-\tilde{w})dx + \int_\Omega u\tilde{w}(p-\tilde{p})dx.\label{eqn18}$$ 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{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} 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{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} 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{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} 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{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} 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 $$\|\partial_tu_0\|_{L^2(\Omega)}^2\leq C(Q,B,k,\Omega) \left[(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)\|\nabla u_0\|_H^1(\Omega)^2 + \|w_0\|_{H^1_0(\Omega)}^2\right] \label{eqn23}$$ 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{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} for $$T<\{2C[(1+\|\bar{v}_0\|_{H^1_0(\Omega)}^2)(\|\nabla\bar{v}_0\|_H^1(\Omega)^2 +\|\nabla u_0\|_H^1(\Omega)^2+\|\nabla w_0\|_H^1(\Omega)^2)+ 1]^2\}^{-1},\label{eqn27}$$ 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{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} using \eqref{eqn9}) of Lemma \ref{lem1}. Simplifying, \eqref{eqn28} yields \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} 2. Differentiating \eqref{eqn3} with respect to $t$ yields $$\frac{\partial}{\partial t}(\partial_tu)-k\Delta(\partial_tu) =-\partial_t\bar{v}.\nabla u+\bar{v}.\nabla(\partial_tu)+Q\partial_tw.f(u) +Qw\partial_tu f'(u)\label{eqn30}$$ Multiply by $\partial_tu$ and integrating by parts over $\Omega$ and use \eqref{eqn1}, \eqref{eqn5} to deduce: \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} using \eqref{eqn10} of Lemma \ref{lem1} and Young's inequality. 3. Combining \eqref{eqn29} and \eqref{eqn31} we deduce \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} 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{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} 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 $$\tau: X^3\to X^3\label{eqn43}$$ 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{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} where we have used some bounds in \eqref{eqn7} and \eqref{eqn8}. 4. Further, we differentiate \eqref{eqn46} with respect $t$ to get \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} 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{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{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} 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{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} 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 $$K:=\{(\bar{v},u,w)|(\bar{v},u,w)-(\bar{v}_0,u_0,w_0)\in X_0^3\text{ and } \|(\bar{v},u,w)\|_{X^3}\leq 2\sqrt{\Sigma}\},\label{eqn54.2}$$ 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 $$\tau[K]\subseteq K,\quad \|\tau[(\bar{v},u,w)]-\tau[(\tilde{\bar{v}},\tilde{u},\tilde{w})]\|_{X^3} \leq \gamma\|(\bar{v},u,w)-(\tilde{\bar{v}},\tilde{u},\tilde{w})\|_{X^3}\label{eqn55}$$ 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 $$\|\tau[(\bar{v}_0,u_0,w_0)]\|_{X^3} =\|(\bar{Q}(x,0),R(x,0),S(x,0))\|_{X^3} =\|(\bar{v}_0,u_0,w_0)\|_{X^3}\leq \sqrt{\Sigma}\label{eqn56}$$ Therefore, for $(\bar{v},u,w)\in K$, using \eqref{eqn54} and \eqref{eqn56}, \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} for $T>0$ sufficiently small such that $$4C\big(T+T^{1/2}\big)^{1/2}\exp[2^{-1}T(1+4\Sigma)] (1+5\Sigma)^{1/2}\leq 1\label{eqn58}$$ 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 $$C\big(T+T^{1/2}\big)^{1/2}\exp[2^{-1}T(1+4\Sigma)](1+5\Sigma)^{1/2} =\gamma<1,\label{eqn59}$$ then, \eqref{eqn54} implies $$\|\tau[(\bar{v},u,w)]-\tau[(\tilde{\bar{v}},\tilde{u},\tilde{w})]\|_{X^3} < \gamma\|(\bar{v},u,w)-(\tilde{\bar{v}},\tilde{u},\tilde{w})\|_{X^3}\label{eqn60}$$ 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 $$(\bar{v}_{k+1},u_{k+1},w_{k+1})=\tau[( \bar{v}_k,u_k,w_k)].\label{eqn67}$$ Consider the series $$(\bar{v}_1,u_1,w_1) + \sum_{r\geq 2}[(\bar{v}_r,u_r,w_r) -(\bar{v}_{r-1},u_{r-1},w_{r-1})]\label{eqn68}$$ The partial sum of the first $k+1$ terms of the series \eqref{eqn68} is $$(\bar{v}_1,u_1,w_1) + \sum_{r= 2}^{k+1}[(\bar{v}_r,u_r,w_r) -(\bar{v}_{r-1},u_{r-1},w_{r-1})]=(\bar{v}_{k+1},u_{k+1},w_{k+1})\label{eqn69}$$ 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{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} 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 $$\sum_{k=0}4\gamma^k\sqrt{\Sigma},$$ which converges, dominates $$\|(\bar{v}_1,u_1,w_1)\|_{X^3} + \sum_{r\geq 2}\|(\bar{v}_r,u_r,w_r) -(\bar{v}_{r-1},u_{r-1},w_{r-1})\|_{X^3}.$$ 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 $$\lim_{k\to \infty}(\bar{v}_{k+1},u_{k+1},w_{k+1})=(\bar{v},u,w) =\lim_{k\to \infty}\tau[( \bar{v}_k,u_k,w_k)]=\tau[(\bar{v},u,w)]\label{eqn73}$$ By \eqref{eqn73}, $(\bar{v},u,w)\in K$ is the unique fixed point of $\tau$. 9. As in \cite{Temam}, define $$V:=\text{The closure of \{\bar{\zeta}\in C_c^\infty(\Omega): \nabla.\bar{\zeta}=0\} in }{H^1_0(\Omega)}.$$ 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 $$|\int_\Omega \nabla.( \bar{v}_k\otimes \bar{v}_k).\bar{\zeta} dx| =|\int_\Omega \bar{v}_k\otimes \bar{v}_k:\nabla\bar{\zeta}dx| \leq \|\zeta\|_{H^1_0(\Omega)}\|u_k\|_{H^1_0(\Omega)}^2,\label{eqn73.11}$$ 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 $$|\int_\Omega w_k f(u_k)\xi dx| \leq \|w_k\|_{H^1_0(\Omega)}\|f(u_k)\|_{L^2(\Omega)}\|\xi\|_{H^1_0(\Omega)},\label{eqn73.20}$$ 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 $$\frac{1}{\rho}\nabla p=\nu\Delta \bar{v}\quad\text{on }\partial\Omega\label{eqn73.32}$$ Following the steps in \cite{Mccomb}, we express \eqref{eqn73.32} in terms of the standard normal derivatives as $$\frac{1}{\rho}\frac{\partial p}{\partial n} =\nu\hat{n}.\frac{\partial^2 \bar{v}}{\partial n^2}\label{eqn73.34}$$ 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 $$\Delta p_{k+1} = \rho \nabla.[\nabla .( \bar{v}_k\otimes \bar{v}_k)],\label{eqn74}$$ 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 $$\frac{1}{\rho}\frac{\partial p_{k+1}}{\partial n} =\nu\hat{n}.\frac{\partial^2 \bar{v}}{\partial n^2}\label{eqn75}$$ Hence, the formal solution of \eqref{eqn74} subject to the condition \eqref{eqn75} is \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} where $G(x,y)$ is the Green's function satisfying the Laplace's equation in the form $$\Delta G(x,y)=\delta(x-y)$$ with the condition $$\frac{\partial G(x,y)}{\partial n}=0\quad (x\text{ on }\partial\Omega).$$ 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{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} which tends to $0$ as $k\to\ \infty$. Therefore $$\lim_{k\to \infty}p_{k+1}^\epsilon(x,t)=p^\epsilon(x,t),\label{eqn81}$$ From whence sending $\epsilon$ to $0$, we obtain $$\lim_{k\to\ \infty}p_{k+1}(x,t)=p(x,t),\label{eqn82}$$ 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 $$\Delta p = \rho \nabla.[\nabla .(\bar{v}\otimes \bar{v})],\label{eqn83}$$ 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. 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