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\begin{document} 
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.\ 1999(1999), No.~02, pp. 1--20.\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{1.5cm}

\title[\hfilneg EJDE--1999/02\hfil flow in a partially fissured medium]
{Two-scale convergence of a model for flow in a partially fissured medium} 

\author[G. W. Clark \& R. E. Showalter \hfil EJDE--1999/02\hfilneg]
{G. W. Clark \& R.E. Showalter}

\address{G. W. Clark \hfill\break
Department of Mathematical Sciences \\
Virginia Commonwealth University \hfill\break
Richmond, VA 23284 USA}
\email{gwclark@saturn.vcu.edu}

\address{R. E. Showalter \hfill\break
Department of Mathematics\\
University of Texas at Austin \hfill\break
Austin, TX 78712 USA}
\email{show@math.utexas.edu}

\date{}
\thanks{Submitted October 28, 1998. Published January 14, 1999.}
\subjclass{35A15, 35B27, 76S05}
\keywords{fissured medium, homogenization, two-scale convergence, 
\hfill\break\indent
dual permeability, modeling, microstructure}


\begin{abstract}
The distributed-microstructure model for the flow of single
phase fluid in a partially fissured composite medium due to
Douglas-Peszy\'{n}ska-Showalter \cite{D-P-S} is extended to a
quasi-linear version. This model contains the geometry of the local
cells distributed throughout the medium, the flux exchange across
their intricate interface with the imbedded fissure system, and the
secondary flux resulting from diffusion paths within the matrix. Both
the exact but highly singular micro-model and the macro-model are
shown to be well-posed, and it is proved that the solution of the
micro-model is two-scale convergent to that of the macro-model as the
spatial parameter goes to zero. In the linear case, the effective
coefficients are obtained by a partial decoupling of the homogenized
system.
\end{abstract}

\maketitle

\theoremstyle{definition}
\theoremstyle{remark}
\numberwithin{equation}{section}
\newcommand{\thmref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lemref}[1]{Lemma~\ref{#1}}
\def\Bbb{\mathbb}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{axiom}{Axiom}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{exercise}{Exercise}[section]
\newtheorem{definition}{Definition}[section]


\section{Introduction}

A fissured medium is a structure consisting of a porous and permeable
matrix which is interlaced on a fine scale by a system of highly
permeable fissures. The majority of fluid transport will occur along
flow paths through the fissure system, and the relative volume and
storage capacity of the porous matrix is much larger than that of the
fissure system. When the system of fissures is so well developed that
the matrix is broken into individual blocks or cells that are isolated
from each other, there is consequently no flow directly from cell to
cell, but only an exchange of fluid between each cell and the
surrounding fissure system. This is the \textit{totally fissured} case
that arises in the modeling of granular materials. In the more general
\textit{partially fissured} case of composite media, not only the
fissure system but also the matrix of cells may be connected, so there
is some flow directly within the cell matrix. The developments below
concern this more general model with the additional component of a
global flow through the matrix.

An exact microscopic model of flow in a fissured medium treats the
regions occupied by the fissure system and by the porous matrix as two
Darcy media with different physical parameters. The resulting
discontinuities in the parameter values across the matrix-fissure
interface are severe, and the characteristic width of the fissures is
very small in comparison with the size of the matrix
blocks. Consequently, any such exact microscopic model, written as a
classical interface problem, is numerically and analytically
intractable.  For the case of a totally fissured medium, these
difficulties were overcome by constructing models which describe the
flow on two scales, macroscopic and microscopic; see \cite{arbo,
adhformal, adhsingle, uhrs, ShoWa}.  A macro-model for flow in a
totally fissured medium was obtained as the limit of an exact
micro-model with properly chosen scaling of permeability in the porous
matrix.  It is an example of a distributed microstructure model.
Derivations of these two--scale models have been based on averaging over
the exact geometry of the region (see \cite{arbo, tamfw}) or by
the construction of a continuous distribution of blocks over the region as
in \cite{ShoWa} or by assuming some periodic structure for the
domain that permits the use of homogenization methods \cite{B-L-M,
ClarkMSM}. (See \cite{Hornung} or \cite{mpthesis} for a
review, and for more information on homogenization see \cite{B-L-P,
S-P}.) This model was extended in \cite{D-P-S} to the partially
fissured case. The novelty in this construction was to represent the flow
in the matrix by a parallel construction in the style of
\cite{B-Z-K, W-R}. Thus, two flows are introduced in the exact
micro-model for the matrix, one is the slow scale flow of
\cite{adhsingle} which leads to local storage, and the additional
one is the global flow within the matrix. A formal asymptotic expansion was
used in \cite{D-P-S} to derive the corresponding distributed
microstructure model. See \cite{Clark, Cook} for another approach
to modeling flow in a partially fissured medium and \cite{Hornung}
for further discussion and related works. Here we extend the considerations
to a quasi-linear version, and we use two-scale convergence to prove the
convergence of the micro-model to the corresponding macro-model.

Our plan for this project is as follows. In the remainder of this
section, we briefly recall the partial differential equations that
describe the flow through a homogeneous medium in order to introduce
some notation. Then we describe in turn various function spaces of
$L^p$ or of Sobolev type, the two-scale convergence procedure, and
basic results for weak and strong formulations of the Cauchy problem
in Banach space.  In Section 2 we describe a nonlinear version of the
\textit{micro-model} from \cite{D-P-S} for flow through a partially
fissured medium and show that this system leads to a well-posed
initial-boundary-value problem. In Section 3 we show that this
micro-model has a two-scale limit as the parameter $\varepsilon
\rightarrow 0$, and this limit satisfies a variational identity. The
point of Section 4 is to establish that this limit satisfies
additional properties which collectively comprise the homogenized
\textit{macro-model}. These results on the well-posedness of the
macro-model are sumarized and completed in Section 5.  There we relate
the weak and strong formulations of the macro-model problem to the
corresponding realizations as a Cauchy problem for a nonlinear
evolution equation in Banach space. We also develop a simpler and
useful reduced system to describe this limit, and we show that it agrees
with the usual homogenized model from \cite{D-P-S} in the linear case.

The authors would like to acknowledge the considerable benefit
obtained from discussions with M. Peszy\'{n}ska \cite{D-P-S,
mpthesis, mpjyv, mpmem, mpnif, mpfem} on the homogenization method for
modeling of flow through porous media . These led to many substantial
improvements in the manuscript.

We begin with a review of notation in the context of the flow of a
single phase slightly compressible liquid through a
\textit{homogeneous medium}. Thus the density $\rho (x,t)$ and
pressure $p(x,t)$ are related by the \textit{state equation} $\rho
=\rho _{0}e^{\kappa p}$, and the equation for conservation of mass is
given by
\begin{equation*}
c(x)\frac{\partial \rho }{\partial t}-\nabla \cdot \sum_{j=1}^{N}(\rho k_{j}(\rho \frac{\partial p}{\partial x_{j}})\frac{\partial p}{\partial x_{j}})=f(x,t).
\end{equation*}
The state equation yields the relationship $\frac{\partial \rho }{\partial
x_{j}} =\kappa \rho \frac{\partial p}{\partial x_{j}}$, so the conservation
equation can be written 
\begin{equation*}
c(x)\frac{\partial \rho }{\partial t}
-\nabla \cdot \sum_{j=1}^{N}(k_{j}(\frac{1}{\kappa } 
\frac{\partial \rho }{\partial x_{j}})\frac{1}{\kappa }\frac{\partial \rho }{
\partial x_{j}})=f(x,t).
\end{equation*}
Finally, by introducing the \textit{flow potential} $u(w)=\int_{0}^{w}\rho
\,dp$, we have 
\begin{equation*}
c(x)\frac{\partial u}{\partial t}
-\nabla \cdot \mu (\nabla u)=f(x,t),
\end{equation*}
where the \textit{flux} is given componentwise by the negative of the
function 
$\mu (\nabla u) \equiv 
\frac{1}{\kappa }\sum_{j=1}^{N}k_{j}( \frac{\partial u}{ \partial
x_{j}})\frac{\partial u}{\partial x_{j}}$. We shall assume below that
this is a monotone function of the gradient.  The classical
Forchheimer-type corrections to the Darcy law for fluids lead to such
functions with growth of order $p=\frac{3}{2}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Various spaces of functions on a bounded (for simplicity) domain
$\Omega $ in $\Bbb{R}^{N}$ with smooth boundary $\partial \Omega
\equiv \Gamma$ will be used. For each $1<p<\infty $, $L^{p}(\Omega )$ is
the usual \textit{ Lebesgue space} of (equivalence classes of) $p$-th
power summable functions, and $W^{1,p}(\Omega )$ is the
\textit{Sobolev space} of functions which belong to $L^{p}(\Omega )$
together with their first order derivatives. The \textit{trace} map
$\gamma :\,W^{1,p}(\Omega )\rightarrow L^{p}(\Gamma )$ is the
restriction to boundary values.

Let $Y=[0,1]^{N}$ denote the unit cube. Corresponding spaces of \textit{$Y$
-periodic} functions will be denoted by a subscript $\#$. For example, $
C_{\#}(Y)$ is the Banach space of functions which are defined on all of $
\Bbb{R}^{N}$ and which are continuous and $Y$-periodic. Similarly, $
L_{\#}^{p}(Y)$ is the Banach space of functions in $L_{\text{loc}}^{p}(\Bbb{R
}^{N})$ which are $Y$-periodic. For this space we take the norm of $L^{p}(Y)$
and note that $L_{\#}^{p}(Y)$ is equivalent to the space of $Y$-periodic
extensions to $\Bbb{R}^{N}$ of the functions in $L^{p}(Y)$. Similarly, we
define $W_{\#}^{1,p}(Y)$ to be the Banach space of $Y$-periodic extensions
to $\Bbb{R}^{N}$ of those functions in $W^{1,p}(Y)$ for which the trace (or
boundary values) agree on opposite sides of the boundary, $\partial Y$, and
its norm is the usual norm of $W^{1,p}(Y)$. The linear space $C_{\#}^{\infty
}(Y)\equiv C_{\#}(Y)\cap C^{\infty }(\Bbb{R}^{N})$ is dense in both of $
L_{\#}^{p}(Y)$ and $W_{\#}^{1,p}(Y)$.

Various spaces of \textit{vector-valued} functions will arise in the
developments below. If $ \Bbb{B}$ is a Banach space and $X$ is a
topological space, then $C(X;\Bbb{B}) $ denotes the space of
continuous $\Bbb{B}$-valued functions on $X$ with the corresponding
supremum norm, and for any measure space $\Omega$ we let $
L^p(\Omega;\Bbb{B})$ denote the space of $p$-th power norm-summable
(equivalence classes of) functions on $\Omega$ with values in
$\Bbb{B}$.  When $X = [0,T]$ or $\Omega = (0,T)$ is the indicated time 
interval, we denote the corresponding \textit{evolution spaces} by
$C(0,T;\Bbb{B})$ and $L^p(0,T;\Bbb{B})$, respectively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Next we quote some definitions and results on 
\textit{two-scale convergence} from \cite{Allaire} slightly
modified to allow for homogenization with a parameter (which we denote by $t$
). These changes do not affect the proofs from \cite{Allaire} in any
essential way.

\begin{definition}
A function, $\psi (x,t,y)\! \in L^{p^{\prime}}(\Omega \times (0,T), C_{\#}(Y))$, which is $Y$-periodic in $y$ and which satisfies 
\begin{equation*}
\lim_{\varepsilon \to 0}\int_{\Omega \times (0,T)}\psi \left( x,t,\frac{x}{%
\varepsilon }\right) ^{p^{\prime }}\,dx\,dt=\int_{\Omega \times
(0,T)}\int_{Y}\psi (x,t,y)^{p^{\prime }}\,dy\,dx\,dt,
\end{equation*}
is called an \emph{admissible} test function. Here $p^{\prime }$ is the
conjugate of $p$, that is, $\frac{1}{p}+\frac{1}{p^{\prime }}=1.$
\end{definition}

\begin{definition}
A sequence $u^{\varepsilon }$ in $L^{p}((0,T)\times \Omega )$ \emph{%
two-scale} converges to $u_{0}(x,t,y)\in L^{p}((0,T)\times \Omega \times Y)$
if for any admissible test function $\psi (x,t,y)$, 
\begin{multline}
\lim_{\varepsilon \to 0}\int_{\Omega }\int_{(0,T)}u^{\varepsilon }(x,t)\psi
\left( x,t,\frac{x}{\varepsilon }\right) \,dt\,dx=  \label{E: twoscale} \\
\int_{\Omega }\int_{(0,T)}\int_{Y}u_{0}(x,t,y)\psi (x,t,y)\,dy\,dt\,dx.
\end{multline}
%\newline
\end{definition}

\begin{theorem}
\label{Lpbound} If $u^{\varepsilon }$ is a bounded sequence in $%
L^{p}((0,T)\times \Omega )$, then there exists a function $u_{0}(x,t,y)$ in $%
L^{p}((0,T)\times \Omega \times Y)$ and a subsequence of $u^{\varepsilon }$
which two-scale converges to $u_{0}$. Moreover, the subsequence $%
u^{\varepsilon }$ converges weakly in $L^{p}((0,T)\times \Omega )$ to $%
u(x,t)=\int_{Y}u_{0}(x,t,y)\,dy$.
\end{theorem}

When the sequence, $u^{\varepsilon }$, is $W^{1,p}$-bounded, we get
more information.

\begin{theorem}
\label{W1pbound1} Let $u^{\varepsilon }$ be a bounded sequence in $%
L^{p}(0,T;W^{1,p}(\Omega ))$ that converges weakly to $u$ in $%
L^{p}((0,T);W^{1,p}(\Omega ))$. Then $u^{\varepsilon }$ two-scale converges
to $u$, and there is a function $U(x,t,y)$ in $L^{p}((0,T)\times \Omega
;W_{\#}^{1,p}(Y)/\Bbb{R})$ such that, up to a subsequence, $\nabla
_{x}u^{\varepsilon }$ two-scale converges to $\nabla _{x}u(x,t)+\nabla
_{y}U(x,t,y)$.
\end{theorem}

\begin{theorem}
\label{W1pbound2}Let $u^{\varepsilon }$ and 
$\varepsilon \nabla _{x}u^{\varepsilon }$ be
two bounded sequences in $L^{p}((0,T)\times \Omega )).$ Then there
exists a function $U\left( x,t,y\right) $ in $L^{p}((0,T)\times
\Omega ;W_{\#}^{1,p}(Y)/\Bbb{R})$ such that, up to a subsequence,
$u^{\varepsilon }$ and $\varepsilon \nabla _{x}u^{\varepsilon }$
two-scale converge to $U\left( x,t,y\right) $ 
and $\nabla_{y}U(x,t,y),$ respectively.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Finally, we formulate the \textit{Cauchy problem} or
\textit{initial-value problem} for an evolution equation in Banach
space in a form that will be convenient for our applications below.
Let $V$ be a reflexive Banach space with dual $V'$; we shall set
$\mathcal{V} = L^p(0,T; V)$ for $1 < p < \infty$, and its dual is
$\mathcal{V}'\cong L^{p'} (0,T;V')$.  
Let $V$ be dense and continuously
embedded in a Hilbert space $H$, so that $V \hookrightarrow
H$ and we can identify $H' \hookrightarrow V'$ by restriction.  
\begin{proposition} \label{Wp}
The Banach space $W_p(0,T) \equiv \{u \in \mathcal{V} : u' \in \mathcal{V}'\}$ 
is contained in $C([0,T], H)$.  
Moreover, if $u \in W_p(0,T)$ then $\vert u(\cdot)\vert^2_H$ is absolutely
continuous on $[0,T]$,
\begin{equation*}
{d \over dt} \vert u(t)\vert^2_H = 2 u'\bigl(t)(u(t)\bigr) \quad 
\text{\rm a.e.}   \quad 
t \in [0,T]\ ,
\end{equation*}
and there is a constant $C$ for which
\begin{equation*}
\|u\|_{C([0,T],H)} \le C \|u \|_{W_p(0,T)}\ ,\quad u \in W_p\ .
\end{equation*}
\end{proposition}

\begin{corollary}  
If $u,v \in W_p(0,T)$ then $(u(\cdot),v(\cdot))_H$ 
is absolutely continuous on $[0,T]$ and 
\begin{equation*}
{d \over dt} \bigl(u(t),v(t)\bigr)_H 
= u'\bigl(t)(v(t)\bigr) + v'\bigl(t)(u(t)\bigr)\ ,\qquad \text{\rm a.e.}
\quad t \in [0,T]\ .
\end{equation*}
\end{corollary}

Suppose
we are given a (not necessarily linear) function $\mathcal{A}  : V \to V'$
and $u_0 \in H$, $f \in \mathcal{V}'$.  Then consider the \textit{Cauchy Problem}
to find
\begin{equation}
u \in \mathcal{V}  : u'(t) + \mathcal{A} \bigl(u(t)\bigr) 
= f(t) \quad\text{ in }\ 
\mathcal{V}'\ , \qquad u(0) = u_0 \ \text{\rm in }H\ .\label{sivp}
\end{equation}
It is understood that $u' \in \mathcal{V}'$ in (\ref{sivp}), 
so it follows from Proposition \ref{Wp} that $u$ is continuous into 
$H$ and the condition on $u(0)$ is meaningful.  If $\mathcal{A}$ is known
to map $\mathcal{V} $ into $\mathcal{V}'$,  i.e., the realization of
$\mathcal{A}  : V \to V'$ as an operator on $\mathcal{V} $ has values in $\mathcal{V}'$,
then (\ref{sivp}) is equivalent to the \textit{variational formulation}
\begin{multline}\label{wivp}
u \in \mathcal{V}  : \text{ for every } v \in   \mathcal{V} 
\text{ with }
v' \in \mathcal{V}'\ \text{ and } v(T)= 0 \\
-\int^T_0 \bigl(u(t), v'(t)\bigr)_H\, dt + \int^T_0\mathcal{A} \bigl(u(t)\bigr)
v(t)\, dt = \int^T_0 f(t)v(t)\, dt +\bigl(u_0,v(0)\bigr)_H\ . 
\end{multline}
The equivalence of the strong and variational formulations of the Cauchy
problem will be used freely in all of our applications below. See
Chapter III of \cite{Show:nonlin} for the above and related results on
the Cauchy problem.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Micro-Model}

We consider a structure consisting of fissures and matrix periodically
distributed in a domain $\Omega$ in $\Bbb{R}^{N}$ with period
$\varepsilon Y, $ where $\varepsilon >0$. Let the unit cube
$Y=[0,1]^{N}$ be given in complementary parts, $ Y_{1} $ and $Y_{2}$,
which represent the local structure of the fissure and matrix,
respectively. Denote by $\chi _{j}\left( y\right) $ the characteristic
function of $Y_{j}$ for $j=1,2$, extended $Y$-periodically to all of
$\Bbb{R} ^{N}$. Thus, $\chi _{1}\left( y\right) +\chi _{2}\left(
y\right) =1$. 
We shall assume that both of the sets $\{y\in
\Bbb{R}^{N}:\chi _{j}\left( y\right) =1\}$, $j=1,2$ are smooth. 
%%%%%
With the assumptions that we make on the coefficients below 
to obtain coercivity estimates, it is not necessary to assume 
further that these sets are also connected. 
%%
The domain $\Omega $ is thus
divided into the two subdomains, $\Omega _{1}^{\varepsilon }$ and
$\Omega _{2}^{\varepsilon }$, representing the \emph{fissures} and
\emph{matrix} respectively, and given by
\begin{equation*}
\Omega _{j}^{\varepsilon }\equiv \{x\in \Omega :\chi _{j}\left( \frac{x}{%
\varepsilon }\right) =1\},\quad j=1,\,2.
\end{equation*}
Let $\Gamma _{1,2}^{\varepsilon }\equiv \partial \Omega _{1}^{\varepsilon
}\cap \partial \Omega _{2}^{\varepsilon }\cap \Omega $ be that part of the
interface of $\Omega _{1}^{\varepsilon }$ with $\Omega _{2}^{\varepsilon }$
that is interior to $\Omega $, and let $\Gamma _{1,2}\equiv \partial
Y_{1}\cap \partial Y_{2}\cap Y$ be the corresponding interface in the local
cell $Y$. Likewise, let $\Gamma _{2,2}\equiv \bar{Y}_{2}\cap \partial Y$ and
denote by $\Gamma _{2,2}^{\varepsilon }$ its periodic extension which forms
the interface between those parts of the matrix $\Omega _{2}^{\varepsilon }$
which lie within neighboring $\varepsilon Y$-cells.

The flow potential of the fluid in the fissures $\Omega _{1}^{\varepsilon }$
is denoted by $u_{1}^{\varepsilon }\left( x,t\right) $ and the corresponding
flux there is given by $-\mu _{1}\left( \frac{x}{\varepsilon },\nabla
u_{1}^{\varepsilon }\right) $. The flow potential in the matrix $\Omega
_{2}^{\varepsilon }$ is represented as the sum of two parts, one component $%
u_{2}^{\varepsilon }\left( x,t\right) $ with flux $-\mu _{2}\left( \frac{x}{
\varepsilon },\nabla u_{2}^{\varepsilon }\right) $ which accounts for the
global diffusion through the pore system of the matrix , and the second
component $u_3^{\varepsilon }\left( x,t\right) $ with flux $-\varepsilon \mu
_{3}\left( \frac{x}{\varepsilon },\varepsilon \nabla u_3^{\varepsilon
}\right) $ and corresponding very high frequency spatial variations which
lead to local storage in the matrix. The \emph{total flow potential} in the
matrix $\Omega _{2}^{\varepsilon }$ is then $\alpha u_{2}^{\varepsilon
}+\beta u_3^{\varepsilon }.$ (Here $\alpha +\beta =1$ with $\alpha \geq 0$
and $\beta > 0$.)

In the following, we shall set $Y_{3}=Y_{2}$ and likewise set $\chi
_{3}=\chi _{2}$ in order to simplify notation. For $j=1,2,3,$ let $\mu _{j}:%
\Bbb{R}^{N} \times \Bbb{R}^{N} \rightarrow \Bbb{R}^{N}$ and assume that for
every $\vec{\xi}\in \Bbb{R}^{N},$ $\mu _{j}\left( \cdot ,\vec{\xi}\right) $
is measurable and $Y$-periodic and for a.e. $y\in Y,$ $\mu _{j}\left(
y,\cdot \right) $ is continuous. In addition, assume that we have positive
constants $k,\,C,\,c_{0}$ 
and $1 < p < \infty$
such that for every $\vec{\xi},\,\vec{\eta}\in \Bbb{R%
}^{N}$ and a.e. $y\in Y$ 
\begin{eqnarray}
\left| \mu _{j}\left( y,\vec{\xi}\right) \right| &\leq &C\left| \vec{\xi}%
\right| ^{p-1}+k  \label{muiscont} \\
\left( \mu _{j}( y,\vec{\xi}) -\mu _{j}( y,\vec{\eta})
\right) \cdot \left( \vec{\xi}-\vec{\eta}\right) &\geq &0  \label{muismon} \\
\mu _{j}\left( y,\vec{\xi}\right) \cdot \vec{\xi} &\geq &c_{0}\left| \vec{\xi%
}\right| ^{p}-k.  \label{muiscoer}
\end{eqnarray}
Let $c_{j}\in C_{\#}\left( Y\right) $ be given such that 
\begin{equation}\label{ciscoer}
0<c_{0}\leq c_{j}\left( y\right) \leq C,\quad 1\le j\le 3.
\end{equation}
Since these are given on $\Bbb{R}^{N}$, we can define for $j=1,2,3$ the
corresponding scaled coefficients at $x\in \Omega_{j}^{\varepsilon }, \,\,%
\vec{\xi}\in \Bbb{R}^{N}$ by 
\begin{equation*}
c_{j}^{\varepsilon }\left( x\right) \equiv c_{j}\left( \frac{x}{\varepsilon }%
\right) ,\,\,\,\mu _{j}^{\varepsilon }\left( x,\vec{\xi}\right) \equiv \,\mu
_{j}\left( \frac{x}{\varepsilon },\vec{\xi}\right) .
\end{equation*}

The exact micro-model introduced in \cite{D-P-S} for
diffusion in a partially fissured medium is given by the system

\begin{eqnarray}
\frac{\partial }{\partial t}\left( c_{1}^{\varepsilon }\left( x\right)
u_{1}^{\varepsilon }\left( x,t\right) \right) -\vec{\nabla}\cdot \mu
_{1}^{\varepsilon }\left( x,\vec{\nabla}u_{1}^{\varepsilon }\left(
x,t\right) \right) =0 &&\text{in }\Omega _{1}^{\varepsilon }  \label{eps1} \\
\frac{\partial }{\partial t}\left( c_{2}^{\varepsilon }\left( x\right)
u_{2}^{\varepsilon }\left( x,t\right) \right) -\vec{\nabla}\cdot \mu
_{2}^{\varepsilon }\left( x,\vec{\nabla}u_{2}^{\varepsilon }\left(
x,t\right) \right) =0 &&\text{in }\Omega _{2}^{\varepsilon }  \label{eps2} \\
\frac{\partial }{\partial t}\left( c_{3}^{\varepsilon }\left( x\right)
u_{3}^{\varepsilon }\left( x,t\right) \right) -\varepsilon \vec{\nabla}\cdot
\mu _{3}^{\varepsilon }\left( x,\varepsilon \vec{\nabla}u_{3}^{\varepsilon
}\left( x,t\right) \right) =0 &&\text{in }\Omega _{2}^{\varepsilon }
\label{eps3} \\
u_{1}^{\varepsilon }=\alpha u_{2}^{\varepsilon }+\beta u_{3}^{\varepsilon }
&&\text{on }\Gamma _{1,2}^{\varepsilon }  \label{eps4} \\
\alpha \mu _{1}^{\varepsilon }\left( x,\vec{\nabla}u_{1}^{\varepsilon
}\left( x,t\right) \right) \cdot \vec{\nu}_{1}=\mu _{2}^{\varepsilon }\left(
x,\vec{\nabla}u_{2}^{\varepsilon }\left( x,t\right) \right) \cdot \vec{\nu}%
_{1} &&\text{on }\Gamma _{1,2}^{\varepsilon }  \label{eps5} \\
\beta \mu _{1}^{\varepsilon }\left( x,\vec{\nabla}u_{1}^{\varepsilon }\left(
x,t\right) \right) \cdot \vec{\nu}_{1}=\varepsilon \mu _{3}^{\varepsilon
}\left( x,\varepsilon \vec{\nabla}u_{3}^{\varepsilon }\left( x,t\right)
\right) \cdot \vec{\nu}_{1} &&\text{on }\Gamma _{1,2}^{\varepsilon }
\label{eps6}
\end{eqnarray}
where $\vec{\nu}_{1}$ is the unit outward normal on $\partial \Omega
_{1}^{\varepsilon}$. We shall similarly let $\vec{\nu}_{2}$ denote the
unit outward normal on $\partial \Omega _{2}^{\varepsilon }$, so
$\vec{\nu}_{1} = -\vec{\nu}_{2}$ on $\Gamma _{1,2}^{\varepsilon}$.
The first equation is the conservation of mass in the fissure system.
In the matrix, $\Omega_2^{\varepsilon}$, we have two components of the
flow potential.  The first is the usual flow through the matrix, and
the second component is scaled by $\varepsilon^p$ to 
represent the very
high frequency variations in flow that result from the relatively very
low permeability of the matrix. Each of these is assumed to satisfy a
corresponding conservation equation. The total flow potential in the
matrix is given by the convex combination $\alpha u_{2}^{\varepsilon
}+\beta u_{3}^{\varepsilon }$ where $\alpha \ge 0,\,\beta > 0$
denote the corresponding fractions of each, so $\alpha + \beta =
1$. Thus, the first interface condition is the continuity of flow
potential, and the remaining conditions determine the corresponding
partition of flux across the interface.  
Since the boundary
conditions will play no essential role in the development, we shall
assume homogeneous Neumann boundary conditions
\begin{eqnarray}
\mu _{1}^{\varepsilon }\left( x,\vec{\nabla}u_{1}^{\varepsilon
}\left( x,t\right) \right) \cdot \vec{\nu}_{1}= 0 &&\text{on }
\partial \Omega_1^{\varepsilon} \cap \partial \Omega \,, \label{eps7} \\
\mu _{2}^{\varepsilon }\left( x,\vec{\nabla}u_{2}^{\varepsilon }\left(
x,t\right) \right) \cdot \vec{\nu}_{2}= 0 &&\text{and }\label{eps8} \\
\mu _{3}^{\varepsilon }\left( x,\vec{\nabla}u_{3}^{\varepsilon }\left(
x,t\right) \right) \cdot \vec{\nu}_{2}= 0 &&\text{on }
\partial \Omega_2^{\varepsilon} \cap \partial \Omega \,.\label{eps9}
\end{eqnarray}
The system is completed  by the \textit{initial conditions}
\begin{equation}
u_{1}^{\varepsilon}\left( \cdot ,0\right) =u_{1}^{0}(\cdot ), \quad
u_{2}^{\varepsilon}\left( \cdot ,0\right) =u_{2}^{0}(\cdot ), \quad
u_{3}^{\varepsilon}\left( \cdot ,0\right) =u_{3}^{0}(\cdot )
  \label{epsIC}
\end{equation}
in $ H^{\varepsilon }$.

Next we develop the \textit{variational formulation} for the 
initial-boundary-value problem (\ref{eps1})-(\ref{epsIC})
 and show that the resulting \textit{Cauchy
problem} is well posed in the appropriate function space.  
Define the \textit{state space}
\begin{equation*}
H^{\varepsilon }\equiv L^{2}\left( \Omega _{1}^{\varepsilon }\right) \times
L^{2}\left( \Omega _{2}^{\varepsilon }\right) \times L^{2}\left( \Omega
_{2}^{\varepsilon }\right) \,,
\end{equation*}
a Hilbert space with the inner product 
\begin{multline*}
\left( \lbrack u_{1},u_{2},u_{3}],[\varphi _{1},\varphi _{2},\varphi
_{3}]\right) _{H^{\varepsilon }}\equiv \\
\int_{\Omega _{1}^{\varepsilon }}c_{1}^{\varepsilon }\left( x\right)
u_{1}\left( x\right) \varphi _{1}\left( x\right) \,dx+\int_{\Omega
_{2}^{\varepsilon }}\left[ c_{2}^{\varepsilon }\left( x\right) u_{2}\left(
x\right) \varphi _{2}\left( x\right) +c_{3}^{\varepsilon }\left( x\right)
u_{3}\left( x\right) \varphi _{3}\left( x\right) \right] \,dx\,.
\end{multline*}
Let $\gamma _{j}^{\varepsilon }:W^{1,p}(\Omega _{j}^{\varepsilon
})\rightarrow L^{p}\left( \partial \Omega _{j}^{\varepsilon }\right) $
be the usual trace maps on the respective spaces for $j=1,2,\
\varepsilon > 0$, and define the \textit{energy space}
\begin{multline*}
V^{\varepsilon }\equiv H^{\varepsilon }\cap \{[u_{1},u_{2},u_{3}]\in
W^{1,p}\left( \Omega _{1}^{\varepsilon }\right) \times W^{1,p}\left( \Omega
_{2}^{\varepsilon }\right) \times W^{1,p}\left( \Omega _{2}^{\varepsilon
}\right) : \\
\gamma _{1}^{\varepsilon }u_{1}=\alpha \gamma _{2}^{\varepsilon }u_{2}+\beta
\gamma _{2}^{\varepsilon }u_{3}\text{ on }\Gamma _{1,2}^{\varepsilon }\}.
\end{multline*}
Note that $V^{\varepsilon }$ is a Banach space when equipped with the norm 
\begin{multline*}
\left\| \lbrack u_{1},u_{2},u_{3}]\right\| _{V^{\varepsilon }}\equiv \left\|
\chi _{1}^{\varepsilon }u_{1}\right\| _{L^{2}\left( \Omega \right) }+\left\|
\chi _{2}^{\varepsilon }u_{2}\right\| _{L^{2}\left( \Omega \right) }+\left\|
\chi _{2}^{\varepsilon }u_{3}\right\| _{L^{2}\left( \Omega \right) }+ \\
\left\| \chi _{1}^{\varepsilon }\vec{\nabla}u_{1}\right\| _{L^{p}\left(
\Omega \right) }+\left\| \chi _{2}^{\varepsilon }\vec{\nabla}u_{2}\right\|
_{L^{p}\left( \Omega \right) }+\left\| \chi _{2}^{\varepsilon }\vec{\nabla}%
u_{3}\right\| _{L^{p}\left( \Omega \right) }.
\end{multline*}

If we multiply each of $\left( \ref{eps1}\right) $, $\left( \ref{eps2}%
\right) $, $\left( \ref{eps3}\right) $ by the corresponding $\varphi
_{1}(x),\,\varphi _{2}(x),\,\varphi _{3}(x)$ for which $[\varphi
_{1},\varphi _{2},\varphi _{3}]\in V^{\varepsilon }$, integrate over the
corresponding domains, and make use of $\left( \ref{eps5}\right) $-$
\left( \ref{eps9}\right) $, we find that the triple of functions $\vec{u}%
^{\varepsilon }(\cdot )\equiv [u_{1}^{\varepsilon }(\cdot
),u_{2}^{\varepsilon }(\cdot ),u_{3}^{\varepsilon }(\cdot )]$ in $%
L^{p}\left( 0,T;V^{\varepsilon }\right) $ satisfies 
\begin{equation*}
\left( \frac{\partial }{\partial t}[u_{1}^{\varepsilon
}(t),u_{2}^{\varepsilon }(t),u_{3}^{\varepsilon }(t)],[\varphi _{1},\varphi
_{2},\varphi _{3}]\right) _{H^{\varepsilon }}+\mathcal{A}^{\varepsilon
}\left( [u_{1}^{\varepsilon }(t),u_{2}^{\varepsilon }(t),u_{3}^{\varepsilon
}(t)]\right) \left( [\varphi _{1},\varphi _{2},\varphi _{3}]\right) =0
\end{equation*}
for all $[\varphi _{1},\varphi _{2},\varphi _{3}]\in V^{\varepsilon }$, 
where we define the operator $\mathcal{A}^{\varepsilon }:V^{\varepsilon
}\rightarrow \left( V^{\varepsilon }\right) ^{\prime }$ by 
\begin{multline*}
\mathcal{A}^{\varepsilon }\left( [u_{1},u_{2},u_{3}]\right) \left( [\varphi
_{1},\varphi _{2},\varphi _{3}]\right) \equiv \int_{\Omega _{1}^{\varepsilon
}}\mu _{1}^{\varepsilon }\left( x,\vec{\nabla}u_{1}\left( x\right) \right)
\cdot \vec{\nabla}\varphi _{1}\left( x\right) \,dx \\
+\int_{\Omega _{2}^{\varepsilon }}\left\{ \mu _{2}^{\varepsilon }\left( x,%
\vec{\nabla}u_{2}\left( x\right) \right) \cdot \vec{\nabla}\varphi
_{2}\left( x\right) +\mu _{3}^{\varepsilon }\left( x,\varepsilon \vec{\nabla}%
u_{3}\left( x\right) \right) \cdot \varepsilon \vec{\nabla}\varphi
_{3}\left( x\right) \right\} \,dx
\end{multline*}
for $[u_{1},u_{2},u_{3}],$ $[\varphi _{1},\varphi _{2},\varphi _{3}]\in
V^{\varepsilon }.$
Thus, the variational form of this problem is to find, for each $\varepsilon
>0$ and $[u_{1}^{0},u_{2}^{0},u_{3}^{0}]\in H^{\varepsilon}$ a triple of
functions $\vec{u}^{\varepsilon }(\cdot )\equiv [u_{1}^{\varepsilon }(\cdot
),u_{2}^{\varepsilon }(\cdot )$, $u_{3}^{\varepsilon }(\cdot )]$ in $%
L^{p}\left( 0,T;V^{\varepsilon }\right) $ such that 
\begin{equation}
\frac{d}{dt}\vec{u}^{\varepsilon }(\cdot )+\mathcal{A}^{\varepsilon }\vec{u}%
^{\varepsilon }(\cdot )=0\text{ in }L^{p^{\prime }}\left(
0,T;(V^{\varepsilon })^{\prime }\right)  \label{VF}
\end{equation}
and 
\begin{equation}
\vec{u}^{\varepsilon }(0) = \vec{u}^{0} \text{ in } H^{\varepsilon }.    
\label{IC}
\end{equation}
Conversely, a solution of (\ref{VF}) will satisfy (\ref{eps1})-(\ref{eps4}), and if that solution is sufficiently smooth, then it will also satisfy
(\ref{eps1})-(\ref{eps9}).

The assumptions $\left( \ref{muiscont}\right) ,\left( \ref{muismon}\right) $
and $\left( \ref{muiscoer}\right) $ guarantee that $\mathcal{A}^{\varepsilon
}$ satisfies the hypotheses of \cite[Proposition III.4.1]{Show:nonlin}, so
there is a unique solution $\vec{u}^{\varepsilon }\equiv [u_{1}^{\varepsilon
},u_{2}^{\varepsilon },u_3^{\varepsilon }]$ in $L^{p}\left(
0,T;V^{\varepsilon }\right) $ of $\left( \ref{VF}\right) $ and $\left( \ref
{IC}\right) $. Note that since $\frac{d }{d t}\vec{u}^{\varepsilon }\in
L^{p^{\prime }}\left( 0,T;\left( V^{\varepsilon }\right) ^{\prime }\right) $
that $\vec{u}^{\varepsilon }\in C\left( [0,T];H^{\varepsilon }\right) $ and
so $\left( \ref{IC}\right) $ is meaningful by Proposition \ref{Wp} above.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Two-scale limits}

We introduce the scaled characteristic functions 
\begin{equation*}
\chi_{j}^{\varepsilon }\left( x\right) \equiv \chi_{j}\left( \frac{x}{%
\varepsilon }\right)\,,\quad j = 1,2.
\end{equation*}
These will be used to denote the \textit{zero-extension} of various
functions. In particular, for any function $w$ defined on $%
\Omega_j^{\varepsilon}$ the product $\chi_{j}^{\varepsilon }\,w$ is
understood to be defined on all of $\Omega$ as the zero extension of $w$.
Similarly, if $w$ is given on $Y_j$, then $\chi_j\,w$ is the corresponding
zero extension to all of $Y$.

Our starting point is a preliminary convergence result for the solutions
described above.

\begin{lemma}
\label{convthm}There exist a pair of functions $u_{j}$ in $L^{p}\left(
0,T;W^{1,p}\left( \Omega \right) \right) ,\ j=1,2,$ and triples of
functions $U_{j}$ in $L^{p}((0,T)\times \Omega ;W_{\#}^{1,p}(Y)/\Bbb{R})$, $%
g_{j}$ in $L^{p^{\prime }}((0,T)\times \Omega \times Y^{N}))$, $%
\,u_{j}^{*}\in L^{2}\left( \Omega \times Y\right) $ for $j=1,2,3$, and a
subsequence taken from the sequence of solutions of $\left( \ref{VF}\right) $%
-$\left( \ref{IC}\right) $ above, hereafter denoted by $\vec{u}^{\varepsilon
}=[u_{1}^{\varepsilon },u_{2}^{\varepsilon },u_{3}^{\varepsilon }]$, which
two-scale converges as follows: 
\begin{eqnarray}
&&\chi _{1}^{\varepsilon }u_{1}^{\varepsilon }\overset{2}{\rightarrow }\chi
_{1}\left( y\right) u_{1}\left( x,t\right)   \label{conv1} \\
&&\chi _{1}^{\varepsilon }\vec{\nabla}u_{1}^{\varepsilon }\overset{2}{%
\rightarrow }\chi _{1}\left( y\right) \left[ \vec{\nabla}u_{1}\left(
x,t\right) +\vec{\nabla}_{y}U_{1}\left( x,y,t\right) \right]   \label{conv2}
\\
&&\chi _{2}^{\varepsilon }u_{2}^{\varepsilon }\overset{2}{\rightarrow }\chi
_{2}\left( y\right) u_{2}\left( x,t\right)   \label{conv3} \\
&&\chi _{2}^{\varepsilon }\vec{\nabla}u_{2}^{\varepsilon }\overset{2}{%
\rightarrow }\chi _{2}\left( y\right) \left[ \vec{\nabla}u_{2}\left(
x,t\right) +\vec{\nabla}_{y}U_{2}\left( x,y,t\right) \right]   \label{conv4}
\\
&&\chi _{2}^{\varepsilon }u_{3}^{\varepsilon }\overset{2}{\rightarrow }\chi
_{2}\left( y\right) U_{3}\left( x,y,t\right)   \label{conv5} \\
&&\varepsilon \chi _{2}^{\varepsilon }\vec{\nabla}u_{3}^{\varepsilon }%
\overset{2}{\rightarrow }\chi _{2}\left( y\right) \vec{\nabla}%
_{y}U_{3}\left( x,y,t\right)   \label{conv6} \\
&&\chi _{1}^{\varepsilon }\mu _{1}^{\varepsilon }\left( \vec{\nabla}%
u_{1}^{\varepsilon }\right) \overset{2}{\rightarrow }\chi _{1}\left(
y\right) \vec{g}_{1}\left( x,y,t\right)   \label{conv7} \\
&&\chi _{2}^{\varepsilon }\mu _{2}^{\varepsilon }\left( \vec{\nabla}%
u_{2}^{\varepsilon }\right) \overset{2}{\rightarrow }\chi _{2}\left(
y\right) \vec{g}_{2}\left( x,y,t\right)   \label{conv8} \\
&&\chi _{2}^{\varepsilon }\mu _{3}^{\varepsilon }\left( \varepsilon \vec{%
\nabla}u_{3}^{\varepsilon }\right) \overset{2}{\rightarrow }\chi _{2}\left(
y\right) \vec{g}_{3}\left( x,y,t\right)   \label{conv9} \\
&&\chi _{j}^{\varepsilon }u_{j}^{\varepsilon }\left( \cdot ,T\right) 
\overset{2}{\rightarrow }\chi _{j}(y)u_{j}^{*}(x),\quad j=1,\,2,\,3.
\label{conv10}
\end{eqnarray}
\end{lemma}

\begin{proof}
Using Proposition \ref{Wp} and $\left( \ref{VF}\right) $ we
can write 
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\left( [u_{1}^{\varepsilon },u_{2}^{\varepsilon
},u_{3}^{\varepsilon }],[u_{1}^{\varepsilon },u_{2}^{\varepsilon
},u_{3}^{\varepsilon }]\right) _{H^{\varepsilon }}+\mathcal{\ A}%
^{\varepsilon }\left( [u_{1}^{\varepsilon },u_{2}^{\varepsilon
},u_{3}^{\varepsilon }]\right) \left( [u_{1}^{\varepsilon
},u_{2}^{\varepsilon },u_{3}^{\varepsilon }]\right) =0.
\end{equation*}
Integrating in $t$ gives 
\begin{equation}
\frac{1}{2}\left\| \vec{u}^{\varepsilon }\left( t\right) \right\|
_{H^{\varepsilon }}^{2}-\frac{1}{2}\left\| \vec{u}^{\varepsilon }\left(
0\right) \right\| _{H^{\varepsilon }}^{2}+\int_{0}^{t}\mathcal{A}%
^{\varepsilon }\left( [u_{1}^{\varepsilon },u_{2}^{\varepsilon
},u_{3}^{\varepsilon }]\right) \left( [u_{1}^{\varepsilon
},u_{2}^{\varepsilon },u_{3}^{\varepsilon }]\right) dt=0
\label{integratedVF}
\end{equation}
which, with the assumption $\left( \ref{muiscoer}\right) $ yields 
\begin{multline}
\frac{1}{2}\left\| \vec{u}^{\varepsilon }\left( t\right) \right\|
_{H^{\varepsilon }}^{2}+c_{0}\int_{0}^{t}\left( \left\| \chi
_{1}^{\varepsilon }\vec{\nabla}u_{1}^{\varepsilon }\right\| _{L^{p}\left(
\Omega \right) }^{p}+\left\| \chi _{2}^{\varepsilon }\vec{\nabla}%
u_{2}^{\varepsilon }\right\| _{L^{p}\left( \Omega \right) }^{p}+\left\|
\varepsilon \chi _{2}^{\varepsilon }\vec{\nabla}u_{3}^{\varepsilon }\right\|
_{L^{p}\left( \Omega \right) }^{p}\right) dt \\
\leq \frac{1}{2}\left\| [\chi _{1}u_{1}^{0},\chi _{2}u_{2}^{0},\chi
_{2}u_{3}^{0}]\right\| _{H^{\varepsilon }}^{2}+t\left| k\right| ,\quad 0\le
t\le T.  \label{apriori}
\end{multline}
Thus, $\vec{u}^{\varepsilon }\left( \cdot \right) $ is bounded in $L^{\infty
}\left( 0,T;H^{\varepsilon }\right) ,$ and so 
$\chi _{1}^{\varepsilon }u_{1}^{\varepsilon }$,
$\chi _{2}^{\varepsilon }u_{2}^{\varepsilon }$, and 
$\chi _{2}^{\varepsilon }u_{3}^{\varepsilon }$ are
bounded in $L^{\infty }( 0,T;L^{2}( \Omega))$. 
Also, $\chi _{1}^{\varepsilon }\vec{\nabla}u_{1}^{\varepsilon }$, 
$\chi _{2}^{\varepsilon }\vec{\nabla}u_{2}^{\varepsilon }$ 
and $\varepsilon \chi _{2}^{\varepsilon }\vec{\nabla}u_{3}^{\varepsilon }$ 
are bounded in $L^{p}( 0,T$; $L^{p}( \Omega ) ^{N})$. 
We obtain $\left( 
\ref{conv1}\right) $ through $\left( \ref{conv4}\right) $ exactly as in 
\cite[Theorem 2.9]{Allaire} by Theorem \ref{W1pbound1}. Statements $\left( 
\ref{conv5}\right) $ and $\left( \ref{conv6}\right) $ follow from Theorem 
\ref{W1pbound2}. Finally, from $\left( \ref{muiscont}\right) $ and the
bounds already established, we have that $\chi _{j}^{\varepsilon }\mu
_{j}^{\varepsilon }\left( x,\vec{\nabla}u_{j}^{\varepsilon }\left(
x,t\right) \right) $ (for $j=1,2$) and $\chi _{2}^{\varepsilon }\mu
_{3}^{\varepsilon }\left( x,\varepsilon \vec{\nabla}u_{3}^{\varepsilon
}\left( x,t\right) \right) $ are bounded in $L^{p^{\prime }}\left(
[0,T],L^{p^{\prime }}(\Omega )\right) $ due to $\left( \ref{muiscoer}\right) 
$, $\left( \ref{apriori}\right) $ and 
\begin{eqnarray*}
\int_{0}^{T}\int_{\Omega }\chi _{j}^{\varepsilon }\left| \mu _{j}\left( 
\frac{x}{\varepsilon },\vec{\xi}\left( x\right) \right) \right| ^{p^{\prime
}}dxdt &\leq &\int_{0}^{T}\int_{\Omega }\chi _{j}^{\varepsilon }\left| \vec{%
\xi}\left( x\right) \right| ^{(p-1)p^{\prime }}dxdt \\
&=&\int_{0}^{T}\int_{\Omega }\chi _{j}^{\varepsilon }\left| \vec{\xi}\left(
x\right) \right| ^{p}dxdt.
\end{eqnarray*}
Thus $\chi _{j}^{\varepsilon }\mu _{j}^{\varepsilon }\left( x,\vec{\nabla}%
u_{j}^{\varepsilon }\left( x,t\right) \right) $ and $\chi _{2}^{\varepsilon
}\mu _{3}^{\varepsilon }\left( x,\varepsilon \vec{\nabla}u_{3}^{\varepsilon
}\left( x,t\right) \right) $ converge as stated.
\end{proof}

Define the \textit{flow potential} $u^{\varepsilon }\equiv \chi
_{1}^{\varepsilon }u_{1}^{\varepsilon }+\chi _{2}^{\varepsilon }\left(
\alpha u_{2}^{\varepsilon }+\beta u_{3}^{\varepsilon }\right) \in
L^{p}\left( 0,T;W^{1,p}\left( \Omega \right) \right) $ for each $%
\varepsilon >0$, and note that on $\Gamma _{1,2}^{\varepsilon }$ 
\begin{equation*}
\gamma _{1}^{\varepsilon }u^{\varepsilon }=\gamma _{1}^{\varepsilon
}u_{1}^{\varepsilon }=\alpha \gamma _{2}^{\varepsilon }u_{2}^{\varepsilon
}+\beta \gamma _{2}^{\varepsilon }u_{3}^{\varepsilon }=\gamma
_{2}^{\varepsilon }u^{\varepsilon }.
\end{equation*}
Thus 
\begin{equation*}
\varepsilon \vec{\nabla}u^{\varepsilon }=\varepsilon \chi _{1}^{\varepsilon }%
\vec{\nabla}u_{1}^{\varepsilon }+\chi _{2}^{\varepsilon }\left( \alpha
\varepsilon \vec{\nabla}u_{2}^{\varepsilon }+\beta \varepsilon \vec{\nabla}%
u_{3}^{\varepsilon }\right) \in L^{p}\left( [0,T]\times \Omega \right)
\end{equation*}
and from Lemma $\left( \ref{convthm}\right) $ we see that 
\begin{equation*}
u^{\varepsilon }\overset{2}{\rightarrow }\chi _{1}\left( y\right)
u_{1}\left( x\right) +\chi _{2}\left( y\right) \left( \alpha u_{2}\left(
x,t\right) +\beta U_{3}\left( x,y,t\right) \right)
\end{equation*}
and 
\begin{equation*}
\varepsilon \vec{\nabla}u^{\varepsilon }\overset{2}{\rightarrow }\chi
_{2}\left( y\right) \beta \vec{\nabla}_{y}U_{3}\left( x,y,t\right) .
\end{equation*}
Now let $\vec{\varphi}\in C_{0}^{\infty }\left( \Omega ,C_{\#}^{\infty
}\left( Y^{N}\right) \right) $ and note that

\begin{multline*}
\int_{\Omega }\varepsilon \vec{\nabla}u^{\varepsilon }\left( x,t\right)
\cdot \vec{\varphi}\left( x,\frac{x}{\varepsilon }\right) dx= \\
-\int_{\Omega }u^{\varepsilon }\left( x,t\right) \left[ \varepsilon \vec{%
\nabla}\cdot \vec{\varphi}\left( x,\frac{x}{\varepsilon }\right) +\vec{\nabla%
}_{y} \cdot \vec{\varphi}\left( x,\frac{x}{\varepsilon }\right) \right] dx.
\end{multline*}
Taking two-scale limits on both sides yields 
\begin{multline}
\int_{\Omega }\int_{Y}\beta \chi _{2}\left( y\right) \vec{\nabla}%
_{y}U_{3}\left( x,y,t\right) \cdot \vec{\varphi}\left( x,y\right) dxdy=
\label{limit} \\
-\int_{\Omega }\int_{Y}\left( \chi _{1}\left( y\right) u_{1}\left(
x,t\right) +\chi _{2}\left( y\right) \left( \alpha u_{2}\left( x,t\right)
+\beta U_{3}\left( x,y,t\right) \right) \right) \vec{\nabla}_{y}\cdot \vec{%
\varphi}\left( x,y\right) dxdy.
\end{multline}
The divergence theorem shows that the left hand side of $\left( \ref{limit}%
\right) $ is simply 
\begin{multline*}
\int_{\Omega }\int_{Y_{2}}\beta \vec{\nabla}_{y}U_{3}\left( x,y,t\right)
\cdot \vec{\varphi}\left( x,y\right) dxdy=-\int_{\Omega }\int_{Y_{2}}\beta
U_{3}\left( x,y,t\right) \vec{\nabla}_{y}\cdot \vec{\varphi}\left(
x,y\right) dxdy \\
+\int_{\Omega }\int_{\partial Y_{2}}\beta U_{3}\left( x,s,t\right) \vec{%
\varphi}\left( x,s\right) \cdot \vec{\nu}_{2}dxds
\end{multline*}
while the right hand side of $\left( \ref{limit}\right) $ can be written 
%\begin{equation*}
\begin{multline*}
-\int_{\Omega }\int_{Y_{1}}u_{1}\left( x,t\right) \vec{\nabla}_{y}\cdot \vec{%
\varphi}\left( x,y\right) dxdy \\
-\int_{\Omega }\int_{Y_{2}}\left( \alpha
u_{2}\left( x,t\right) +\beta U_{3}\left( x,y,t\right) \right) \vec{\nabla}%
_{y}\cdot \vec{\varphi}\left( x,y\right) dxdy.
%\end{equation*}
\end{multline*}
We see that $\left( \ref{limit}\right) $ yields 
\begin{multline*}
\int_{\Omega }\int_{\partial Y_{2}}\beta U_{3}\left( x,s,t\right) \vec{%
\varphi}\left( x,s\right) \cdot \vec{\nu}_{2}dxds= \\
-\int_{\Omega }\int_{Y_{1}}u_{1}\left( x,t\right) \vec{\nabla}_{y}\cdot \vec{%
\varphi}\left( x,y\right) dxdy-\int_{\Omega }\int_{Y_{2}}\alpha u_{2}\left(
x,t\right) \vec{\nabla}_{y}\cdot \vec{\varphi}\left( x,y\right) dxdy \\
=-\int_{\Omega }\int_{\partial Y_{1}}u_{1}\left( x,t\right) \vec{\varphi}%
\left( x,s\right) \cdot \vec{\nu}_{1}dxds-\int_{\Omega }\int_{\partial
Y_{2}}\alpha u_{2}\left( x,t\right) \vec{\varphi}\left( x,s\right) \cdot 
\vec{\nu}_{2}dxds.
\end{multline*}
Since $U_{3}$ and $\vec{\varphi}$ are periodic on $\Gamma _{2,2}$, this
shows that 
\begin{equation}
\beta U_{3}+\alpha u_{2}=u_{1}\text{ \thinspace \thinspace \thinspace
\thinspace \thinspace on }\partial Y_{1}\cap \partial Y_{2}\equiv \Gamma
_{1,2}\,.  \label{constr}
\end{equation}

%%%%%%%%%%%%%%%%%%%%
Next we seek a variational statement which is satisfied by the limits obtained in Lemma 3.1.
Choose smooth functions 
\begin{equation*}
\varphi _{j}\in L^{p}\left( 0,T;W^{1,p}\left( \Omega \right) \right) ,\
j=1,2,\quad \Phi _{j}\in L^{p}\left( (0,T)\times \Omega ;W_{\#}^{1,p}\left(
Y\right) \right) ,\ j=1,2,3,
\end{equation*}
such that 
\begin{equation*}
\frac{\partial \varphi _{j}}{\partial t}\in L^{p^{\prime }}\left(
0,T;W^{1,p}\left( \Omega \right)^{\prime }\right) ,\ j=1,2,\ \frac{
\partial \Phi _{3}}{\partial t}\in L^{p^{\prime }}\left( (0,T)\times \Omega
;W_{\#}^{1,p}\left( Y\right)^{\prime } \right),
\end{equation*}
and $\beta \Phi_{3}\left( x,y,t\right) 
=\varphi _{1}\left( x,t\right) -\alpha \varphi _{2}\left( x,t\right)
 \quad \text{for }y\in \Gamma _{1,2}.$ %%%%%%%%%%%%%%%%
In the following we shall use the notation $(\cdot)_{,t}$ to represent 
the time
derivative $\frac{\partial }{\partial t}(\cdot).$ 
Apply $\left( \ref{VF}\right) $ to the triple 
$[\varphi _{1}\left( x,t\right) +\varepsilon \Phi _{1}\left( x,
\frac{x}{\varepsilon },t\right) ,\,\varphi _{2}\left( x,t\right)
+\varepsilon \Phi _{2}\left( x,\frac{x}{\varepsilon },t\right) ,\,\Phi
_{3}^{\varepsilon}\left( x,\frac{x}{\varepsilon },t\right) ]$
in $L^p\left(0,T; V^\varepsilon \right)$, 
where we define 
$\Phi_{3}^{\varepsilon}\left( x,y,t\right) 
\equiv \Phi_{3}\left( x,y,t\right)
+ \frac{\varepsilon}{\beta} \Phi_{1}\left( x,y,t\right)
- \frac{\varepsilon \alpha}{\beta} \Phi_{2}\left( x,y,t\right)$.
Then integrate by parts in $t$ to obtain 
\begin{multline}
-\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega _{j}^{\varepsilon
}}c_{j}^{\varepsilon }u_{j}^{\varepsilon }\left( \varphi _{j,t}
+\varepsilon \Phi _{j,t}\right) dxdt-\int_{0}^{T}\int_{\Omega
_{2}^{\varepsilon }}c_{3}^{\varepsilon }u_{3}^{\varepsilon }\Phi
_{3,t}^{\varepsilon}\, dxdt  \label{VF2} \\
+\sum_{j=1}^{2}\int_{\Omega _{j}^{\varepsilon }}\!\!\!\!
c_{j}^{\varepsilon
}u_{j}^{\varepsilon }\left( x,T\right) \left( \varphi _{j}\left( x,T\right)
+\varepsilon \Phi _{j}\left( x,\frac{x}{\varepsilon },T\right) \right)
dx+\int_{\Omega _{2}^{\varepsilon }}\!\!\!\!
c_{3}^{\varepsilon }u_{3}^{\varepsilon
}\left( x,T\right) \Phi_{3}^{\varepsilon}\left( x,\frac{x}{\varepsilon },T\right) dx \\
-\sum_{j=1}^{2}\int_{\Omega _{j}^{\varepsilon }}c_{j}^{\varepsilon
}u_{j}^{0}\left( \varphi _{j}\left( x,0\right) +\varepsilon \Phi _{j}\left(
x,\frac{x}{\varepsilon },0\right) \right) dx-\int_{\Omega _{2}^{\varepsilon
}}c_{3}^{\varepsilon }u_{3}^{0}\Phi_{3}^{\varepsilon}\left( x,\frac{x}{\varepsilon },0\right)\, dx \\
+\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega _{j}^{\varepsilon }}\mu
_{j}^{\varepsilon }\left( x,\vec{\nabla}u_{j}^{\varepsilon }\left(
x,t\right) \right) \cdot \vec{\nabla}\left( \varphi _{j}\left( x,t\right)
+\varepsilon \Phi _{j}\left( x,\frac{x}{\varepsilon },t\right) \right) dxdt
\\
+\int_{0}^{T}\int_{\Omega _{2}^{\varepsilon }}\mu _{3}^{\varepsilon }\left(
x,\varepsilon \vec{\nabla}u_{3}^{\varepsilon }\left( x,t\right) \right)
\cdot \varepsilon \left[ \vec{\nabla}\Phi_{3}^{\varepsilon}\left( x,\frac{x}{\varepsilon }
,t\right) +\frac{1}{\varepsilon }\vec{\nabla}_{y}\Phi_{3}^{\varepsilon}\left( x,\frac{x}{\varepsilon},t\right) \right] \,dxdt=0.
\end{multline}
Letting $\varepsilon \rightarrow 0$ in $\left( \ref{VF2}\right) $ now yields 
\begin{multline}
-\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
u_{j}\left( x,t\right) \varphi _{j,t}\left( x,t\right) \,dydxdt
\label{VFlimit} \\
-\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}c_{3}\left( y\right) U_{3}\left(
x,y,t\right) \Phi_{3,t}\left( x,y,t\right) dydxdt \\
+\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
u_{j}^{*}\left( x\right) \varphi _{j}\left( x,T\right) \,dydx+\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) u_{3}^{*}\left( x\right) \Phi _{3}\left(
x,y,T\right) dydx \\
-\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
u_{j}^{0}\left( x\right) \varphi _{j}\left( x,0\right) \,dydx-\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) u_{3}^{0}\left( x\right) \Phi _{3}\left(
x,y,0\right) dydx \\
+\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega }\int_{Y_{j}}\vec{g}_{j}\left(
x,y,t\right) \cdot \left[ \vec{\nabla}\varphi _{j}\left( x,t\right) +\vec{%
\nabla}_{y}\Phi _{j}\left( x,y,t\right) \right] \,dydxdt \\
+\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\vec{g}_{3}\left( x,y,t\right) \cdot 
\vec{\nabla}_{y}\Phi _{3}\left( x,y,t\right) \,dydxdt=0.
\end{multline}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We can summarize the preceding as follows. Define the 
\textit{energy space} 
\begin{multline*}
W \equiv \{ [u_1,\,u_2,\,U_1,\,U_2,\,U_3] \in W^{1,p}(\Omega)^2 \times
L^{p}\left(\Omega ;W_{\#}^{1,p}(Y)\right)^3 :	\\
\beta U_3(x,y) =
u_1(x) - \alpha u_2(x) \text{ for } y \in \Gamma_{1,2} \}.
\end{multline*}
We have shown that the limit obtained in Lemma 3.1 satisfies
$$[u_1,\,u_2,\,U_1,\,U_2,\,U_3] \in L^{p}\left( 0,T; W \right)$$
and by density, (\ref{VFlimit}) holds for all
$[\varphi_1,\,\varphi_2,\,\Phi_1,\,\Phi_2,\,\Phi_3] \in 
L^{p}\left( 0,T; W \right)$ such that
$\frac{d}{dt}[\varphi_1,\,\varphi_2,\,0,\,0,\,\Phi_3] 
\in  L^{p^{\prime}}\left( 0,T; W^{\prime} \right)$.
It remains to find the strong form of the problem and to identify 
the \textit{flux} terms $\vec{g}_{j}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Homogenized Problem}

We shall decouple the variational identity (\ref{VFlimit}) in order to 
obtain the strong form of our homogenized system. This will be accomplished by making special choices of the test functions 
$[\varphi_1,\,\varphi_2,\,\Phi_1,\,\Phi_2,\,\Phi_3]$ as above, 
and the strong form will be displayed below in Corollary \ref{strongvar}. 
First we choose  
$\varphi _{1},$ $\varphi _{2}$, $\Phi _{1}$, $\Phi _{2}$ 
all equal to zero, and choose $\Phi _{3}$ as above and 
to vanish at $t=0$ and $t=T$ and on $\Gamma_{1,2}$. 
Together with the identity (\ref{constr}) from above, 
this gives at \textit{a.e.} $x \in \Omega$ the \textit{cell system}
\begin{gather}
c_{3}\left( y\right) \frac{\partial U_{3}\left( x,y,t\right) }{\partial t}-%
\vec{\nabla}_{y}\cdot \vec{g}_{3}\left( x,y,t\right) =0\,,\quad y\in Y_{2}\,,
\label{veqn} \\
U_{3}\text{ and }\vec{g}_{3}\cdot \vec{\nu} \text{ are $Y$-periodic on }\Gamma
_{2,2}\,,  \label{U3eqn2} \\
\beta U_{3}=u_{1}-\alpha u_{2}\text{ on }\Gamma _{1,2}\,.  \label{U3eqn3}
\end{gather}
Next let $\varphi _{1}$ be as above and vanish at $t=0$ and $t=T$, and choose
$\Phi _{3}$ by the requirement that 
$\beta \Phi _{3}(x,y,t) = \varphi _{1}(x,t)$ for $y \in Y_2$. With the remaining test functions all zero, this yields the \textit{macro-fissure equation}
\begin{multline}
\left( \int_{Y_{1}}c_{1}(y)\,dy\right) \frac{\partial u_{1}\left( x,t\right) 
}{\partial t}
+ \frac{1}{\beta} \frac{\partial}{\partial t}
\int_{Y_{2}}c_{3}(y)U_3(x,y,t)\,dy	\\
=\vec{\nabla}\cdot \int_{Y_{1}}\vec{g}_{1}\left( x,y,t\right)dy\,.  
\label{u1eqn}
\end{multline}
Similarly we choose $\varphi _{2}$ as above and vanishing 
at $t=0$ and $t=T$ and let $\Phi _{3}$ be determined by 
$\beta \Phi _{3}(x,y,t) = - \alpha \varphi _{2}(x,t)$ for $y \in Y_1$
to obtain the \textit{macro-matrix equation}
\begin{multline}
\left( \int_{Y_{2}}c_{2}(y)\,dy\right) \frac{\partial u_{2}\left( x,t\right) 
}{\partial t}
- \frac{\alpha}{\beta} \frac{\partial}{\partial t}
\int_{Y_{2}}c_{3}(y)U_3(x,y,t)\,dy	\\
=\vec{\nabla}\cdot \int_{Y_{2}}\vec{g}_{2}\left( x,y,t\right)dy\,.  
\label{u2eqn}
\end{multline}
Finally, by setting the test functions 
$\varphi _{1},$ $\varphi _{2}$, $\Phi _{3}$
all equal to zero and by choosing $\Phi _{1}$, $\Phi _{2}$ as above, 
we obtain 
the pair of systems
\begin{gather}
\vec{\nabla}_{y}\cdot \vec{g}_{j}\left( x,y,t\right) =0\quad y\in Y_{j}\,,
\label{geqn} \\
\vec{g}_{j}\cdot \vec{\nu} =0\text{ on }\Gamma _{1,2}
\text{ and } \vec{g}_{j}\cdot \vec{\nu}  \text{ is $Y$-periodic on }
\partial Y_j \cap \partial Y\text{ for }j=1,2.  \label{geqn2}
\end{gather}

Note that $\left( \ref{veqn}\right) $ and $\left( \ref{u1eqn}\right) $ 
and $\left( \ref{u2eqn}\right) $ hold in $
L^{p^{\prime }}((0,T)\times \Omega ;W_{\#}^{1,p}(Y)')$ and 
$L^{p^{\prime }}( 0,T$; $W^{1,p}( \Omega )' ) ,\ $respectively. 
Substituting $\left( \ref{veqn}\right) $-$\left( \ref{geqn2}
\right) $ in $\left( \ref{VFlimit}\right) $ gives the boundary conditions
\begin{gather}
\int_{Y_{1}}\vec{g}_{1}\left( x,y,t\right)dy \cdot \vec{\nu_1} = 0 
\quad \text{ and }\\
\int_{Y_{2}}\vec{g}_{2}\left( x,y,t\right)dy \cdot \vec{\nu_2} = 0 
\quad \text{ on } \partial \Omega
\end{gather}
and the initial and final conditions 
\begin{equation}
U_{3}\left( x,y,0\right) =u_{3}^{0}\left( x \right) ,\text{ }U_{3}\left(
x,y,T\right) =u_{3}^{*}\left( x,y\right)  \label{vIC}
\end{equation}
and 
\begin{equation}
u_{j}\left( x,0\right) =u_{j}^{0}\left( x\right) ,\text{ }u_{j}\left(
x,T\right) =u_{j}^{*}\left( x\right) \,\,\,\,\,\,\,\,\,\text{for }j=1,2
\label{ujIC}
\end{equation}
in $L^{2}\left( \Omega \times Y_{2}\right) $ and $L^{2}\left( \Omega \right) 
$ respectively. 
The final conditions appearing above will be used only to identify the 
functions $\vec{g}_{i}\left( x,y,t\right)$ below; 
they are {\it not} part of the problem.
Note also that using $\left( \ref{vIC}\right) $ and $\left( 
\ref{ujIC}\right) $, integrating by parts in $t$ in $\left( \ref{VFlimit}%
\right) ,$ and replacing the test functions $\varphi _{j}$ (for $j=1,2$) and 
$\Phi _{j}$ (for $j=1,2,3$) with sequences converging to $u_{j}$ and $U_{j}$
gives the following ``homogenized'' version of $\left( \ref{integratedVF}%
\right) ,$ 
\begin{multline}
\frac{1}{2}\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
\left| u_{j}\left( x,T\right) \right| ^{2}\,dydx+\frac{1}{2}\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) \left| U_{3}\left( x,y,T\right) \right|
^{2}dydx  \label{homintVF} \\
-\left[ \frac{1}{2}\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left(
y\right) \left| u_{j}^{0}\left( x\right) \right| ^{2}\,dydx+\frac{1}{2}%
\int_{\Omega }\int_{Y_{2}}c_{3}\left( y\right) \left| u_{3}^{0}\left(
x\right) \right| ^{2}dydx\right] \\
+\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega }\int_{Y_{j}}\vec{g}_{j}\left(
x,y,t\right) \cdot \left[ \vec{\nabla}u_{j}\left( x,t\right) +\vec{\nabla}%
_{y}U_{j}\left( x,y,t\right) \right] \,dydxdt \\
+\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\vec{g}_{3}\left( x,y,t\right) \cdot 
\vec{\nabla}_{y}U_{3}\left( x,y,t\right) \,dydxdt=0.
\end{multline}

It remains to find $\vec{g}_{1,\,\,\,}\vec{g}_{2}$ and $\vec{g}_{3}$ in
terms $u_{1},$ $u_{2},$ $U_{1,\text{ }}U_{2}$ and $U_{3}.$ To this end, let $%
\vec{\phi}$ and $\vec{\xi}$ be in $C_{0}^{\infty }\left( [0,T]\times \Omega
;C_{\#}^{\infty }\left( Y\right) \right) ^{N}$ and $\Phi _{1},\Phi _{2},\Phi
_{3}\in C_{0}^{\infty }\left( [0,T]\times \Omega ;C_{\#}^{\infty }\left(
Y\right) \right) $ and for $\varepsilon >0,$ define the triple of functions 
\begin{equation*}
\eta _{j}^{\varepsilon }\left( x,t\right) =\chi _{j}\left( \frac{x}{%
\varepsilon }\right) \vec{\nabla}u_{j}\left( x,t\right) +\varepsilon \chi
_{j}\left( \frac{x}{\varepsilon }\right) \vec{\nabla}\Phi _{j}\left( x,\frac{%
x}{\varepsilon },t\right) +\lambda \vec{\phi}\left( x,\frac{x}{\varepsilon }%
,t\right) \,, \quad j=1,2,
\end{equation*}
and 
\begin{equation*}
\eta _{3}^{\varepsilon }\left( x,t\right) =\chi _{2}\left( \frac{x}{%
\varepsilon }\right) \left( \varepsilon \vec{\nabla}\Phi _{3}\left( x,\frac{x%
}{\varepsilon },t\right) +\lambda \vec{\xi}\left( x,\frac{x}{\varepsilon }%
,t\right) \right)\,.
\end{equation*}
Note that each $\eta _{j}^{\varepsilon }\left( x,t\right) $ and (because of
the continuity assumption) $\mu _{j}\left( \frac{x}{\varepsilon },\eta
_{j}^{\varepsilon }\left( x,t\right) \right) $ ($j=1,2,3$) arises from an
 admissible
test function, and we have the two-scale convergence
\begin{equation*}
\eta _{j}^{\varepsilon }\overset{2}{\rightarrow }\eta _{j}\left(
x,y,t\right) \equiv \chi _{j}\left( y\right) \vec{\nabla}u_{j}\left(
x,t\right) +\chi _{j}\left( y\right) \vec{\nabla}_{y}\Phi _{j}\left(
x,y,t\right) +\lambda \vec{\phi}\left( x,y,t\right), \,\,\,\,\,j=1,2,
\end{equation*}
\begin{equation*}
\eta _{3}^{\varepsilon }\overset{2}{\rightarrow }\eta _{3}\left(
x,y,t\right) \equiv \chi _{2}\left( y\right) \left( \vec{\nabla}_{y}\Phi
_{3}\left( x,y,t\right) \right) +\lambda \vec{\xi}\left( x,y,t\right) .
\end{equation*}
By $\left( \ref{muismon}\right) $ we have 
\begin{multline}
\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega _{j}^{\varepsilon }}\left( \mu
_{j}^{\varepsilon }\left( x,\vec{\nabla}u_{j}^{\varepsilon }\right) -\mu
_{j}^{\varepsilon }\left( x,\eta _{j}^{\varepsilon }\right) \right) \left( 
\vec{\nabla}u_{j}^{\varepsilon }-\eta _{j}^{\varepsilon }\right) dxdt
\label{monotonicity} \\
+\int_{0}^{T}\int_{\Omega _{2}^{\varepsilon }}\left( \mu _{3}^{\varepsilon
}\left( x,\varepsilon \vec{\nabla}u_{3}^{\varepsilon }\right) -\mu
_{3}^{\varepsilon }\left( x,\eta _{3}^{\varepsilon }\right) \right) \left(
\varepsilon \vec{\nabla}u_{3}^{\varepsilon }-\eta _{3}^{\varepsilon }\right)
dxdt\geq 0.
\end{multline}
Expanding $\left( \ref{monotonicity}\right) $ and employing $\left( \ref
{integratedVF}\right) $ at $t=T$ gives 
\begin{multline*}
\frac{1}{2}\sum_{j=1}^{2}\left\{ \int_{\Omega _{j}^{\varepsilon
}}c_{j}^{\varepsilon }\left( \left| u_{j}^{0}\right| ^{2}-\left|
u_{j}^{\varepsilon }\left( x,T\right) \right| ^{2}\right) dx\right\} +\frac{1%
}{2}\int_{\Omega _{2}^{\varepsilon }}c_{3}^{\varepsilon }\left( \left|
u_{3}^{0}\right| ^{2}-\left| u_{3}^{\varepsilon }\left( x,T\right) \right|
^{2}\right) dx \\
-\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega _{j}^{\varepsilon }}\left\{ \mu
_{j}^{\varepsilon }\left( x,\eta _{j}^{\varepsilon }\right) \cdot \left( 
\vec{\nabla}u_{j}^{\varepsilon }-\eta _{j}^{\varepsilon }\right) +\mu
_{j}^{\varepsilon }\left( x,\vec{\nabla}u_{j}^{\varepsilon }\right) \cdot
\eta _{j}^{\varepsilon }\right\} dxdt \\
-\int_{0}^{T}\int_{\Omega _{2}^{\varepsilon }}\left\{ \mu _{3}^{\varepsilon
}\left( x,\eta _{3}^{\varepsilon }\right) \cdot \left( \varepsilon \vec{%
\nabla}u_{3}^{\varepsilon }-\eta _{3}^{\varepsilon }\right) +\mu
_{3}^{\varepsilon }\left( x,\varepsilon \vec{\nabla}u_{3}^{\varepsilon
}\right) \cdot \eta _{3}^{\varepsilon }\right\} dxdt\geq 0.
\end{multline*}
Let $\varepsilon \rightarrow 0$ and apply the two-scale convergence results
above to obtain 
\begin{multline}
\frac{1}{2}\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
\left| u_{j}^{0}\left( x\right) \right| ^{2}dydx+\frac{1}{2}\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) \left| u_{3}^{0}\left( x\right) \right|
^{2}dydx  \label{monlimit} \\
-\frac{1}{2}\underset{\varepsilon \rightarrow 0}{\lim }\left[
\sum_{j=1}^{2}\int_{\Omega _{j}^{\varepsilon }}c_{j}^{\varepsilon }\left|
u_{j}^{\varepsilon }\left( x,T\right) \right| ^{2}dx+\frac{1}{2}\int_{\Omega
_{2}^{\varepsilon }}c_{3}^{\varepsilon }\left| u_{3}^{\varepsilon }\left(
x,T\right) \right| ^{2}dx\right] \\
-\sum_{j=1}^{2}\int_{0}^{T}\!\!\!\int_{\Omega }\int_{Y_{j}}\!\!\!
\mu _{j}\left( y,\eta
_{j}\left( x,y,t\right) \right) \cdot \left( \vec{\nabla}_{y}U_{j}\left(
x,y\right) -\vec{\nabla}_{y}\Phi _{j}\left( x,y,t\right) -\lambda \vec{\phi}%
\left( x,y,t\right) \right) dydxdt \\
-\sum_{j=1}^{2}\int_{0}^{T}\!\!\!\int_{\Omega }\int_{Y_{j}}\!\!\!
\vec{g}_{j}\left(
x,y,t\right) \cdot \left( \vec{\nabla}u_{j}\left( x,t\right) +\vec{\nabla}%
_{y}\Phi _{j}\left( x,y,t\right) +\lambda \vec{\phi}\left( x,y,t\right)
\right) dydxdt \\
-\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\mu _{3}\left( y,\eta _{3}\left(
x,y,t\right) \right) \cdot \left( \vec{\nabla}_{y}U_{3}\left( x,y,t\right) -%
\vec{\nabla}_{y}\Phi _{3}\left( x,y,t\right) -\lambda \vec{\xi}\left(
x,y,t\right) \right) dydxdt \\
-\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\vec{g}_{3}\left( x,y,t\right) \cdot
\left( \vec{\nabla}_{y}\Phi _{3}\left( x,y,t\right) +\lambda \vec{\xi}\left(
x,y,t\right) \right) dydxdt\geq 0.
\end{multline}
Set $\vec{\phi}=\chi _{1}\vec{\theta}_{1}+\chi _{2}\vec{\theta}_{2}$ where $%
\vec{\theta}_{j}\in C_{0}^{\infty }\left( [0,T]\times \Omega ,C^{\infty
}\left( Y_{j}\right) \right) $ and $\chi _{j}\vec{\theta}_{j}$ is $Y$%
-periodic. Following \cite{Allaire} we note that since each $\mu _{j}$ is
continuous in the second variable, we may replace $\Phi _{j}$ by sequences
converging strongly in $L^{p}((0,T)\times \Omega ;W_{\#}^{1,p}(Y)/\Bbb{R})$
to $\chi _{1}\left( y\right) U_{1}\left( x,y,t\right) $, $\chi _{2}\left(
y\right) U_{2}\left( x,y,t\right) $ and $\chi _{2}\left( y\right)
U_{3}\left( x,y,t\right) $ for $j=1,2$ and $3$, respectively. Thus $\left( 
\ref{monlimit}\right) $ becomes 
%\newpage
\begin{multline}
\frac{1}{2}\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
\left| u_{j}^{0}\left( x\right) \right| ^{2}dydx+\frac{1}{2}\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) \left| u_{3}^{0}\left( x\right) \right|
^{2}dydx  \label{monlimit2} \\
-\frac{1}{2}\underset{\varepsilon \rightarrow 0}{\lim }\left[
\sum_{j=1}^{2}\int_{\Omega _{j}^{\varepsilon }}c_{j}^{\varepsilon }\left|
u_{j}^{\varepsilon }\left( x,T\right) \right| ^{2}dx+\frac{1}{2}\int_{\Omega
_{2}^{\varepsilon }}c_{3}^{\varepsilon }\left| u_{3}^{\varepsilon }\left(
x,T\right) \right| ^{2}dx\right] \\
\end{multline}
\begin{multline*}
+\sum_{j=1}^{2}\int_{0}^{T}\!\!\!\int_{\Omega }\int_{Y_{j}}\!\!\!
\mu _{j}\left( y,\vec{%
\nabla}u_{j}\left( x,t\right) +\vec{\nabla}_{y}U_{j}\left( x,y,t\right)
+\lambda \vec{\theta}_{j}\left( x,y,t\right) \right) \cdot \lambda \vec{%
\theta}_{j}\left( x,y,t\right) dydxdt \\
-\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega }\int_{Y_{j}}\vec{g}_{j}\left(
x,y,t\right) \cdot \left( \vec{\nabla}u_{j}\left( x,t\right) +\vec{\nabla}%
_{y}U_{j}\left( x,y,t\right) +\lambda \vec{\theta}_{j}\left( x,y,t\right)
\right) dydxdt \\
+\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\mu _{3}\left( y,\vec{\nabla}%
_{y}U_{3}\left( x,y,t\right) +\lambda \vec{\xi}\left( x,y,t\right) \right)
\cdot \left( \lambda \vec{\xi}\left( x,y,t\right) \right) dydxdt \\
-\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\vec{g}_{3}\left( x,y,t\right) \cdot
\left( \vec{\nabla}_{y}U_{3}\left( x,y,t\right) +\lambda \vec{\xi}\left(
x,y,t\right) \right) dydxdt\geq 0.
\end{multline*}
We now employ $\left(\ref{vIC}\right) ,$ $\left(
\ref{ujIC}\right) $ and $\left(\ref{homintVF}\right) $ in
$\left(\ref{monlimit2}\right) $
to obtain 
\begin{multline}
\sum_{j=1}^{2}\int_{0}^{T}\!\!\!\int_{\Omega }\int_{Y_{j}}\!\!\!
\mu _{j}\left( y,\vec{%
\nabla}u_{j}\left( x,t\right) +\vec{\nabla}_{y}U_{j}\left( x,y,t\right)
+\lambda \vec{\theta}_{j}\left( x,y,t\right) \right) \cdot \lambda \vec{%
\theta}_{j}\left( x,y,t\right) dydxdt  \label{theend} \\
+\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\mu _{3}\left( y,\vec{\nabla}%
_{y}U_{3}\left( x,y,t\right) +\lambda \vec{\xi}\left( x,y,t\right) \right)
\cdot \lambda \vec{\xi}\left( x,y,t\right) dydxdt \\
-\sum_{j=1}^{2}\int_{0}^{T}\!\!\!\int_{\Omega }\int_{Y_{j}}\!\!\!
\vec{g}_{j}\left(
x,y,t\right) \cdot \lambda \vec{\theta}_{j}\left( x,y,t\right)
dydxdt-\int_{0}^{T}\!\!\!\int_{\Omega }\int_{Y_{2}}\!\!\!
\vec{g}_{3}\left( x,y,t\right)
\cdot \lambda \vec{\xi}\left( x,y,t\right) dydxdt \\
\geq \frac{1}{2}\underset{\varepsilon \rightarrow 0}{\lim }\left[
\sum_{j=1}^{2}\int_{\Omega _{j}^{\varepsilon }}c_{j}^{\varepsilon }\left|
u_{j}^{\varepsilon }\left( x,T\right) \right| ^{2}dx+\frac{1}{2}\int_{\Omega
_{2}^{\varepsilon }}c_{3}^{\varepsilon }\left| u_{3}^{\varepsilon }\left(
x,T\right) \right| ^{2}dx\right] \\
-\frac{1}{2}\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
\left| u_{j}\left( x,T\right) \right| ^{2}\,dydx+\frac{1}{2}\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) \left| U_{3}\left( x,y,T\right) \right|
^{2}dydx.
\end{multline}
The right hand side of $\left( \ref{theend}\right) $ is non-negative by 
\cite[Proposition 1.6]{Allaire}, so dividing by $\lambda $ and letting $%
\lambda \rightarrow 0$ gives 
\begin{multline}
\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega }\int_{Y_{j}}\left[ \mu _{j}\left( y,%
\vec{\nabla}u_{j}\left( x,t\right) +\vec{\nabla}_{y}U_{j}\left( x,y,t\right)
\right) -\vec{g}_{j}\left( x,y,t\right) \right] \cdot \vec{\theta}_{j}\left(
x,y,t\right) dydxdt  \label{lastline} \\
+\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\left[ \mu _{3}\left( y,\vec{\nabla}%
_{y}U_{3}\left( x,y,t\right) \right) -\vec{g}_{3}\left( x,y,t\right) \right]
\cdot \vec{\xi}\left( x,y,t\right) dydxdt\geq 0.
\end{multline}
This holds for all $\vec{\theta}_{1},$ $\vec{\theta}_{2},$ and $\vec{\xi}$,
so 
\begin{equation*}
\mu _{j}\left( y,\vec{\nabla}u_{j}\left( x,t\right) +\vec{\nabla}%
_{y}U_{j}\left( x,y,t\right) \right) =\vec{g}_{j}\left( x,y,t\right) \text{
in }Y_{j}, \qquad j = 1,2,
\end{equation*}
and 
\begin{equation*}
\mu _{3}\left( y,\vec{\nabla}_{y}U_{3}\left( x,y,t\right) \right) =\vec{g}%
_{3}\left( x,y,t\right) \text{ in }Y_{2}.
\end{equation*}
These identities complete the strong form of the homogenized problem.
We shall summarize and complement these results in the following section.

\section{The Main Result}
\begin{theorem}\label{weakvar}
Assume that (\ref{muiscont})-(\ref{ciscoer}) hold, that $\beta > 0$,
and that $u_1^0, u_2^0, \text{ and }u_3^0 \in L^2(\Omega)$ are given. Then
the limits $[u_{1},\ u_{2},\ U_{1},\ U_{2},\ U_{3}]$ established above
in Lemma \ref{convthm} are the unique solution
$$
u_{j} \in L^{p}(0,T;W^{1,p}(\Omega )),\ j=1,2, \quad U_{j} \in 
L^{p}((0,T)\times \Omega ;W_{\#}^{1,p}(Y_j)/\Bbb{R}),\ j=1,2,3,
$$
with 
$\beta U_3(x,y,t) =
u_1(x,t) - \alpha u_2(x,t) \text{ for } y \in \Gamma_{1,2}$
of the \textbf{homogenized system}
\begin{multline}
-\sum_{j=1}^{2}\int_{0}^{T}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
u_{j}\left( x,t\right) \varphi _{j,t}\left( x,t\right) \,dydxdt
\label{finalVF} \\
-\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}c_{3}\left( y\right) U_{3}\left(
x,y,t\right) \Phi _{3,t}\left( x,y,t\right) dydxdt \\
-\sum_{j=1}^{2}\int_{\Omega }\int_{Y_{j}}c_{j}\left( y\right)
u_{j}^{0}\left( x\right) \varphi _{j}\left( x,0\right) \,dydx-\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) u_{3}^{0}\left( x\right) \Phi _{3}\left(
x,y,0\right) dydx \\
+\sum_{j=1}^{2}\int_{0}^{T}\!\!\!\int_{\Omega }\int_{Y_{j}}\!\!\!\!
\mu _{j}\left( y,\vec{%
\nabla}u_{j}\left( x,t\right) +\vec{\nabla}_{y}U_{j}\left( x,y,t\right)
\right) \cdot \left[ \vec{\nabla}\varphi _{j}\left( x,t\right) +\vec{\nabla}%
_{y}\Phi _{j}\left( x,y,t\right) \right] dydxdt \\
+\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\mu _{3}\left( y,\vec{\nabla}%
_{y}U_{3}\left( x,y,t\right) \right) \cdot \vec{\nabla}_{y}\Phi _{3}\left(
x,y,t\right) \,dydxdt=0,
\end{multline}
for all 
\begin{equation*}
\varphi _{j}\in L^{p}\left( 0,T;W^{1,p}\left( \Omega \right) \right) ,\
j=1,2,\quad \Phi _{j}\in L^{p}\left( (0,T)\times \Omega ;W^{1,p}_{\#}\left(
Y_{j}\right) \right) ,\,\,\,\,j=1,2,3,
\end{equation*}
for which 
\begin{multline*}
\frac{\partial \varphi _{j}}{\partial t}\in L^{p^{\prime }}\left(
0,T;(W^{1,p}\left( \Omega \right))^{\prime} \right) ,\ j=1,2,\quad  
\frac{\partial \Phi _{3}}{\partial t}\in L^{p^{\prime }}\left( (0,T)\times
\Omega ;(W_{\#}^{1,p}\left( Y_{j}\right))^{\prime} \right) , \\
\text{ }\beta \Phi _{3}\left( x,y,t\right) =\varphi _{1}\left( x,t\right)
-\alpha \varphi _{2}\left( x,t\right) \quad \text{for }y\in \Gamma _{1,2},
\end{multline*}
and 
\begin{equation*}
\varphi _{1}\left( x,T\right) =\varphi _{2}\left( x,T\right)
=\Phi _{3}\left(x,y,T\right) =0.
\end{equation*}
\end{theorem}

Only the uniqueness needs yet to be verified, and this will follow
below.  In particular, $U_1$ and $U_2$ are determined within a
constant for each $t \in (0,T)$, so each of these is unique up to a
corresponding function of $t$.  We shall check that (\ref{finalVF}) is
just the \textit{variational form} of the Cauchy problem for an
appropiate evolution equation in Banach space, and that the
corresponding \textit{strong problem} is described as follows.  The
state space is given by
\begin{equation*}
H \equiv \{ [\varphi_1,\,\varphi_2,\,\Phi_3] \in L^2(\Omega) \times 
L^2(\Omega) \times L^{2}\left(\Omega ; L^2(Y_2)\right)
\end{equation*}
with the scalar product
\begin{equation*}
\left(\mathbf{u}, \mathbf{\varphi}\right)_H \equiv 
\sum_{j=1}^2 \int_{\Omega } \int_{Y_{j}} c_j(y)\,dy\, u_j(x)\, \varphi_j(x)\,dx
+ \int_{\Omega } \int_{Y_{2}} c_3(y)\, U_3(x,y)\, \Phi_3(x,y)\,dy\,dx.
\end{equation*}
Define the \textit{energy space}
\begin{multline*}
V \equiv \{ [\varphi_1,\,\varphi_2,\,\Phi_3] \in 
H \cap \left(W^{1,p}(\Omega) \times W^{1,p}(\Omega) \times
L^{p}\left(\Omega ;W_{\#}^{1,p}(Y_2)\right)\right) :	\\
\beta \Phi_3(x,y) =
\varphi_1(x) - \alpha \varphi_2(x) \text{ for } y \in \Gamma_{1,2} \}
\end{multline*}
and the corresponding \textit{evolution space} by $\mathcal{V} = L^p(0,T; V)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary} \label{strongvar}
The triple $\mathbf{u(\cdot)} \equiv 
[u_{1}(\cdot),\ u_{2}(\cdot),\ U_{3}(\cdot)]$ is the unique
solution 
$\mathbf{u(\cdot)} \in \mathcal{V}$ 
with 
$\mathbf{u'(\cdot)} \in \mathcal{V}'$
of the \textbf{strong homogenized system}
\begin{multline*}
\left( \int_{Y_{1}}c_{1}(y)\,dy\right) \frac{\partial u_{1}\left( x,t\right) 
}{\partial t}
+ \frac{1}{\beta} \frac{\partial}{\partial t}
\int_{Y_{2}}c_{3}(y)U_3(x,y,t)\,dy	\\
=\vec{\nabla}\cdot \int_{Y_{1}}\mu _{1}\left( y,\vec{\nabla}u_{1}\left( x,t\right) 
+\vec{\nabla}_{y}U_{1}\left( x,y,t\right) \right) dy\,,  
%\label{fissure}
\end{multline*}
\begin{multline*}
\left( \int_{Y_{2}}c_{2}(y)\,dy\right) \frac{\partial u_{2}\left( x,t\right) 
}{\partial t}
- \frac{\alpha}{\beta} \frac{\partial}{\partial t}
\int_{Y_{2}}c_{3}(y)U_3(x,y,t)\,dy	\\
=\vec{\nabla}\cdot \int_{Y_{2}}\mu _{2}\left( y,\vec{\nabla}u_{2}\left( x,t\right) +\vec{\nabla}_{y}U_{2}\left( x,y,t\right) \right) dy\,,
%\label{matrix}
\end{multline*}
\begin{gather*}
c_{3}\left( y\right) \frac{\partial U_{3}\left( x,y,t\right) }{\partial t}-%
\vec{\nabla}_{y}\cdot \mu _{3}\left( y,\vec{\nabla}_{y}U_{3}\left( x,y,t\right) \right) =0\,,\quad y\in Y_{2}\,, 	\\
U_{3}\left( x,y,t\right)\text{ and }
\mu _{3}\left( y,\vec{\nabla}_{y}U_{3}\left( x,y,t\right) \right)
\cdot \vec{\nu} \text{ are $Y$-periodic on }\Gamma
_{2,2}\,,  \label{U3eqn2a} \\
\beta U_{3}=u_{1}-\alpha u_{2}\text{ on }\Gamma _{1,2}\,,  \label{cell}
\end{gather*}
with the boundary conditions
\begin{gather*}	
\int_{Y_{1}}\mu _{1}\left( y,\vec{\nabla}u_{1}\left( x,t\right) +\vec{\nabla}_{y}U_{1}\left( x,y,t\right) \right) dy \cdot \vec{\nu_1} = 0 
\text{ and }\\
\int_{Y_{2}}\mu _{2}\left( y,\vec{\nabla}u_{2}\left( x,t\right) +\vec{\nabla}_{y}U_{2}\left( x,y,t\right) \right) dy \cdot \vec{\nu_2} = 0 
\text{ on } \partial \Omega,
%\label{localbc}
\end{gather*}
and the initial conditions
\begin{equation*}
u_{j}\left( x,0\right) =u_{j}^{0}\left( x\right) \text{ for }j=1,2,\qquad  U_{3}\left( x,y,0\right) =u_{3}^{0}\left( x \right)\,, \label{localic}
\end{equation*}
where $U_1 \in L^{p}((0,T)\times \Omega ;W_{\#}^{1,p}(Y_1)/\Bbb{R}),\ 
U_2 \in L^{p}((0,T)\times \Omega ;W_{\#}^{1,p}(Y_2)/\Bbb{R})$ are solutions of the \textit{local problems}
\begin{gather*}
\vec{\nabla}_{y}\cdot \mu _{j}\left( y,\vec{\nabla}_{y}U_{j}\left( x,y,t\right) \ + \vec{\nabla}u_{j}\left( x,t\right)\right)=0\,,\quad y\in Y_{j}\,,
\label{local} \\
\mu _{j}\left( y,\vec{\nabla}_{y}U_{j}\left( x,y,t\right) +
\vec{\nabla}u_{j}\left( x,t\right) \right) \cdot \nu =0\text{ on }\Gamma _{1,2},
\qquad \text{$Y$-periodic on }\Gamma _{2,2}\text{ }j=1,2.  \label{localbc}
\end{gather*}
\end{corollary}

\begin{proof}
	Define an operator $\mathcal{A}: V \rightarrow V'$ by
\begin{multline}
\langle \mathcal{A} \mathbf{u},\mathbf{\varphi} \rangle \equiv 
\sum_{j=1}^2 \int_{\Omega } \int_{Y_{j}}\{\mu _{j}(y,\vec{\nabla}u_{j}\left(
x\right) +\vec{\nabla}_{y}U_{j}(x,y)\}\cdot 
(\vec{\nabla}\varphi_{j}\left( x\right))\,dydx \\
+ \int_{\Omega } \int_{Y_{3}}\{\mu _{3}(y,\vec{\nabla}_{y}U_{3}(x,y)\}\cdot 
(\vec{\nabla}_{y}\Phi_{3}(x,y))\,dydx,\\ 
\qquad 
\mathbf{u} = [u_1,u_2,U_3],\ \mathbf{\varphi}=[\varphi_1,\varphi_2,\Phi] \in V,
\end{multline}
where $U_1(x,y)$ and $U_2(x,y)$ are determined by
\begin{multline}
U_j \in L^{p}\left(\Omega ;W_{\#}^{1,p}(Y_j)\right): \\
\int_{\Omega } \int_{Y_{j}}\{\mu _{j}(y,\vec{\nabla}_{y}U_{j}(x,y)
+ \vec{\nabla}u_{j}\left(x\right)\}\cdot 
(\vec{\nabla}_{y}\Phi(x,y))\,dydx = 0,\\ 
\qquad \Phi \in L^{p}\left(\Omega ;W_{\#}^{1,p}(Y_j)\right),
\end{multline}
for $j = 1,\,2$.  It has been already shown in Section 4 that
$[u_1,u_2,U_1,U_2,U_3]$ satisfies the homogenized system of Theorem
\ref{weakvar}, and from this it follows that $\mathbf{u(\cdot)}$
satisfies the variational form of the Cauchy problem (\ref{wivp}). It
is easy to check that $\mathcal{A}$ is monotone and bounded
$\mathcal{V} \rightarrow \mathcal{V}'$, so $\mathbf{u(\cdot)}$ is the
unique solution as well of the strong problem (\ref{sivp}), and this
is realized as the strong homogenized system of Corollary \ref{strongvar}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}
Theorem \ref{weakvar} describes the limiting form of the original
micro-model from Section 2 as a system for the five unknowns $u_{1}$,
$u_{2}$, $U_{1}$, $U_{2} $, and $U_{3}$. This system can also be realized
from an evolution equation based on the variational identity (\ref{VFlimit})
on the space $W$,
but this would be of \textit{degenerate} type: the time derivatives of
$U_{1}$ and $U_{2}$ do not occur in the system. However, by following
the suggestion implicit in Corollary \ref{strongvar} we were able to
incorporate the local functions $U_{1}$ and $U_{2}$ in the definition
of the operator $\mathcal{A}$ and thereby to write our limiting system
as a non-degenerate evolution equation on the space $V$ with three
components.
\end{remark}

In the linear case, one can carry this decoupling even further and
represent each of the functions $U_{1}$ and $U_{2}$ in terms of the
corresponding $u_{1}$ or $u_{2}$ in order to obtain a closed system for the
remaining three unknowns. Suppose that we have symmetric $Y$-periodic
coefficient functions $a_{ij}^{1}(y)\in C\left( Y_{1}\right) $ and
$a_{ij}^{2}(y)\in C\left( Y_{2}\right) $ (which are zero off their
respective domains). We assume that there is a $c_0 >0$,
independent of $y$, such that
\begin{equation*}
\sum_{i,j=1}^{N}a_{ij}^{k}(y)\xi _{i}\xi _{j}\geq c_0 |\xi |^{2},\quad
y\in Y_{k}\,\,\,\,\,\,\text{for }k=1,2.
\end{equation*}
Extending each $a_{ij}^{k}$ to all of $\Bbb{R}$ by periodicity, we define
for $k=1,2$ and $\vec{\xi}\in \Bbb{R}^{N}$%
\begin{equation*}
\mu _{k}^{\varepsilon }\left( x,\vec{\xi}\right)_i
=\sum_{j=1}^{N}a_{ij}^{k}\left( \frac{x}{\varepsilon }\right) \xi _{j}.
\end{equation*}
Then with $p=2,$ the results developed above apply. For each of $k=1,2$ we
isolate from $\left( \ref{finalVF}\right) $ the following problem for $
U_{k}(x,y,t)$:

Find $U_{k}\in 
L^{2}\left( (0,T)\times \Omega ;W^{1,2}_{\#}\left(Y_{k}\right) \right)$ 
such that 
\begin{multline}
\int_{0}^{T}\int_{\Omega }\int_{Y_{k}}\mu _{k}\left( y,\vec{\nabla}
u_{k}\left( x,t\right) +\vec{\nabla}_{y}U_{k}\left( x,y,t\right) \right)
\cdot \vec{\nabla}_{y}\Phi _{k}\left( x,y,t\right) \,dydxdt=0  \label{LP1} \\
\text{for all }\Phi _{k}\in L^{2}\left( (0,T)\times \Omega ;W^{1,2}_{\#}\left(
Y_{k}\right) \right) {.}
\end{multline}
The ``input'' to this problem is $\vec{\nabla}u_{k}\left( x,t\right)
$, independent of $y$, so this permits us to separate variables with
the following construction:

For $1\le i\le N$, define $W_{i}^{k}(y)$ to be the solution of 
\begin{gather*}
-\vec{\nabla}_{y}\cdot \left[ \mu _{k}(y,\vec{\nabla}_{y}W_{i}^{k}(y)+\vec{e}%
_{i})\right] =0\,\,\,\,\,\,\,\,\text{in }\ Y_{k}\ , \\
\mu _{k}(y,\vec{\nabla}_{y}W_{i}^{k}(y)+\vec{e}_{i})\cdot \vec{n}=0\,\,\,\,%
\text{on }\ \partial Y_{k}\sim \partial Y \\
W_{i}^{k}(\cdot )\ \text{is $Y$-periodic.}
\end{gather*}

Then by linearity we can write 
\begin{equation*}
U_{k}(x,y,t)=\sum_{j=1}^{N}\frac{\partial u_{k}}{\partial x_{j}}%
(x,t)W_{j}(y)\ .
\end{equation*}
If we substitute this into $\left( \ref{finalVF}\right) $ with $\Phi
_{k}(x,y,t)=\sum_{j=1}^{N}{\frac{\partial \varphi _{k}(x,t)}{\partial x_{j}}}%
W_{j}^{k}(y)$, we obtain the \textbf{decoupled homogenized system\/} 
\begin{multline*}
-\sum_{k=1}^{2}\int_{0}^{T}\int_{\Omega }\tilde{c}_{k}u_{k}\left( x,t\right)
\varphi _{k,t}\left( x,t\right) \,dxdt-\sum_{k=1}^{2}\int_{\Omega }%
\tilde{c}_{k}u_{k}^{0}\left( x\right) \varphi _{k}\left( x,0\right) \,dx \\
+\sum_{k=1}^{2}\int_{0}^{T}\int_{\Omega
}\int_{Y_{k}}\sum_{i,j=1}^{N}A_{ij}^{k}\frac{\partial u_{k}}{\partial x_{i}}%
(x,t){{\frac{\partial \varphi _{k}}{\partial x_{j}}}}\left( x,t\right)
\,dydxdt=0\,\,\,\,\,\,\text{for }k=1,2
\end{multline*}
\begin{multline*}
-\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}c_{3}\left( y\right) U_{3}\left(
x,y,t\right) \Phi _{3,t}\left( x,y,t\right) dydxdt \\ 
-\int_{\Omega
}\int_{Y_{2}}c_{3}\left( y\right) u_{3}^{0}\left( x\right) \Phi _{3}\left(
x,y,0\right) dydx \\
+\int_{0}^{T}\int_{\Omega }\int_{Y_{2}}\mu _{3}\left( y,\vec{\nabla}%
_{y}U_{3}\left( x,y,t\right) \right) \cdot \vec{\nabla}_{y}\Phi _{3}\left(
x,y,t\right) \,dydxdt=0.
\end{multline*}
where the coefficients are given by 
\begin{equation*}
\tilde{c}_{k}=\int_{Y_{k}}c_{k}\,dy\quad ,\quad A_{ij}^{k}=\int_{Y_{k}}\mu
_{k}(y,\vec{\nabla}_{y}W_{i}^{k}\left( y\right) +\vec{e}_{i})\cdot (\vec{%
\nabla}_{y}W_{j}^{k}\left( y\right) +\vec{e}_{j})\,dy\,.
\end{equation*}
These are the usual \textit{effective coefficients} which are
constants that result from the ``averaging'' due to the homogenization
procedure.

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\end{document}