\documentstyle{amsart} \begin{document} {\noindent\small {\em Electronic Journal of Differential Equations}, Vol.\ 1999(1999), No.~27, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)} \thanks{\copyright 1999 Southwest Texas State University and University of North Texas.} \vspace{1cm} \title[\hfilneg EJDE--1999/27\hfil Correctors for flow] {Correctors for flow in a partially fissured medium } \author[M. Rajesh\hfil EJDE--1999/27\hfilneg] {M. Rajesh} \address{ M. Rajesh \hfil\break The Institute of Mathematical sciences \hfil\break C.I.T. Campus, Taramani \hfil\break Madras-600 113, INDIA } \email{rajesh@@imsc.ernet.in } \date{} \thanks{Submitted May 21, 1999. Published August 25, 1999.} \subjclass{76S05, 35B27, 73B27. } \keywords{ Partially fissured medium, flow, homogenization, correctors.} \begin{abstract} We prove a corrector result for the homogenization of flow in a partially fissured medium. The homogenization problem was studied by Clark and Showalter~\cite{cs} using the two-scale convergence technique. \end{abstract} \maketitle \newtheorem{theo}{Theorem}[section] \newtheorem{lem}{Lemma}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{defi}{Definition}[section] \newtheorem{rem}{Remark:}[section] \newcommand{\ve}{\varepsilon} \newcommand{\xe}{\frac{x}{\varepsilon}} \newcommand{\abs}[1]{|#1|} \newcommand{\om}[1]{\Omega^{\ve}_{#1}} \newcommand{\ra}{\longrightarrow} \newcommand{\ce}[1]{c_{#1} (\xe)} \newcommand{\ue}[1]{u_{#1}^{\ve}} \newcommand{\der}[1]{\frac{\partial u_{#1}}{\partial t}} \newcommand{\dpe}[1]{\frac{\partial u_{#1}^{\ve}}{\partial t}} \newcommand{\eo}{\ve \rightarrow 0} \newcommand{\leb}[1]{L^{2}(#1)} \newcommand{\intot}{\int_{0}^{T}} \newcommand{\intw}{\int_{\Omega}} \newcommand{\intn}[1]{\int_{Y_{#1}}} \newcommand{\wpe}[1]{W^{1,p}({#1})} \newcommand{\ora}[1]{\overrightarrow{#1}} \newcommand{\lime}{\lim_{\ve \ra 0}} \newcommand{\norm}[1]{\|#1 \|} \newcommand{\ot}{[0,T] \times \Omega} \newcommand{\scale}{\stackrel{2-s}{\ra}} \newcommand{\dert}[1]{\frac{\partial #1}{\partial t}} \catcode`\@=11 \def\theequation{\@arabic{\c@section}.\@arabic{\c@equation}} \catcode`\@=12 \section{Introduction} A fissured medium consists of a porous and permeable {\em matrix} interlaced, on a fine scale, by a system of highly permeable {\em fissures}. Fluid flow in such a medium takes place, primarily, through the fissures. The fissured medium is said to be {\em totally fissured} if the matrix is broken up into disjoint cells by the fissures. In this case, there is no direct flow through the matrix but only an exchange of fluids between the cells and the surrounding fissures. If, on the other hand, the matrix is connected there is a global flow through the matrix as well. This is the {\em partially fissured case}. As remarked by Clark and Showalter~\cite{cs}, an exact microscopic model for flow in a fissured medium, written as a classical interface problem, is both analytically and numerically intractable. One way to get around this difficulty is to model the flow on two separate scales, one microscopic and the other macroscopic. The problem, then, can be studied as a problem in homogenization. Such a model for flow in a partially fissured medium was considered by Douglas, Peszy\'nska and Showalter~\cite{dps} assuming the diffusion operator to be linear. Clark and Showalter~\cite{cs} extend the results of~\cite{dps} to the case where the diffusion operator is quasilinear. The corresponding homogenization problem was solved under weak monotonicity conditions and using the two-scale convergence method.\par Correctors for the homogenization of quasilinear equations \begin{equation} -\operatorname{div}\left( a \left( \xe, \nabla u_{\ve} \right) \right)= f \end{equation} were obtained by Dal Maso and Defranceschi~\cite{dmdf} under some strong monotonicity conditions on the function $a$. Later, the proof of the corrector result was greatly simplified using the two-scale convergence method by Allaire~\cite{all}. Based on these ideas we prove a corrector result for the flow in a partially fissured medium under strong monotonicity conditions on the diffusion operator. The plan of the paper is as follows. In Section 2, we describe the micro-model for flow in a partially fissured medium. In Section 3, we recall the homogenization results obtained by Clark and Showalter in \cite{cs} under weak monotonicity of the diffusion operator. In Section 4, we present our results on correctors. Strong monotonicity conditions are required here. \section{The micro-model} \addtocounter{equation}{-1}% We present, here, the micro-model for flow in a partially fissured medium as described in~\cite{cs}. Let $\Omega$ be a bounded open set in $R^N$. $Y=[0,1]^N$ denotes the unit cube and $Y=Y_{1} \bigcup Y_{2}$, where $Y_{1}$ and $Y_{2}$ represent the local structure of the fissure and matrix respectively. Let $\chi_{j}(y)$ denote the characteristic function of $Y_{j}$ (j=1, 2) extended Y-periodically to all of $R^N$. We shall assume that the sets $\{ y \in R^N : \chi_{j}(y)=1 \}$ (j=1, 2) are smooth (connectedness will not be required in view of the coercivity conditions to be assumed on the coefficients in the differential operators). The domain $\Omega$ is thus divided into the two subdomains, $\om{1}$ and $\om{2}$, representing the fissures and the matrix respectively, and are given by $$\om{j}=\left \{ x \in \Omega : \chi_{j} (\xe)=1 \right\}, \, \, j=1, 2 .$$ Henceforth, we will denote $\chi_{j} (\xe) \mbox{ by } \chi_{j}^{\ve}$.\\ Let $ \Gamma_{1,2}^{\ve} = \partial \om{1} \bigcap \partial \om{2} \bigcap \Omega$ denote the interface of $\om{1}$ with $\om{2}$ which is interior to $\Omega$. $\Gamma_{1,2} = \partial Y_{1} \bigcap \partial Y_{2} \bigcap Y$ denotes the corresponding interface in the reference cell Y. We set $\om{3} \equiv \om{2}, Y_{3} \equiv Y_{2}$, and $\chi_{3} \equiv \chi_{2}$, to be used to simplify notation at times.\par Let $\mu_{j} : R^N \times R^N \ra R^N \,( \, j=1, 2, 3)$ be functions which satisfy the following hypothesis: \begin{enumerate} \item $\mu_{j} ( \, . \, , \, \xi)$ is measurable and Y-periodic for all $\xi \in R^N.$ \label{a1} \item $\mu_{j} (y \, , \, . \,)$ is continuous for a.e. $y \in Y$. \label{a2} \item There exist positive constants $k, C, c_{0}$ and $1 < p < \infty$ such that for every $\xi, \, \eta \in R^N$ and a.e. $y \in Y$ \end{enumerate} \begin{eqnarray} | \mu_{j} (y \,, \,\xi) | &\leq& C |\, \xi \,| ^{p-1} + k \label{growth}\\ ( \mu_{j}( y\, , \, \xi ) - \mu_{j}( y \, , \, \eta))\, . \, (\xi -\eta)) &\ge& 0 \label{wm}\\ \mu_{j}(y \, , \, \xi) \,. \, \xi &\ge& c_{0} |\, \xi \,|^{p}-k. \label{coercivity} \end{eqnarray} $q$ will denote the conjugate exponent of $p$, viz. $ q= p/p-1$. Let $c_{j} \in C_{\sharp}(Y) \, (\, j = 1, 2, 3)$ be continuous Y-periodic functions on $R^N$ such that \begin{equation} 0< c_{0} \le c_{j} \le C. \label{coercivity2} \end{equation} The flow potential of the fluid in the fissure $\om{1}$ is denoted by the function $u_{1}^{\ve}(x, t)$ and the corresponding flux by $- \mu_{1} \left( \xe, \nabla u_{1}^{\ve} \right)$. The flow potential in the matrix is represented as the sum of two parts, one component $u_{2}^{\ve}(x, t)$ with the flux $-\mu_{2} \left( \xe, \nabla u_{2}^{\ve} \right)$ which accounts for the global diffusion through the pore system of the matrix, and the second component $u_{3}^{\ve}(x, t)$ with flux $-\ve \mu_{3} \left( \xe, \ve \nabla u_{3}^{\ve} \right)$ and corresponding high frequency spatial variations which lead to local storage in the matrix. The ``total flow potential'' in the matrix is then $ \alpha u_{2}^{\ve} + \beta u_{3}^{\ve}$ (here $\alpha + \beta = 1$ with $\alpha \ge 0, \beta > 0$). The exact microscopic model for diffusion in a partially fissured medium is given by the system \begin{eqnarray} \ce{1} \dpe{1} - \operatorname{div} \mu_{1} \left( \xe \, , \, \nabla \ue{1} \right) & = & 0 \; \; \; \, \mbox{ in } \om{1} \label{ff}\\ \ce{2} \dpe{2} - \operatorname{div} \mu_{2} \left( \xe \, , \, \nabla \ue{2} \right) & = & \mbox{ 0 $\, \, \, \,$ in $\om{2}$} \label{fm1}\\ \ce{3} \dpe{3} -\ve \operatorname{div} \mu_{3} \left( \xe \, , \, \ve \nabla \ue{3} \right) & = & \mbox{ 0 $\;$ in $\om{2}$} \label{fm2}\\ \alpha \ue{2} + \beta \ue{3} & = & \ue{1} \; \mbox{ on } \Gamma_{1,2}^{\ve} \label{cty}\\ \alpha \mu_{1} \left( \xe \, , \, \nabla \ue{1} \right) \, . \, \nu_{1}^{\ve} & = & \mu_{2} \left( \xe \, , \, \nabla \ue{2} \right) . \nu_{1}^{\ve} \label{partflux1} \\ \beta \mu_{1} \left( \xe \, , \, \nabla \ue{1} \right) \, . \, \nu_{1}^{\ve} & = & \ve \mu_{3} \left( \xe \, , \, \nabla \ve \ue{3} \right) . \nu_{1}^{\ve} \; \label{partflux2} \end{eqnarray} where the last two conditions hold on $\Gamma_{1,2}^{\ve}$. We have the homogeneous Neumann condition on the external boundary \begin{eqnarray} \mu_{1} \left( \xe \, , \, \nabla \ue{1} \right) \, . \, \nu_{1}^{\ve}& = & 0 \; \mbox{ on } \partial \om{1} \cap \partial \Omega \label{bdy1}\\ \mu_{2} \left( \xe \, , \, \nabla \ue{2} \right) \, . \, \nu_{2}^{\ve}& = & 0 \; \mbox{ on } \partial \om{2} \cap \partial \Omega \label{bdy2}\\ \mu_{3} \left( \xe \, , \, \ve \nabla \ue{3} \right) \, . \, \nu_{2}^{\ve}& = & 0 \; \mbox{ on } \partial \om{2} \cap \partial \Omega \label{bdy3} \end{eqnarray} where $\nu_{j}^{\ve}$ denotes the outward normal on $\partial \om{j}, j=1, 2$.\\ \noindent The system is completed by the initial conditions \begin{equation} \ue{j} (0 \, , \, . ) = u_{j}^{0} \in \leb{\Omega}, \, \, 1 \le j \le 3. \label{in} \end{equation} \noindent {\bf Remark 2.1:} Condition (\ref{cty}) is the continuity of flow potential across the interface. Conditions (\ref{partflux1}), (\ref{partflux2}) determine the partition of flux across the interface.$\Box$ We now describe the variational formulation needed to study the well posedness of the Cauchy problem. The {\em state space} is the Hilbert space \begin{eqnarray*} H_{\ve} \equiv \leb{\om{1}} \times \leb{\om{2}} \times \leb{\om{2}}\ ( = \leb{\om{1}} \times \leb{\om{2}}^{2}) \end{eqnarray*} equipped with the inner product \begin{eqnarray*} ( [u_{1}, \, u_{2}, \, u_{3}],[\phi_{1}, \, \phi_{2}, \, \phi_{3}])_{H_{\ve}} = \sum_{j=1}^{3} \int_{\om{j}} \ce{j} \, u_{j}(x) \, \phi_{j}(x) \, dx . \end{eqnarray*} Define the {\em energy space} \begin{eqnarray*} B_{\ve} \equiv H_ {\ve} \, \cap \{ [\ora{u}] \in \wpe{\om{1}} \times \wpe{\om{2}}^{2} : u_{1}= \alpha u_{2}+\beta u_{3} \mbox{ on } \Gamma_{1,2}^{\ve} \} \end{eqnarray*} where $\ora{u}=(u_{1},u_{2},u_{3})$. $B_{\ve}$ is a Banach space with the norm \begin{eqnarray*} \|\, [u_{1},\,u_{2},\,u_{3}] \|_{B_{\ve}} \equiv \sum_{j=1}^{3} \|\,\chi_{j}^{\ve} \, u_{j} \,\|_{\leb{\Omega}} + \sum_{j=1}^{3} \|\,\chi_{j}^{\ve} \,\nabla \,u_{j}\, \|_{L^{p}(\Omega)}. \end{eqnarray*} Define the operator $A_{\ve}\, : \, B_{\ve} \ra B_{\ve}^{'}$ (where $B_{\ve}^{'}$ denotes the dual of $B_{\ve}$) by, \begin{eqnarray*} \begin{array}{ll} A_{\ve}\,( [\, u_1, \, u_2, \, u_3\,] )([\, \phi_1,\, \phi_2, \phi_3 \,]) \equiv \sum_{j=1}^{2} &\int_{\om{j}} \, \mu_{j}( \xe, \, \nabla u_{j} \,)\, . \, \nabla \phi_{j} \, dx \\ &+ \int_{\om{2}} \, \mu_{3}(\, \xe, \, \ve \nabla u_{3} \,)\, . \, \ve \nabla \phi_{3} \, dx \end{array} \end{eqnarray*} for $[u_1, \, u_2, \, u_3 ], \, [\phi_1, \, \phi_2, \, \phi_3 ] \in B_{\ve}$. Let \begin{eqnarray*} V_{\ve} \equiv \{ \ora{u^{\ve}} \in L^{p}([0,T]; B_{\ve}): (\ora{u^{\ve}}) \, ^{'} \in L^{q}([0,T]; B_{\ve}') \}\, . \end{eqnarray*} For $\ve > 0$, the Cauchy problem is equivalent to finding a solution $ \ora{u^{\ve}} \in V_{\ve}$ to the problem \begin{eqnarray} \frac{ {\rm d} \ora{u^\ve} }{\rm dt} + A_{\ve} \ora{u^{\ve}}& = & 0\mbox{ in } L^{q}([0,T]; B_{\ve}^{'})\\ \ora{u^{\ve}}(0) & = & \ora{u^{0}} \mbox{ in } H_{\ve} \end{eqnarray} and this problem is well-posed, thanks to the conditions (\ref{growth})-(\ref{coercivity}) (cf. Showalter~\cite{s}). We end with an identity(cf.~\cite{cs}), \begin{eqnarray} \frac12 \norm{\ora{u^{\ve}}(T)}^{2}_{H_{\ve}} - \frac12 \norm{\ora{u^{\ve}}(0)}^{2}_{H_{\ve}} + \intot A_{\ve}(\ora{u^{\ve}})(\ora{u^{\ve}}) dt = 0. \label{id} \end{eqnarray} \section{Homogenization} \addtocounter{equation}{-17}% We recall the results on the homogenization of flow in a partially fissured medium here (cf.~\cite{cs}) in the form of propositions. For a proof of these results, refer Clark and Showalter (cf.~\cite{cs}). We recall the definition of two-scale convergence (cf. \cite{all}, \cite{cs}). \begin{defi} A function, $\psi(t,x,y) \in L^{q}(\ot, C_{\sharp}(Y))$, which is Y-periodic in y and satisfies \begin{eqnarray*} \lime \intot \!\intw \psi \left( t,x, \xe \right)^{q} \, dx \, dt = \intot \!\intw \!\int_{Y} \psi(t,x,y)^q \, dy \, dx \, dt \end{eqnarray*} is called an {\em admissible} test function. $\Box$ \end{defi} \begin{defi} A sequence $f^{\ve} \mbox{ in } L^{p}(\ot))$ two-scale converges to a function $f(t, x, y) \in L^{p}(\ot \times Y)$ if for any admissible test function $\psi(t,x, y)$, \begin{eqnarray*} \lime \intot \!\intw \, f^{\ve}(t,x) \, \psi\left(t,x, \xe \right) \, dx \, dt = \intot \!\intw \!\int_{Y} f(t,x,y) \, \psi(t,x,y) \, dy \, dx \, dt \end{eqnarray*} We write $f^{\ve} \scale f$.$\Box$ \end{defi} \begin{prop} Let $\ora{u^{\ve}}$ be the solution of the Cauchy problem (\ref{ff})-(\ref{in}). The following estimate holds \begin{equation} \sum_{j=1}^{2} \norm{\chi_{j}^{\ve} \nabla \ue{j}}^{p}_{p, \Omega_{T}} + \norm{\chi_{2}^{\ve} \ve \nabla \ue{3}}^{p}_{p, \Omega_{T}} \le \frac{C}{2c} \sum_{j=1}^{3} \norm{u_{j}^{0}}^{2}_{2, \Omega}.\label{estimate} \end{equation} \end{prop} $\Box$ \begin{prop}Let $\ora{u^{\ve}}$ be the solution of the Cauchy problem (\ref{ff})-(\ref{in}). There exist functions $u_{j} \mbox{ in } L^{p}([0,T];W^{1,p}(\Omega)), \, j=1, 2$ and functions $ U_{j} \mbox{ in } L^{p}(\ot; W^{1,p}_{\sharp}(Y_{j}) / R), \, j=1, 2, 3$ such that, for a subsequence of $\ora{u^\ve}$, (to be indexed by $\ve$ again) the following hold: \begin{eqnarray*} \chi_{j}^{\ve} \ue{j}& \scale& \chi_{j}(y) u_{j}(t,x), \, \, \, j = 1,2\\ \chi_{2}^{\ve} \ue{3}& \scale& \chi_{2}(y) U_{3}(t,x,y) \\ \chi_{j}^{\ve} \nabla \ue{j} & \scale & \chi_{j}(y) (\nabla_{x} u_{j}(t,x)+ \nabla_{y} U_{j}(t,x,y)), \, \, \,j=1,2\\ \ve \chi_{2}^{\ve} \nabla \ue{3} & \scale & \chi_{2}(y) \nabla_{y} U_{3}(t,x,y)\\ \chi_{j}^{\ve} \mu_{j}(\xe, \nabla \ue{j})&\scale& \chi_{j}(y) \mu_{j}(y, \nabla_{x} u_{j} + \nabla_{y} U_{j}), \, \, \, j = 1,2\\ \chi_{2}^{\ve} \mu_{3}(\xe, \ve \nabla \ue{3})&\scale& \chi_{2}(y) \mu_{3}(y, \nabla_{y} U_{3})\\ \chi_{j}^{\ve} \ue{j}(T,x)& \scale & \chi_{j}(y) u_{j}(T,x), \, \, \, j=1,2\\ \chi_{2}^{\ve} \ue{3}(T,x) &\scale & \chi_{2}(y) U_{3}(T,x, y) \mbox{ and }\\ u_{1}(t,x)& = &\alpha u_{2}(t,x) + \beta U_{3}(t,x, y) \mbox{ for all } y \in \Gamma_{1,2}.\ \ \ \Box \end{eqnarray*} \end{prop} \begin{prop} The functions $u_{1}, u_{2}, U_{1}, U_{2}, U_{3}$ satisfy the homogenized system \begin{equation} \begin{array}{l} {\displaystyle - \sum_{j=1}^{2} \intot \!\intw \!\intn{j} \, c_{j}(y) u_{j} \dert{\phi_{j}} \, dy \, dx \, dt - \intot \!\intw \!\intn{2}\, c_{3}(y) U_{3}\, \dert{\Phi_{3}}\, dy \, dx \, dt} \\ {\displaystyle -\sum_{j=1}^{2} \intw \!\intn{j} c_{j}(y) u_{j}^{0} \, \phi_{j}(0,x)\, dy \, dx - \intw \!\intn{2} \, c_{3}(y) u_{3}^{0} \, \Phi_{3}(0,x, y) \, dy \, dx} \\ {\displaystyle + \sum_{j=1}^{2} \intot\! \intw \!\intn{j} \, \mu_{j}(y, \nabla_{x} u_{j} + \nabla_{y} U_{j}). (\nabla_{x} \phi_{j} + \nabla_{y} \Phi_{j}) \, dy \, dx \, dt} \\ {\displaystyle + \intot \!\intw \!\intn{2} \, \mu_{3}(y, \nabla_{y} U_{3}) .(\nabla_{y} \Phi_{3}) \, dy\, dx \, dt}= 0 \end{array} \end{equation} for all \begin{eqnarray*} \phi_{j}(t,x) &\in& L^{p}([0,T]; \, W^{1,p}(\Omega)), \, \, j=1,2 \\ \Phi_{j}(t,x,y) &\in& L^{p}([0,T] \times \Omega ;W^{1,p}_{\sharp}(Y_{j})), \, \,j=1, 2, 3 \end{eqnarray*} satisfying \begin{eqnarray*} \dert{\phi_{j}} & \in& L^{q}([0,T]; W^{-1,q}(\Omega)), \, \, j=1,2 \\ \dert{\Phi_{j}} &\in& L^{q}([0,T] \times \Omega; (W^{1,p}_{\sharp}(Y_{j}))^{'}), \, \, j=1,2,3 \end{eqnarray*} \begin{equation} \beta \Phi_{3}(t,x, y) = \phi_{1}(t,x) - \alpha \phi_{2}(t,x) \mbox { for all } y \in \Gamma_{1,2} \mbox{ and, }\nonumber \end{equation} \begin{equation} \phi_{1}(T,x) = \phi_{2}(T,x) = \Phi_{3}(T,x, y) = 0. \nonumber \ \ \ \Box \end{equation} \end{prop} The strong form of the homogenized problem has the following description. Define the {\em state space}, \begin{eqnarray*} H \equiv \leb{\Omega} \times \leb{\Omega} \times \leb{\Omega \times Y_{2}} \end{eqnarray*} with the scalar product \begin{eqnarray*} (\ora{\psi},\ora{\phi})_{H}&= &\sum_{j=1}^{2} \intw \! \int_{Y_{j}} \, c_{j}(y) \psi_{j}(x)\phi_{j}(x) \, dy \, dx\\ & &+\intw \!\int_{Y_{2}} \, c_{3}(y) \Psi_{3}(x,y) \Phi_{3}(x,y) \, dy \, dx \end{eqnarray*} for every $\ora{\psi}=[\psi_{1},\psi_{2}, \Psi_{3}], \ora{\phi}=[\phi_{1},\phi_{2},\Phi_{3}] \in H$. Define the {\em energy space}, \begin{eqnarray*} B \equiv \{ [\phi_{1},\phi_{2}, \Phi_{3}] \in& H \cap \wpe{\Omega} \times \wpe{\Omega} \times \leb{\Omega; W^{1,p}_{\sharp}(Y_{2})/R}\\ :& \beta \Phi_{3}(x, y) = \phi_{1}(x) - \alpha \phi_{2}(x,y) \mbox{ for all } y \in \Gamma_{1,2} \} \end{eqnarray*} and the corresponding {\em evolution space} $V \equiv \L^{p}([0,T];B)$. \begin{prop} $\ora{u} = [u_{1}, u_{2}, U_{3}] \in V$ and is the solution of the strong homogenized system, \begin{eqnarray} (\int_{Y_{1}} \, c_{1}(y) \, dy) \der{1}(t,x) &+ & \frac{1}{\beta} \dert{ }(\int_{Y_{2}} \, c_{3}(y) U_{3}(t,x,y)\, dy ) \nonumber \\ & = &\operatorname{div}_{x}( \int_{Y_{1}} \mu_{1}(y, \nabla_{x} u_{1} + \nabla_{y} U_{1}) \, dy) \label{sh1} \end{eqnarray} \begin{eqnarray} (\int_{Y_{2}} \, c_{2}(y) \, dy) \der{2}(t,x) & - & \frac{\alpha}{\beta} \dert{ }(\int_{Y_{2}} \, c_{3}(y) U_{3}(t,x,y)\, dy ) \nonumber \\ & = &\operatorname{div}_{x}( \int_{Y_{2}} \mu_{2}(y, \nabla_{x} u_{2} + \nabla_{y} U_{2}) \, dy) \label{sh2} \end{eqnarray} \begin{eqnarray} c_{3}(y) \frac{\partial U_{3}(t,x,y)}{\partial t} - \operatorname{div}_{y} \mu_{3}(y, \nabla U_{3}(t,x,y))=0 \end{eqnarray} where $U_{3}(t,x,y) \mbox{ and } \mu_{3}(y, \nabla_{y} U_{3}(t,x,y)). \nu$ are Y-periodic and, \begin{eqnarray} \beta U_{3}(t,x,y) = u_{1}(t,x) - \alpha u_{2}(t,x) \mbox{ for } y \in \Gamma_{1,2} \end{eqnarray} with {\em boundary conditions} \begin{eqnarray} \int_{Y_{1}} \, \mu_{1}(y, \nabla_{x} u_{1} + \nabla_{y} U_{1}) \, dy . \nu & = & 0 \mbox{ on } \partial \Omega \label{bdy4}\\ \int_{Y_{2}} \, \mu_{2}(y, \nabla_{x} u_{2} + \nabla_{y} U_{2}) \, dy . \nu & = & 0 \mbox{ on } \partial \Omega \label{bdy5} \end{eqnarray} and {\em initial conditions} \begin{equation} u_{j}(0,x) = u_{j}^{0}(x) \, \, j=1,2; \, U_{3}(0,x,y)= u_{3}^{0}(x). \end{equation} The functions $U_{j}(t,x,y)$ solve the {\em cell problems}, \begin{equation} \operatorname{div}_{y} \mu_{j}(y, \nabla_{x} u_{j}(t,x)+ \nabla_{y} U_{j}(t,x,y))= 0 \mbox{ for } y \in Y_{j} \label{cell1} \end{equation} \begin{equation} \mu_{j}(y, \nabla_{x} u_{j}(t,x) + \nabla_{y} U_{j}(t,x,y)). \nu = 0 \mbox{ on } \Gamma_{1,2} \mbox{ and } \label{cell2} \end{equation} $Y$-periodic on $\Gamma_{2,2}$, for $j =1,2$. In the above, $ t, x$ are treated as parameters and the cell equations are solved.$\Box$ \end{prop} For $\xi \in R^{N}$, define the following functions; \begin{equation} \lambda_{j}({\xi}) = \intn{j} \, \mu_{j}(y, \xi + \nabla_{y} V_{j}^{\xi}(y) \, dy, \, \, j=1,2 \end{equation} where $V_{j}^{\xi}$ is the Y-periodic solution of \begin{eqnarray} \operatorname{div}_{y} \mu_{j}(y, \xi+ \nabla_{y} V_{j}^{\xi}(y))& =& 0 \mbox{ in } Y_{j}\\ \mu_{j}(y, \xi + \nabla_{y} V_{j}^{\xi}(y)). \nu& =& 0 \mbox{ on } \Gamma_{1,2} \end{eqnarray} Then, because of (\ref{cell1}), (\ref{cell2}), the righthandsides in (\ref{sh1}), (\ref{sh2}) can be replaced by the functions $\operatorname{div}_{x}\lambda_{1}(\nabla_{x} u_{1}(t,x))$ and ${\rm div}_{x}\lambda_{2}(\nabla_{x} u_{2}(t,x))$ respectively. Also the lefthandsisdes of (\ref{bdy4}), (\ref{bdy5}) can be replaced by $\lambda_{1}(\nabla_{x} u_{1}). \nu$ and $\lambda_{2}(\nabla_{x} u_{2}). \nu$ respectively. \smallskip \noindent {\bf Remark 3.1:} We note that the functions $\lambda_{j}$ can be interpreted as the integrands in the $\Gamma- \mbox{limit}$ of the functionals $$ F_{j,\ve}(\nabla v) = \intw \, \chi_{j}^{\ve} \mu_{j}(\xe, \nabla v) \, dx \,.$$ In fact, $\Gamma-\lim F_{j,\ve}(\nabla v) = \intw \, \lambda_{j}(\nabla v) \, dx \, $(cf. DalMaso~\cite{dm}). Further, the functions $\lambda_{j}, \, \, j=1,2$ satisfy conditons (\ref{growth})-(\ref{coercivity}) for the same $p$ but maybe for different constants $\tilde{k}, \tilde{C}, \tilde{c_{0}}$ (cf.~\cite{dmdf},~\cite{cddf}). $\Box$ \begin{prop} The following {\em energy identity} holds (cf.~\cite{cs}), \begin{eqnarray*} \frac{1}{2} \sum_{j=1}^{2} \intw \!\intn{j} \,c_{j}(y) |u_{j}(T,x)|^{2} \, dy \, dx + \frac{1}{2}\intw \!\intn{2} \, c_{3}(y) |U_{3}(T, x, y)|^{2} \, dy \, dx & &\\ - \frac{1}{2} \sum_{j=1}^{2} \intw \!\intn{j} \,c_{j}(y) |u_{j}^{0}(x)|^{2} \, dy \, dx - \frac{1}{2}\intw \!\intn{2} \, c_{3}(y) |u_{3}^{0}(x)|^{2} \, dy \,dx & &\\ +\sum_{j=1}^{2} \intot \!\intw \!\intn{j} \, \mu_{j}(y,\nabla_{x} u_{j} +\nabla_{y} U_{j}).(\nabla_{x} u_{j} + \nabla_{y} U_{j}) \, dy \, dx \, dt& &\\ + \intot \!\intw \!\intn{2} \, \mu_{3}(y, \nabla_{y} U_{3}). \nabla_{y} U_{3} \, dy \, dx \, dt = 0. \end{eqnarray*} \end{prop} \section{Correctors} \addtocounter{equation}{-14} We now prove corrector results for the gradient of flows under stronger hypotheses on $\mu_{j}$'s than (\ref{growth})-(\ref{coercivity}). Let $k_{1}, k_{2} > 0$ be constants and assume for j=1,2, 3: \begin{equation} \mu_{j}(\, . \, \xi) \mbox{ is measurable and Y-periodic for all } \xi \in R^{N} \label{per} \end{equation} For $\xi, \eta \in R^{N}$ with $|\xi|+|\eta|>0$ and a.e. $y \in Y$, \begin{eqnarray} \mu_{j}(y,0)&=&0 \label{at0}\\ |\mu_{j}(y,\xi) - \mu_{j}(y, \eta)|&\le& k_{1}(|\xi|+|\eta|)^{p-2}|\xi -\eta| \label{lipcty}\\ (\mu_{j}(y,\xi) - \mu_{j}(y, \eta)).(\xi -\eta)&\ge& k_{2}(\abs{\xi} + \abs{\eta})^{p-2}\abs{\xi -\eta}^{2} \label{sm} \end{eqnarray} The above hypotheses will, henceforth, be known as ({\bf H}).\\ {\bf Remark 4.1: } Note that (\ref{at0}) and (\ref{lipcty}) imply \begin{eqnarray} \abs{\mu_{j}(y, \xi)} \le k_{1} \abs{\xi}^{p-1}\label{lip*} \end{eqnarray} and, (\ref{at0}) and (\ref{sm}) imply \begin{eqnarray} \mu_{j}(y,\xi). \xi \ge k_{2} \abs{\xi}^{p}.\label{sm*} \end{eqnarray} Thus, the new hypotheses are indeed stronger than the original hypotheses on $\mu_{j}$'s. Moreover, \begin{eqnarray} (\mu_{j}(y, \xi) -\mu_{j}(y, \eta)).(\xi -\eta) &\ge& k_{2} \abs{\xi - \eta}^{p} \label{i1} \, \, \mbox{ if } p \ge 2\\ \abs{\mu_{j}(y, \xi) - \mu_{j}(y, \eta)} & \le & k_{1} \abs{\xi - \eta}^{p-1} \mbox{ if } 1 < p < 2.\label{i2} \end{eqnarray} These inequalities follow from (\ref{sm}) and (\ref{lipcty}) and triangle inequality in $R^{N}. \, \, \Box$ \smallskip \noindent {\bf Remark 4.2:} An example of $\mu_{j}$ satisfying (\ref{at0})- (\ref{sm}) is $\mu_{j} = \abs{\xi}^{p-2} \xi$, i.e. the corresponding diffusion operator is the p-Laplacian. More generally, let ${\bf C}$ denote the class of functions $$f \in C^{0}(\overline{\Omega} \times R^{N};R^{N}) \cap C^{1}(\Omega \times R^{N}\setminus\{0\};R^{N})$$ which satisfy condition (\ref{at0}) and the following \begin{eqnarray*} \sum_{j,j =1}^{N} \abs{\frac{\partial f_{j}}{\partial \eta_{i}}}(x, \eta) & \le & \Gamma \abs{\eta}^{p-2} \\ \sum_{j,j=1}^{N} \abs{\frac{\partial f_{j}}{\partial \eta_{i}}} (x, \eta) \xi_{i} \xi_{j} & \ge & \gamma \abs{\eta}^{p-2} \abs{\xi}^{2} \end{eqnarray*} for all $ x \in \Omega, \eta \in R^{N}\setminus \{0\} \mbox{ and } \xi \in R^{N}$ and $\Gamma, \gamma$ are positive constants. Then for $\mu_{j}$ 's in the class ${\bf C}$ the conditions ({\bf H}) are satisfied (cf. Damascelli~\cite{ld}).$\Box$ Let $\ue{1},\ue{2},\ue{3}$ be the solution of the Cauchy problem (\ref{ff})- (\ref{in}) and let \newline $[u_{1},u_{2}, U_{1},U_{2}, U_{3}]$ be as in Section 3. We will denote $\ot$ by $\Omega_{T}$. Define the sequence of functions \begin{eqnarray} \xi_{j}(t,x,y) & \equiv &\chi_{j}(y) ( \nabla_{x} u_{j}(t,x) + \nabla_{y} U_{j} (t,x,y)), \, \, j =1,2,\\ \xi_{3}(t,x,y) & \equiv & \chi_{2}(y) \nabla_{y} U_{3} (t,x,y) \end{eqnarray} and let, \begin{equation} \xi_{j}^{\ve}(t,x) \equiv \xi_{j}(t,x,\xe), \, \, j=1,2,3. \end{equation} Our main theorems are the following: \begin{theo} If the functions, $\nabla_{y} U_{j},\ j=1,2,3$ are admissible (cf. Definition 2.1), then \begin{eqnarray*} \varlimsup_{\eo}\norm{\chi_{j}(\xe)\left(\nabla \ue{j}(t,x)- \xi_{j}^{\ve}(t,x) \right) }_{p, \Omega_{T}} & \ra& 0 \mbox{, j=1,2,} \\ \varlimsup_{\eo} \norm{\chi_{2}(\xe)\left( \ve \nabla \ue{3}(t,x) -\xi_{3}^{\ve}(t,x) \right)}_{p,\Omega_{T}} & \ra& 0. \end{eqnarray*} \end{theo} \begin{theo} If the functions, $\nabla_{y} U_{j}, \ j=1,2,3$ are admissible (cf. Definition 2.1), then \begin{eqnarray*} \varlimsup_{\eo} \norm{\chi_{j}(\xe)\left( \mu_{j} \left(\xe, \nabla \ue{j}\right) - \mu_{j}\left( \xe,\xi_{j}^{\ve}(t,x) \right) \right)}_{q,\Omega_{T}} & \ra & 0 \mbox{, j=1,2, }\\ \varlimsup_{\eo} \norm{\chi_{2}(\xe)\left(\mu_{3}\left(\xe, \ve \nabla \ue{3}\right) - \mu_{3}\left(\xe,\xi_{3}^{\ve}(t,x) \right) \right)}_{q,\Omega_{T}} & \ra & 0. \end{eqnarray*} \end{theo} \noindent {\bf Remark 4.3: } Theorem 4.1 gives strong convergence of the gradients of the flow of the Cauchy problem to the gradients of the flow of the homogenized problem by adding a corrector, whereas the gradients of the flow of the Cauchy problem, themselves, only weakly converge to the gradients of the flow of the homogenized problem in $L^{p}$. Theorem 4.2 gives an analogous result for the flux terms.$\Box$\par We first calculate some limits and prove some estimates in order to prove the theorems. For that we need some more notations, functions and quantities which we will use hereafter. We will use M to denote a generic constant which does not depend on $\ve$, but probably on $p, k_{1},k_{2}, c_{0}, C$, and the $L^{2}$ norm of the initial vector $\ora{u^{0}}$. We will also set $\om{3} \equiv \om{2}$ and $ Y_{3} \equiv Y_{2}$. Let $0 < \kappa < 1$ be a constant and $\Phi_{j}(t,x,y)$ be admissible test functions such that $$\sum_{j=1}^{3}\norm{\nabla_{y} U_{j}- \Phi_{j}}^{p}_{p, \ot \times Y_{j}} \le \kappa.$$ Note that, $$\Phi_{j}(t,x,\xe) \scale \Phi_{j}(t,x,y)$$ for j=1,2,3. Define the functions: \begin{eqnarray} \eta_{j}^{\ve}(t,x) & = & \chi_{j}(\xe)(\nabla_{x}u_{j}(t,x)+ \Phi_{j} (t,x,\xe)), \, \,j=1, 2\label{def1}\\ \eta_{3}^{\ve}(t,x) & = & \chi_{2}(\xe) \Phi_{3}(t,x,\xe).\label{def2} \end{eqnarray} Then we note that the functions $\eta_{j}^{\ve}(t,x)$ and $ \mu_{j}^{\ve}(\xe, \eta_{j}^{\ve}(t,x))$ arise from admissible test functions and we have the following two-scale convergence (cf.~\cite{cs}), \begin{eqnarray*} \eta_{j}^{\ve} &\scale& \eta_{j}(t,x,y) \equiv \chi_{j}(y)(\nabla_{x} u_{j}(t,x)+ \Phi_{j}(t,x,y)), \, \, \, j=1,2,\\ \eta_{3}^{\ve} &\scale& \eta_{3}(t,x,y) \equiv \chi_{2}(y) \Phi_{3}(t,x,y)\\ \mu_{j}(\xe, \eta_{j}^{\ve}) & \scale & \chi_{j}(y) \, \mu_{j}(y\, ,\, \eta_{j}(t,x,y)), \, \, j=1,2,3. \end{eqnarray*} \begin{lem}(cf. ~\cite{dmdf}, lemma 3.1.) Let $1

0 \}$$ \end{lem} \noindent {\bf Proof: } Multiply and divide the integrand in left hand side by $(\abs{\phi_{1}}+\abs{\phi_{2}})^{(2-p)p/2}$ and apply H$\ddot{o}$lder's inequality to get the result. $\Box$ \begin{lem} \begin{eqnarray*} \sum_{j=1}^{2}\norm{\chi_{j}(y)(\nabla_{x} u_{j} + \nabla_{y} U_{j})}^{p}_{p} + \norm{\chi_{2}(y) \nabla_{y} U_{3}}^{p}_{p} \le \frac{C}{2k_{2}} \sum_{j=1}^{3} \norm{u_{j}^{0}}^{2}_{2,\Omega} \nonumber \end{eqnarray*} \end{lem} \noindent {\bf Proof: } Follows from the energy identity (Proposition 3.5) and (\ref{sm*}). $\Box$ \begin{lem} Let $ \xi_{i}, \eta_{i}, \xi_{i}^{\ve}, \eta_{i}^{\ve}, i =1,2,3$ be functions as defined above. Then, \begin{eqnarray*} \lefteqn{\varlimsup_{\eo} \intot \!\int_{\om{i}} \left(\mu_{i}(\xe,\nabla \ue{i}) - \mu_{i}(\xe, \eta_{i}^{\ve})\right). (\nabla \ue{i} - \eta_{i}^{\ve})\, dx \, dt}\\ &\le \sum_{j=1}^{3} \intot \!\intw \! \int_{Y_{j}} \left( \mu_{j}(y, \xi_{j}) - \mu_{j}(y,\eta_{j})\right) .(\xi_{j} -\eta_{j})\, dy \, dx \, dt \end{eqnarray*} for i=1,2 and \begin{eqnarray*} \lefteqn{ \varlimsup_{\eo} \intot \!\int_{\om{2}} \left(\mu_{3}(\xe,\ve \nabla \ue{3}) - \mu_{3}(\xe, \eta_{3}^{\ve})\right). (\ve \nabla \ue{3} - \eta_{3}^{\ve})\, dx \, dt}\\ &\le \sum_{j=1}^{3} \intot \!\intw \! \int_{Y_{j}} \left( \mu_{j}(y, \xi_{j}) - \mu_{j}(y,\eta_{j})\right) .(\xi_{j} -\eta_{j})\, dy \, dx \, dt \end{eqnarray*} \end{lem} \noindent {\bf Proof: } Denote the integrals appearing in the left-handsides of the above relations by $l_{1}^{\ve}, l_{2}^{\ve} \mbox{ and } l_{3}^{\ve}$ respectively. Then for i=1,2,3, using (\ref{id}), we obtain, \begin{eqnarray*} l_{i}^{\ve} &\le&\sum_{j=1}^{3}l_{j}^{\ve} \\ & = & \frac12 \sum_{j=1}^{3} \int_{\om{j}} c_{j}(\xe) \abs{u_{j}^{0}(x)}^{2} \, dx -\frac12 \sum_{j=1}^{3}\int_{\om{j}} c_{j}(\xe) \abs{u_{j}^{\ve}(T,x)}^{2}\, dx\\ & & -\sum_{j=1}^{2} \intot \!\int_{\om{j}} \mu_{j}(\xe, \eta_{j}^{\ve}).(\nabla \ue{j} - \eta_{j}^{\ve})\, dx\, dt\\ & & -\intot \!\int_{\om{j}} \mu_{3}(\xe, \eta_{3}^{\ve}).(\ve \nabla \ue{3} - \eta_{3}^{\ve})\, dx\, dt\\ & &-\sum_{j=1}^{2} \intot \!\int_{\om{j}} \mu_{j}(\xe, \nabla \ue{j}).\eta_{j}^{\ve} \, dx \, dt - \intot \! \int_{\om{2}} \mu_{3}(\xe, \ve \nabla \ue{3}).\eta_{3}^{\ve} \,dx\, dt \end{eqnarray*} We now use the two-scale convergence properties of various functions discussed so far to pass to the limit. We get, \begin{eqnarray*} \varlimsup_{\eo} \sum_{j=1}^{3} l_{j}^{\ve} &= & \frac12 \sum_{j=1}^{3}\intw \! \int_{Y_{j}} c_{j}(y) \abs{u_{j}^{0}(x)}^{2}\, dy \, dx\\ & &-\varliminf_{\eo}\frac12 \sum_{j=1}^{3} \int_{\om{j}} c_{j}(\xe) \abs{u_{j}^{\ve}(T,x)}^{2}\, dx \\ & &-\sum_{j=1}^{3} \intot \!\intw \!\int_{Y_{j}} \mu_{j}(y, \eta_{j}).(\xi_{j} -\eta_{j})\, dy \, dx \, dt\\ & &-\sum_{j=1}^{3} \intot \!\intw \!\int_{Y_{j}} \mu_{j}(y, \xi_{j}).\eta_{j}\, dy \, dx \, dt \end{eqnarray*} The right hand side can be written as \begin{eqnarray*} \frac12 \sum_{j=1}^{3} \intw \!\int_{Y_{j}} c_{j}(y) \abs{u_{j}^{0}(x)}^{2}\, dy \, dx -\varliminf_{\eo}\frac12 \sum_{j=1}^{3} \int_{\om{j}} c_{j}(\xe) \abs{u_{j}^{\ve}(x,T)}^{2}\, dx & &\\ +\sum_{j=1}^{3} \intot \!\intw \!\int_{Y_{j}} \left( \mu_{j}(y, \xi_{j}) - \mu_{j}(y,\eta_{j})\right) .(\xi_{j} -\eta_{j})\, dy \, dx \, dt & &\\ -\sum_{j=1}^{3} \intot \!\intw \!\int_{Y_{j}} \mu_{j}(y, \xi_{j}) .\xi_{j} \, dy \, dx \, dt & & \end{eqnarray*} which, using Proposition 3.5 to replace the last expression, is nothing but, \begin{eqnarray*} \frac12 \sum_{j=1}^{2} \intw \!\int_{Y_{j}} c_{j}(y) \abs{u_{j}(T,x)}^{2}\, dy \, dx +\frac12 \intw \!\int_{Y_{2}} c_{3}(y) \abs{U_{3}(T,x,y)}^{2}\, dy \, dx & &\\ -\varliminf_{\eo}\frac12 \sum_{j=1}^{3} \int_{\om{j}} c_{j}(\xe) \abs{u_{j}^{\ve}(x,T)}^{2}\, dx & &\\ +\sum_{j=1}^{3} \intot \!\intw \!\int_{Y_{j}} \left( \mu_{j}(y, \xi_{j}) - \mu_{j}(y,\eta_{j})\right) .(\xi_{j} -\eta_{j})\, dy \, dx \, dt \end{eqnarray*} However, by standard arguments, \begin{eqnarray*} \sum_{j=1}^{2} \intw \!\int_{Y_{j}} c_{j}(y) \abs{u_{j}(T,x)}^{2}\, dy \, dx +\intw \!\int_{Y_{2}} c_{3}(y) \abs{U_{3}(T,x,y)}^{2}\, dy \, dx & &\\ \le \varliminf_{\eo} \sum_{j=1}^{3} \int_{\om{j}} c_{j}(\xe) \abs{u_{j}^{\ve}(x,T)}^{2}\, dx & & \end{eqnarray*} This completes the proof. $\Box$ \begin{lem} Let $\xi_{j},\eta_{j}, \kappa$ be as before. Then, \begin{eqnarray*} \sum_{j=1}^{3} \intot \!\intw \! \int_{Y_{j}} \left( \mu_{j}(y, \xi_{j}) - \mu_{j}(y,\eta_{j})\right) .(\xi_{j} -\eta_{j})\, dy \, dx \, dt \le M \kappa^{\delta(p)} \end{eqnarray*} where $$\delta(p) = \left\{ \begin{array}{ll} 1 & \mbox{ if } 1