\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 115, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/115\hfil
 Compactness result for periodic structures]
{Compactness result for periodic structures and its application
to the homogenization of a diffusion-convection equation}

\author[A. Meirmanov, R. Zimin \hfil EJDE-2011/115\hfilneg]
{Anvarbek Meirmanov, Reshat Zimin}  % in alphabetical order

\address{Anvarbek M. Meirmanov \newline
Department of mahtematics \\
Belgorod State University \\
ul.Pobedi 85, 308015 Belgorod, Russia}
\email{meirmanov@bsu.edu.ru}

\address{Reshat Zimin \newline
Department of mahtematics \\
Belgorod State University \\
ul.Pobedi 85, 308015 Belgorod, Russia}
\email{reshat85@mail.ru}

\thanks{Submitted April 10, 2011. Published September 6, 2011.}
\subjclass[2000]{35B27, 46E35, 76R99}
\keywords{Weak, strong and two-scale convergence;
homogenization; \hfill\break\indent diffusion-convection}

\begin{abstract}
 We prove the strong compactness of the sequence
 $\{c^{\varepsilon}(\mathbf{x},t)\}$ in $L_2(\Omega_T)$,
 $\Omega_T=\{(\mathbf{x},t):\mathbf{x}\in\Omega
 \subset  \mathbb{R}^3, t\in(0,T)\}$, bounded in
 $W^{1,0}_2(\Omega_T)$ with the sequence of time derivative
 $\{\partial/\partial t\big(\chi(\mathbf{x}/\varepsilon)
 c^{\varepsilon}\big)\}$  bounded in the space 
 $L_2\big((0,T); W^{-1}_2(\Omega)\big)$. 
 As an application we consider the  homogenization of a
 diffusion-convection equation with a
 sequence of divergence-free velocities
 $\{\mathbf{v}^{\varepsilon}(\mathbf{x},t)\}$
 weakly convergent in $L_2(\Omega_T)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{assumption}[theorem]{Assumption}

\section{Introduction} \label{Introduction}

There are several compactness criteria and among them
Tartar's method of compensated compactness \cite{Tartar} and the
method suggested by  Aubin in \cite{Aubin} (see also \cite{Lions}).
These methods intensively used in the theory of nonlinear
differential equations. As a rule, the first one has applications
in stationary problems, while the second method is used in
non-stationary nonlinear equations.

In the present publication we discuss the method, closed
to the Aubin compactness lemma.   In its simplest
setting, this result provides the strong compactness in
$L_2(\Omega_T)$ (throughout the article, we use the customary
notation of function spaces and norms  \cite{Lions,LSU})
to the sequence of functions
$\{c^{\varepsilon}(\mathbf{x},t)\}$ bounded in
$L_{\infty}\big((0,T);L_2(\Omega)\big)\cap
W^{1,0}_2(\Omega_T)$ with the sequence of the time derivatives
$\{\partial c^{\varepsilon}/\partial t\}$  bounded in
$L_2\big((0,T);  W^{-1}_2(\Omega)\big)$. But in many
applications (especially in homogenization), the second condition
on a boundedness of the time derivatives in some dual space is not
always satisfied. Sometimes, instead of the last condition, one
has the boundedness of time derivatives in a dual space
$L_2\big((0,T); W^{-1}_2(\Omega^{\varepsilon}_f)\big)$,
defined on some periodic subdomain
$\Omega^{\varepsilon}_f\subset \Omega$. Using  new ideas of
Nguetseng's two-scale convergence method \cite{NGU}  we prove that
even under this weak condition the sequence
$\{c^{\varepsilon}(\mathbf{x},t)\}$ still remains strongly
compact in $L_2(\Omega_T)$. The main point here is the fact,
that if for some $t_0\in(0,T)$,
\[
\lim_{\varepsilon\to 0}\varepsilon^2
\int_{\Omega}|\nabla
c^{\varepsilon}(\mathbf{x},t_0)|^2\,dx=0,
\]
then the bounded in $L_2(\Omega)$ sequence
$\{c^{\varepsilon}(\mathbf{x},t_0)\}$ contains a
subsequence, which two-scale converges in $L_2(\Omega)$ to
some function $\bar{c}(\mathbf{x},t_0)$.

Recall that, in general, any bounded in $L_2(\Omega)$ sequence
$\{u^{\varepsilon}\}$ contains a two-scale convergent
subsequence $\{u^{\varepsilon_k}\}$, where the limiting
function $U(\mathbf{x},\mathbf{y})$ is 1-periodic in
variable $\mathbf{y} \in Y=(0,1)^{n}$:
\[
\int_{\Omega}u^{\varepsilon_k}(\mathbf{x})
\varphi(\mathbf{x},\frac{\mathbf{x}}{\varepsilon_k})dx
\to\iint
_{\Omega Y}U(\mathbf{x},\mathbf{y})
\varphi(\mathbf{x},\mathbf{y})dydx
\]
for any smooth function $\varphi(\mathbf{x},\mathbf{y})$,
1-periodic in the variable $\mathbf{y}$. In particular,
for $\varphi(\mathbf{x},\mathbf{y})=\varphi_0(\mathbf{y})
\cdot h(\mathbf{x})$, where $\varphi_0 \in L_2(Y)$ and
$h  \in L_{\infty}(\Omega)$.

A similar compactness result has been proved in \cite{AAPP}
under different assumptions on the sequence
$\{c^{\varepsilon}(\mathbf{x},t)\}$. More precisely,
the corresponding \cite[Lemma 4.2]{AAPP} states,
that if for all $\varepsilon>0$
\[
0\leqslant c^{\varepsilon}(\mathbf{x},t)\leqslant M_0,
 \int_{\Omega_T} |c^{\varepsilon}(\mathbf{x}
 +\triangle \mathbf{x},t)-c^{\varepsilon}(\mathbf{x},t)|^2\,dx\,dt
\leqslant M_0 \omega(|\triangle \mathbf{x}|),
\]
with some $\omega(\xi)$, such that $\omega(\xi)\to 0$ as $\xi\to 0$,
and
\[
\|\frac{\partial}{\partial t}(\chi^{\varepsilon} c^{\varepsilon})\|
_{L_2\big((0,T); W^{-1}_2(\Omega)\big)}\leqslant M_0,
\]
where $0<\chi^{-}\leqslant\chi^{\varepsilon}\leqslant\chi^{+}<1 $,
$\chi^{\pm}=const$, then the family $\{c^{\varepsilon}\}$ is a
compact set in $L_2(\Omega_T)$.

As an application of our result we consider  the homogenization of
diffusion-convection equation
\begin{equation}
\frac{\partial c^{\varepsilon}}{\partial t}+
\mathbf{v}^{\varepsilon}\cdot\nabla c^{\varepsilon}=
\triangle c^{\varepsilon}, \quad \mathbf{x}\in
\Omega^{\varepsilon},\;   t\in(0,T), \label{0.1}
\end{equation}
with boundary and initial conditions
\begin{gather}
\big(\nabla c^{\varepsilon}-\mathbf{v}^{\varepsilon}
c^{\varepsilon}\big) \cdot \boldsymbol{\nu}=0, \quad \mathbf{x}\in
\partial\Omega^{\varepsilon}\backslash S,\;  t\in(0,T), \label{0.2}
\\
c^{\varepsilon}(\mathbf{x},t)=0, \quad
\mathbf{x}\in S\cap\partial\Omega^{\varepsilon},\; t\in(0,T),
\label{0.3}
\\
c^{\varepsilon}(\mathbf{x},0)=c_0(\mathbf{x}), \quad
\mathbf{x}\in \Omega^{\varepsilon}. \label{0.4}
\end{gather}
In \eqref{0.2},  $\boldsymbol{\nu}$ is the unit outward normal
vector to the boundary  $\partial\Omega^{\varepsilon}$ and
$S=\partial\Omega$.

We assume that velocities $\mathbf{v}^{\varepsilon}$ are
uniformly bounded in  $L_8\big((0,T);L_4(\Omega)\big)$:
\begin{equation}
\int_0^{T}\Big(\int_{\Omega}
|\mathbf{v}^{\varepsilon}|^{4}dx\Big)^2dt\leqslant
M_0^2, \label{0.5}
\end{equation}
and
\begin{equation}
\nabla\cdot \mathbf{v}^{\varepsilon}=0,
\mathbf{x}\in\Omega_T. \label{0.6}
\end{equation}

As usual, the solution to the problem \eqref{0.1}--\eqref{0.4}
is understood in a weak sense as a solution of the integral
identity
\begin{equation}
\int_{\Omega^{\varepsilon}_T}\Big(c^{\varepsilon}
\frac{\partial\phi}{\partial t}-\big(\nabla c^{\varepsilon}
-\mathbf{v}^{\varepsilon} c^{\varepsilon}\big)\cdot
\nabla \phi\Big)\,dx\,dt
=-\int_{\Omega^{\varepsilon}}c_0(\mathbf{x})\phi(\mathbf{x},0)\,dx
\label{0.7}
\end{equation}
for any smooth functions $\phi$, such that $\phi(\mathbf{x},T)=0$.

Homogenization means the limiting procedure in \eqref{0.7} as
$\varepsilon\to 0$ and the main problem here  is how to
pass to the limit in the nonlinear term
\[
c^{\varepsilon} \mathbf{v}^{\varepsilon}\cdot
\nabla \phi.
\]
It has been done for velocities  with a special structure
\[
\mathbf{v}^{\varepsilon}=\mathbf{v}^{\varepsilon}
(\mathbf{x}),  \text{or}
\mathbf{v}^{\varepsilon}=\mathbf{v}(\mathbf{x},t,
\frac{\mathbf{x}}{\varepsilon})
\]
(see, for example, \cite{AGP, AC,BLP,BJP,Hornung,HJ}).
However, in the  general case we need the
strong compactness in $L_2(\Omega_T)$ of the sequence
 $\{c^{\varepsilon}\}$. Our compactness result and the
energy estimate
\[
\max_{0<t<T}\int_{\Omega^{\varepsilon}}
|c^{\varepsilon}(\mathbf{x},t)|^2dx+
\int_{\Omega_T^{\varepsilon}}
|\nabla c^{\varepsilon}(\mathbf{x},t)|^2\,dx\,dt\leqslant
M_1^2
\]
provide this compactness.

Note, that to apply any compactness result we must consider
sequences in a fixed domain. To do that we use the well-known
extension result \cite{ACE} and restrict ourself with  special
domains $\Omega^{\varepsilon}$:

\begin{assumption} \label{assum1} \rm
Let $\chi(\mathbf{y})$ be 1-periodic in the variable
$\mathbf{y}$ function, such that $\chi(\mathbf{y})=1,
\mathbf{y}\in Y_f\subset Y$,  $\chi(\mathbf{y})=0,
\mathbf{y}\in Y_{s}= Y\backslash\overline{Y}_f$.
\begin{itemize}
\item[(1)] The set $Y_f$ is an open one and  $\gamma=\partial
Y_f\cap\partial Y_{s}$ is a Lipschitz continuous  surface.

\item[(2)] Let $Y_f^{\varepsilon}$ be a periodic repetition in $\mathbb{R}^n$ of
the elementary cell $\varepsilon Y_f$. Then
$Y_f^{\varepsilon}$ is a connected set with a  Lipschitz
continuous  boundary $\partial Y_f^{\varepsilon}$.

\item[(3)] $\Omega\subset \mathbb{R}^n$ is a bounded domain with a Lipschitz
continuous  boundary $S=\partial\Omega$ and $\Omega^{\varepsilon}=
\Omega\cap Y_f^{\varepsilon}$.
\end{itemize}
\end{assumption}

Due to periodicity of $Y^{\varepsilon}_f$ the characteristic
function of the domain $\Omega^{\varepsilon}$ in $\Omega$ has a form:
\[
\chi^{\varepsilon}(\mathbf{x})=\chi(\frac{\mathbf{x}}{\varepsilon}).
\]

For such domains $\Omega^{\varepsilon}$  the extension theorem
\cite{ACE} allows us to construct a linear operator
$\mathbb{A}^{\varepsilon}$
\begin{equation}
\mathbb{A}^{\varepsilon}:
W^1_2(\Omega^{\varepsilon})\to W^1_2(\Omega),
\tilde{c} ^{\varepsilon}=\mathbb{A}^{\varepsilon}(c^{\varepsilon}),
\label{0.8}
\end{equation}
such that
\begin{gather}
\int_{\Omega} |\tilde{c} ^{\varepsilon}(\mathbf{x},t)|^2dx
\leqslant C_0 \int_{\Omega^{\varepsilon}}
|c^{\varepsilon}(\mathbf{x},t)|^2dx,
\label{0.9}
\\
\int_{\Omega}
|\nabla \tilde{c} ^{\varepsilon}(\mathbf{x},t)|^2dx
\leqslant C_0 \int_{\Omega^{\varepsilon}}
|\nabla c^{\varepsilon}(\mathbf{x},t)|^2dx.
\label{0.10}
\end{gather}
where the constant $C_0=C_0(\Omega,Y_f)$ does not depend on
$\varepsilon$ and $t\in(0,T)$.


\section{Main results} \label{Main results}


Our principal result is the following

\begin{theorem}\label{thm1.1}
Let   $\{\tilde{c} ^{\varepsilon}(\mathbf{x},t)\}$
 be a bounded sequence in
$L_{\infty}\big((0,T);L_2(\Omega)\big)\cap W^{1,0}_2(\Omega_T)$
and weakly convergent in
$L_2\big((0,T);L_2(\Omega)\big)\cap W^{1,0}_2(\Omega_T)$
to a function $c(\mathbf{x},t)$. Also let  the sequence
$\{\partial/\partial t(\chi^{\varepsilon}(\mathbf{x})
\tilde{c} ^{\varepsilon}(\mathbf{x},t))\}$  be bounded in
 $L_2\big((0,T); W^{-1}_2(\Omega) \big)$, where
$\chi^{\varepsilon}(\mathbf{x})=\chi(\mathbf{x}/\varepsilon)$,
 $\chi(\mathbf{y})$ is 1-periodic in the variable $\mathbf{y}$
measurable bounded function, such that
\[
\langle   \chi   \rangle_{Y}=\int_{Y}\chi(\mathbf{y}) dy=m\neq 0,
\]
and  $Y$ is the unit cube in $\mathbb{R}^n$.
Then the sequence $\{\tilde{c} ^{\varepsilon}(\mathbf{x},t)\}$
converges strongly in $L_2(\Omega_T)$ to its weak limit
$c(\mathbf{x},t)$.
\end{theorem}

As an application of this result we consider a homogenization
of the problem \eqref{0.1}--\eqref{0.4}.

We prove the following result.

\begin{theorem}\label{thm1.2}
Under conditions \eqref{0.5}--\eqref{0.6} and Assumption \ref{assum1}  let
$c^{\varepsilon}(\mathbf{x},t)$ be the solution to the
problem \eqref{0.1}--\eqref{0.4}, $c_0\in L_2(\Omega)$,
\begin{equation}
\int_{\Omega} |c_0|^2dx\leqslant M_0^2,
\label{1.1}
\end{equation}
and
\begin{equation}
\tilde{\mathbf{v}}^{\varepsilon}\rightharpoonup
\mathbf{v}  \quad  \text{weakly in }  L_2(\Omega_T),
\label{1.2}
\end{equation}
where $\tilde{\mathbf{v}}^{\varepsilon}(\mathbf{x},t)=
\chi^{\varepsilon}(\mathbf{x})
\mathbf{v}^{\varepsilon}(\mathbf{x},t)$.
Then  the sequence $\{\tilde{c} ^{\varepsilon}\}$, where
$\tilde{c} ^{\varepsilon}=\mathbb{A}^{\varepsilon}(c^{\varepsilon})$,
converges strongly in $L_2(\Omega_T)$  and weakly in
$W^{1,0}_2(\Omega_T)$ to the solution $c(\mathbf{x},t)$ of
the homogenized equation
\begin{equation}
m \frac{\partial c}{\partial t}=
\nabla\cdot\big(\mathbb{B}\cdot\nabla c+(\mathbf{v}_0-\mathbf{v}) c\big),
\quad \mathbf{x}\in \Omega,  t\in(0,T),
\label{1.3}
\end{equation}
with boundary and initial conditions
\begin{gather}
c(\mathbf{x},t)=0,  \mathbf{x}\in S,  t\in(0,T),
\label{1.4}
\\
c(\mathbf{x},0)= c_0(\mathbf{x}),
\quad \mathbf{x}\in \Omega.
\label{1.5}
\end{gather}
In \eqref{1.3} the symmetric strictly positively defined constant
matrix $\mathbb{B}$ and the vector $\mathbf{v}_0$ are given below
by formulas \eqref{3.13} and \eqref{3.14}.
\end{theorem}

\section{Proof of Theorem \ref{thm1.1}}

 We split the proof into several independent steps.
As a first step we prove the following.

\begin{lemma}\label{lem2.1}
Under conditions of Theorem \ref{thm1.1} the sequence
$\{\chi^{\varepsilon}(\mathbf{x})
 \tilde{c}^{\varepsilon}(\mathbf{x},t)\}$ converges weakly
in $L_2(\Omega)$ to the function $m c(\mathbf{x},t)$ for almost
 all $t\in(0,T)$.
\end{lemma}

\begin{proof}
By the properties of the two-scale convergence \cite{NGU,AM} the
sequence $\{\tilde{c}^{\varepsilon}\}$ two-scale converges in
$L_2(\Omega_T)$ to the function $c(\mathbf{x},t)$. That
is, for any 1-periodic in variable $\mathbf{y}$ smooth function
$\varphi(\mathbf{x},\mathbf{y},t)$
\[
\int_{\Omega_T}\tilde{c}^{\varepsilon}(\mathbf{x},t)
\varphi(\mathbf{x},\frac{\mathbf{x}}{\varepsilon},t)\,dx\,dt
\to \int_{\Omega_T}c(\mathbf{x},t)\big(\int_{Y}
\varphi(\mathbf{x},\mathbf{y},t)dy\big) dx dt.
\]
In particular, this relation holds true for
$\varphi=\varphi_0(\mathbf{x},t)\varphi_1(\mathbf{y})$ with
$\varphi_0 \in L_{\infty}(\Omega_T)$ and $\varphi_1\in L_2(Y)$.
If we choose
\[
\varphi(\mathbf{x},\frac{\mathbf{x}}{\varepsilon},t)
=\chi(\frac{\mathbf{x}}{\varepsilon}) \eta(t)\psi(\mathbf{x})
=\chi^{\varepsilon}(\mathbf{x}) \eta(t)\psi(\mathbf{x}),
\]
then
\begin{equation}
\int_{\Omega_T}\tilde{c}^{\varepsilon}(\mathbf{x},t)
\chi^{\varepsilon}(\mathbf{x})
\eta(t)\psi(\mathbf{x})\,dx\,dt\to
\int_{\Omega_T}m c(\mathbf{x},t)
\eta(t)\psi(\mathbf{x})\,dx\,dt. \label{2.1}
\end{equation}
Let
\[
f_{\psi}^{ \varepsilon}(t)=
\int_{\Omega}\chi^{\varepsilon}(\mathbf{x})
\tilde{c}^{\varepsilon}(\mathbf{x},t)
\psi(\mathbf{x})dx,
\quad
f_{\psi}(t)=\int_{\Omega}m c(\mathbf{x},t)
\psi(\mathbf{x})dx.
\]
Then the above relation means that
\begin{equation}
\int_0^{T}\eta(t)f_{\psi}^{\varepsilon}(t)
dt\to\int_0^{T}\eta(t)f_{\psi}(t)   dt,
\label{2.2}
\end{equation}
for any functions $\eta\in L_{\infty}(0,T)$  and
$\psi\in L_{\infty}(\Omega)$.

To prove the lemma  we have to show that for almost all  $t\in
(0,T)$ functions $f_{\psi}^{ \varepsilon}(t)$ pointwise converge
to the function $f_{\psi}(t)$. First of all,  we restrict ourself
with functions $\psi\in {\mathaccent"7017 W}^1_2 (\Omega)$.

By the assumptions in Theorem \ref{thm1.1}, the time derivatives
$\partial/\partial t\big(\chi^{\varepsilon}(\mathbf{x})
\tilde{c}^{\varepsilon}\big)$ belong to the space
$L_2\big((0,T);{\mathaccent"7017 W}^{-1}_2 (\Omega)\big)$
and uniformly bounded there.  This means that there exists a
sequence $\{\mathbf{F}^{\varepsilon}(\mathbf{x},t)\}$,
such that
\[
\int_{\Omega_T}|\mathbf{F}^{\varepsilon}|^2\,dx\,dt
\leqslant M^2_0,
\]
and
\begin{equation}
\int_{\Omega_T}\frac{d\varphi(t)}{dt}
\chi^{\varepsilon}(\mathbf{x})
\tilde{c}^{\varepsilon}(\mathbf{x},t)
\psi(\mathbf{x})\,dx\,dt=
\int_{\Omega_T}\varphi(t)
\mathbf{F}^{\varepsilon}(\mathbf{x},t)\cdot
\nabla\psi(\mathbf{x})\,dx\,dt
\label{2.3}
\end{equation}
for any  $\varphi\in  ^1_2 (0,T)$
and  $\psi\in {\mathaccent"7017 W}^1_2 (\Omega)$.
If we put
\[
g^{\varepsilon}(t)=-\int_{\Omega}
\mathbf{F}^{\varepsilon}(\mathbf{x},t)
\cdot\nabla\psi(\mathbf{x})d\mathbf{x},
\]
then
\[
\int_0^{T}|g^{\varepsilon}|^2dt\leqslant
M^2_0 \|\nabla\psi\|_{2,\Omega}^2=M_{\psi}^2,
\]
and identity \eqref{2.3} rewrites as
\begin{equation}
\int_0^{T}\big(f_{\psi}^{\varepsilon}(t)\frac{d\varphi(t)}{d
t}+ \varphi(t)g^{\varepsilon}(t)\big)dt=0.
\label{2.4}
\end{equation}
Therefore by \cite{Adams}, the function  $f_{\psi}^{\varepsilon}(t)$
possesses the generalized time derivative
$g^{\varepsilon}(t)\in L_2(0,T)$ and takes place a representation
\[
f_{\psi}^{\varepsilon}(t)=f_{\psi}^{\varepsilon}(t_{\varepsilon})+
\int_{t_{\varepsilon}}^{t} g^{\varepsilon}(\tau)d\tau,
|f_{\psi}^{\varepsilon}(t_{\varepsilon})|\leqslant M_{\psi}.
\]
In particular,
\begin{equation}
|f_{\psi}^{\varepsilon}(t)|\leqslant  M_{\psi},
|f_{\psi}^{\varepsilon}(t_1)-f_{\psi}^{\varepsilon}(t_2)|
\leqslant M_{\psi} |t_2-t_1|^{1/2}.
\label{2.5}
\end{equation}
Thus, we may apply the Ascoli-Arzela theorem \cite{KF} and
state that there exists some subsequence $\{\varepsilon_{m}\}$,
such that the sequence of continuous functions
$\{f_{\psi}^{\varepsilon_{m}}(t)\}$ uniformly converges to some
continuous function  $\overline{f}_{\psi}(t)$:
\begin{equation}
f_{\psi}^{\varepsilon_{m}}(t)\Rightarrow\overline{f}_{\psi}(t),
\quad \text{as }  \varepsilon_{m}\to 0,  \forall t\in (0,T).
\label{2.6}
\end{equation}
Therefore,
\begin{equation}
\int_0^{T}\eta(t)f_{\psi}^{\varepsilon_{m}}(t)   dt\to
\int_0^{T}\eta(t)\overline{f}_{\psi}(t)dt, \quad
\text{as }  \varepsilon_{m}\to 0.
\label{2.7}
\end{equation}
But, on the other hand, according to \eqref{2.1}
\begin{equation}
\int_0^{T}\eta(t)f_{\psi}^{\varepsilon_{m}}(t)
dt\to\int_0^{T}\eta(t)f_{\psi}(t)   dt,
\text{as}  \varepsilon_{m}\to 0.
\label{2.8}
\end{equation}
By the arbitrary choice of $\eta(t)$ \eqref{2.6}-\eqref{2.8}
result
\[
f_{\psi}^{\varepsilon_{m}}(t)\to f_{\psi}(t)\quad
\text{as }  \varepsilon_{m}\to 0,
 \text{ for almost all }  t\in[0,T].
\]
Due to the uniqueness of the limit, the last relation holds
for the entire sequence  $\{f_{\psi}^{\varepsilon}(t)\}$:
\[
f_{\psi}^{ \varepsilon}(t)=
\int_{\Omega}\chi^{\varepsilon}(\mathbf{x})
c^{\varepsilon}(\mathbf{x},t)
\psi(\mathbf{x})dx\to
\int_{\Omega}m c(\mathbf{x},t)\psi(\mathbf{x})dx=
f_{\psi}(t)
\]
as $\varepsilon\to 0$  for almost all $t\in(0,T)$.
\end{proof}

As a next step we prove the following result.

\begin{lemma}\label{lem2.2}
Under conditions of Theorem \ref{thm1.1} there exists
a subsequence $\{\varepsilon_k\}$, such that
\begin{equation}
\lim_{\varepsilon_k\to 0}\varepsilon_k^2
\int_{\Omega}|\nabla \tilde{c} ^{\varepsilon_k}
(\mathbf{x},t_0)|^2dx=0
\label{2.9}
\end{equation}
for almost all  $t_0\in(0,T)$.
\end{lemma}

\begin{proof}
In fact,  the boundedness  of the sequence
$\{\nabla \tilde{c} ^{\varepsilon}\}$ in  $L_2(\Omega_T)$
implies
\begin{equation}
\lim_{\varepsilon\to 0}\varepsilon^2
\int_{\Omega_T}|\nabla
\tilde{c} ^{\varepsilon}(\mathbf{x},t)|^2\,dx\,dt=0.
\label{2.10}
\end{equation}
Let
\[
u^{\varepsilon}(t)=\varepsilon^2 \int_{\Omega}|\nabla
\tilde{c}^{\varepsilon}(\mathbf{x},t)|^2dx.
\]
Then the relation \eqref{2.10} means that the sequence
$\{u^{\varepsilon}\}$ converges to zero in $L_1(0,T)$. Due to
the well-known theorem of functional analysis \cite{KF}
there exists some subsequence $\{\varepsilon_k\}$,
such that the sequence $\{u^{\varepsilon_k}(t_0)\}$
pointwise converge to zero for  almost all  $t_0\in (0,T)$:
\[
u^{\varepsilon_k}(t_0)\to 0\quad
 \text{for almost all }    t_0\in (0,T).
\]
The above relation  proves \eqref{2.9}.
\end{proof}

The following statement is a crucial one and essentially
uses the notion of two-scale convergence.

\begin{lemma}\label{lem2.3}
Under the conditions of Theorem \ref{thm1.1}, the sequence
$\{\tilde{c} ^{\varepsilon_k}(\mathbf{x},t_0)\}$ two-scale
converges in $L_2(\Omega)$ to the function $c(\mathbf{x},t_0)$
for almost all $t_0\in (0,T)$.
\end{lemma}

\begin{proof}
Let $Q\subset (0,T)$ be the set of full measure  in  $(0,T)$,
where hold true conditions of the Lemma \ref{lem2.1} and
condition \eqref{2.9}.

By hypothesis, the sequence
$\{\tilde{c} ^{\varepsilon_k}(\mathbf{x},t_0)\}$
for $t_0\in Q$ is bounded in  $L_2(\Omega)$.
Therefore, there exists some subsequence which  two-scale
converges in  $L_2(\Omega)$ to some 1-periodic in
variable $\mathbf{y}$ function
$\overline{C}(\mathbf{x},\mathbf{y},t_0)
 \in L_2(\Omega\times Y)$.
Applying integration by parts
\begin{align*}
&\varepsilon_k \int_{\Omega}\nabla
c^{\varepsilon_k}(\mathbf{x},t_0)\cdot
\boldsymbol{\varphi}(\frac{\mathbf{x}}{\varepsilon_k})
\psi(\mathbf{x})dx\\
&=-\varepsilon_k\int_{\Omega}
c^{\varepsilon_k}(\mathbf{x},t_0)
\boldsymbol{\varphi}(\frac{\mathbf{x}}{\varepsilon_k})
\cdot\nabla\psi(\mathbf{x})dx
-\int_{\Omega}c^{\varepsilon_k}(\mathbf{x},t_0)
\big(\nabla_{y}\cdot
\boldsymbol{\varphi}(\frac{\mathbf{x}}{\varepsilon_k})\big)
\psi(\mathbf{x})dx
\end{align*}
for arbitrary functions  $\boldsymbol{\varphi}\in W^1_2(Y)$
and $\psi\in {\mathaccent"7017 W}^1_2 (\Omega)$,
and relation \eqref{2.9} we arrive at the equality
\begin{equation}
\int_{\Omega}\psi(\mathbf{x})\Big(\int_{Y}
\overline{C}(\mathbf{x},\mathbf{y},t_0)\nabla_{y}\cdot
\boldsymbol{\varphi}(\mathbf{y})dy\Big)dx=0
\label{2.11}
\end{equation}
after passing to the limit as $\varepsilon_k\to 0$.

By the arbitrary choice of test functions $\boldsymbol{\varphi}$
and $\psi$, the last integral identity  implies
\begin{equation}
\overline{C}(\mathbf{x},\mathbf{y},t_0)=
\overline{c}(\mathbf{x},t_0).
\label{2.11b}
\end{equation}
Thus, the chosen  subsequence of the sequence
$\{c^{\varepsilon_k}(\mathbf{x},t_0)\}$ two-scale
converges in  $L_2(\Omega)$ to the function
$\overline{c}(\mathbf{x},t_0)$. In particular,
 by the properties of  two-scale convergent sequences \cite{NGU}
the same subsequence of  $\{\chi^{\varepsilon_k}(\mathbf{x})
c^{\varepsilon_k}(\mathbf{x},t_0)\}$, where
$\chi^{\varepsilon_k}(\mathbf{x})=
\chi(\mathbf{x}/\varepsilon_k)$,  weakly converges in
$L_2(\Omega)$ to the function
$m \overline{c}(\mathbf{x},t_0)$. On the other hand, due to
Lemma \ref{lem2.1} this subsequence weakly converges in
$L_2(\Omega)$ to the function $m c(\mathbf{x},t_0)$. The
uniqueness of the weak limit results the equality
\[
\overline{c}(\mathbf{x},t_0)=c(\mathbf{x},t_0)
\]
and the convergence of the entire sequence
$\{c^{\varepsilon_k}(\mathbf{x},t_0)\}$ to the same limit.
\end{proof}

\begin{lemma}\label{lem2.4}
Under the conditions of Theorem \ref{thm1.1},
 the sequence $\{\tilde{c} ^{\varepsilon_k}\}$ converges strongly
in $L_2(\Omega_T)$ to the function  $c(\mathbf{x},t)$.
\end{lemma}

\begin{proof}
Let
\[
\mathbb{H}^1=W^1_2(\Omega)\subset\mathbb{H}^{0}=
L_2(\Omega)\subset\mathbb{H}^{-1}=W^{-1}_2(\Omega).
\]
It is well known that $\mathbb{H}^1$ is compactly imbedded
 in $\mathbb{H}^{0}$,  and $\mathbb{H}^{0}$ is compactly imbedded
in $\mathbb{H}^{-1}$ (\cite{Lions}, \cite{Adams}).
The first imbedding provides for any $\eta>0$ an existence of
some constant $C_{\eta}$ such that
\[
\|\tilde{c} ^{\varepsilon_k}-c\|_{\mathbb{H}^{0}}(t)\leqslant
\eta\|\tilde{c} ^{\varepsilon_k}-c\|_{\mathbb{H}^1}(t)+
C_{\eta}\|\tilde{c} ^{\varepsilon_k}-c\|_{\mathbb{H}^{-1}}(t)
\]
for all $k$ and for all $t\in[0,T]$ (see \cite{Lions}). Therefore,
\begin{align*}
\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\|
_{\mathbb{H}^{0}}^2(t)dt
&\leqslant
\eta\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\|
_{\mathbb{H}^1}^2(t)dt+
C_{\eta}\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\|
_{\mathbb{H}^{-1}}^2(t)dt\\
&\leqslant 2 \eta M_0^2+
C_{\eta}\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\|
_{\mathbb{H}^{-1}}^2(t)dt.
\end{align*}
Due to the compact imbedding $\mathbb{H}^{0}\to\mathbb{H}^{-1}$,
the weak convergence  in  $\mathbb{H}^{0}$ of the sequence
$\{\tilde{c} ^{\varepsilon_k}(\mathbf{x},t_0)\}$ to the function
 $c(\mathbf{x},t_0)$ for all $t_0\in Q$,  and the dominated
convergence theorem \cite{KF} one has
\[
\int_0^{T}\|\tilde{c} ^{\varepsilon_k}-c\|
_{\mathbb{H}^{-1}}^2(t)dt \to 0 \quad
\text{as }  k\to\infty.
\]
This last fact and the arbitrary choice of the constant $\eta$
prove the statement of the lemma.
\end{proof}

\section{Proof of Theorem \ref{thm1.2}}

To simplify the proof we additionally suppose that

\begin{assumption} \label{assum2}\rm
\begin{itemize}
\item[(1)]  $Y_{s}\subset Y,  \gamma \cap \partial Y=\emptyset$;

\item[(2)]  the domain $\Omega$ is a unit cube;

\item[(3)]  $1/\varepsilon$ is an integer.
\end{itemize}
\end{assumption}

As before, we divide the proof by several steps.
As a first step we state the
well-known existence and uniqueness result for solutions of the
problem \eqref{0.1}--\eqref{0.3} (see  \cite{LSU}).

\begin{lemma}\label{lem3.1}
Under conditions of Theorem \ref{thm1.2} for all
$\varepsilon >0$ the problem \eqref{0.1}--\eqref{0.4}
has a unique solution
\[
c^{\varepsilon}\in
L_{\infty}\big((0,T);L_2(\Omega^{\varepsilon})\big)\cap
W^{1,0}_2(\Omega_T^{\varepsilon})
\]
and
\begin{equation}
\max_{0<t<T}\int_{\Omega^{\varepsilon}}
|c^{\varepsilon}(\mathbf{x},t)|^2dx+
\int_{\Omega^{\varepsilon}_T}|\nabla
c^{\varepsilon}|^2\,dx\,dt \leqslant M^2_1.
\label{3.1}
\end{equation}
\end{lemma}

To get the basic estimate \eqref{3.1} we first rewrite  \eqref{0.1}
in the form
\[
\frac{\partial c^{\varepsilon}}{\partial t}=
\nabla\cdot(\nabla c^{\varepsilon}-
\mathbf{v}^{\varepsilon}c^{\varepsilon}),
\]
multiply by $c^{\varepsilon}$ and integrate by parts over
domain $\Omega^{\varepsilon}$:
\[
\frac{1}{2}\frac{d}{dt}\int_{\Omega^{\varepsilon}}
|c^{\varepsilon}(\mathbf{x},t)|^2dx+
\int_{\Omega^{\varepsilon}}|\nabla c^{\varepsilon}|^2dx
= \int_{\Omega^{\varepsilon}}c^{\varepsilon}
\mathbf{v}^{\varepsilon}\cdot\nabla c^{\varepsilon}dx.
\]
Let $\tilde{c} ^{\varepsilon}(.,t)=\mathbb{A}^{\varepsilon}
\big(c^{\varepsilon}(.,t)\big)$ be an extension of the function
$c^{\varepsilon}$ onto domain $\Omega$. Then
\begin{equation}
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\chi^{\varepsilon}
|\tilde{c}^{\varepsilon}(\mathbf{x},t)|^2dx+
\int_{\Omega}\chi^{\varepsilon}|
\nabla\tilde{c}^{\varepsilon}|^2dx
= \int_{\Omega}\chi^{\varepsilon}\tilde{c}^{\varepsilon}
\mathbf{v}^{\varepsilon}\cdot
\nabla \tilde{c}^{\varepsilon}dx\equiv J_1.
\label{3.2}
\end{equation}
To estimate $J_1$ we use the H\"{o}lder inequality:
\begin{align*}
|J_1|&\leqslant\big(\int_{\Omega}\chi^{\varepsilon}
|\mathbf{v}^{\varepsilon}|^{4}dx\big)^{1/4}\cdot
\big(\int_{\Omega}\chi^{\varepsilon}
|\tilde{c}^{\varepsilon}|^{4}dx\big)^{1/4}\cdot
\big(\int_{\Omega}\chi^{\varepsilon}
|\nabla \tilde{c}^{\varepsilon}|^2dx\big)^{1/2}\\
&\leqslant \big(\int_{\Omega}\chi^{\varepsilon}
|\mathbf{v}^{\varepsilon}|^{4}dx\big)^{1/4}\cdot
\big(\int_{\Omega}|\tilde{c}^{\varepsilon}|^{4}dx
\big)^{1/4}\cdot\big(\int_{\Omega}
|\nabla \tilde{c}^{\varepsilon}|^2dx\big)^{1/2}.
\end{align*}
Due to Assumption \ref{assum2}
\[
\tilde{c}^{\varepsilon}\in {\mathaccent"7017 W}^1_2(\Omega),
\]
and we may apply the well-known interpolation inequality (see \cite{LSU})
\[
\big(\int_{\Omega}|\tilde{c}^{\varepsilon}|^{4}dx
\big)^{1/4}\leqslant\beta\big(\int_{\Omega}
|\tilde{c}^{\varepsilon}|^2dx\big)^{1/8}\cdot
\big(\int_{\Omega}|\nabla \tilde{c}^{\varepsilon}|^2dx
\big)^{3/8}.
\]
Therefore (see \eqref{0.9} and \eqref{0.10})
\begin{align*}
|J_1|& \leqslant\beta\big(\int_{\Omega}\chi^{\varepsilon}
|\mathbf{v}^{\varepsilon}|^{4}dx\big)^{1/4}\cdot
\big(\int_{\Omega}|\tilde{c}^{\varepsilon}|^2dx
\big)^{1/8}\cdot\big(\int_{\Omega}
|\nabla \tilde{c}^{\varepsilon}|^2dx\big)^{7/8}\\
&\leqslant C_0\beta\big(\int_{\Omega}\chi^{\varepsilon}
|\mathbf{v}^{\varepsilon}|^{4}dx\big)^{1/4}\cdot
\big(\int_{\Omega}\chi^{\varepsilon}|\tilde{c}^{\varepsilon}
|^2dx\big)^{1/8}\cdot\big(\int_{\Omega}\chi^{\varepsilon}
|\nabla \tilde{c}^{\varepsilon}|^2dx\big)^{7/8}.
\end{align*}
Applying Young's  and  Gronwall inequalities and using
assumption \eqref{0.5} and properties of the extension
operator  $\mathbb{A}^{\varepsilon}$  we arrive at
\begin{equation}
\max_{0<t<T}\int_{\Omega}
|\tilde{c} ^{\varepsilon}(\mathbf{x},t)|^2dx+
\int_{\Omega_T}|\nabla
\tilde{c} ^{\varepsilon}|^2\,dx\,dt
\leqslant M^2_1, \label{3.3}
\end{equation}
which is obviously equivalent to \eqref{3.1}.

The integral identity for the function $\tilde{c} ^{\varepsilon}$
with test functions $\phi=\varphi(t)\psi(\mathbf{x})$,
 $\varphi\in {\mathaccent"7017 W}^1_2 (0,T)$,
$\psi\in {\mathaccent"7017 W}^1_2 (\Omega)$ takes a form
\[
\int_{\Omega_T}\frac{d\varphi}{d t}(t) \chi^{\varepsilon}
\tilde{c} ^{\varepsilon}\psi(\mathbf{x})\,dx\,dt=
\int_{\Omega_T}\varphi(t) \chi^{\varepsilon}
\big(\nabla \tilde{c} ^{\varepsilon}
-\mathbf{v}^{\varepsilon} \tilde{c} ^{\varepsilon}\big)\cdot
\nabla \psi(\mathbf{x})\,dx\,dt.
\]
Thus,
\[
\frac{\partial}{\partial t}\big(\chi^{\varepsilon}(\mathbf{x})
\tilde{c} ^{\varepsilon}\big)
\in L_2\big((0,T); W^{-1}_2(\Omega) \big),
\]
and we may apply Theorem \ref{thm1.1}  and
 Nguetseng's Theorem \cite{NGU} to state, that up to some
subsequence the sequence
$\{\tilde{c} ^{\varepsilon}\}$ weakly in
${\mathaccent"7017 W}^{1,0}_2(\Omega_T)$ and strongly in
$L_2(\Omega_T)$ converges to the function $c(\mathbf{x},t)$,
and the sequence $\{\nabla\tilde{c} ^{\varepsilon}\}$ two-scale
converges in $L_2(\Omega_T)$ to  1-periodic in variable $\mathbf{y}$
function $\nabla c(\mathbf{x},t)+\nabla_{y}C(\mathbf{x},\mathbf{y},t)$.

We may also assume that the sequence
$\{\mathbf{v}^{\varepsilon}\}$ two-scale converges
to  1-periodic in variable $\mathbf{y}$ function
$\mathbf{V}(\mathbf{x},\mathbf{y},t)$.

The next lemmas are  standard. We derive the macro-and
microscopic equations and find the solution of microscopic
equation.

\begin{lemma}\label{lem3.2}
Under conditions of Theorem \ref{thm1.2},
 the two-scale limits $c(\mathbf{x},t)$ and
$C(\mathbf{x},\mathbf{y},t)$ satisfy the macroscopic
integral identity
\begin{equation}
\int_{\Omega_T}\Big(m c \frac{\partial\phi}{\partial
t}-\big(m \nabla c +\langle\nabla_{y}C\rangle_{Y_f}-
\mathbf{v} c\big)\cdot \nabla \phi\Big)\,dx\,dt
=-\int_{\Omega} m c_0(\mathbf{x})\phi(\mathbf{x},0)dx
\label{3.4}
\end{equation}
for arbitrary smooth functions $\phi(\mathbf{x},t)$, such that
$\phi(\mathbf{x},T)=0$, which is equivalent to the macroscopic
equation
\begin{equation}
m \frac{\partial c}{\partial t}
=\nabla\cdot\big(m \nabla c+\langle\nabla_{y}C\rangle_{Y_f}
-c \mathbf{v}\big),
\quad \mathbf{x}\in \Omega,  t\in(0,T),
\label{3.5}
\end{equation}
with boundary and initial conditions
\begin{gather}
c(\mathbf{x},t)=0,  \mathbf{x}\in S,  t\in(0,T), \label{3.6}
\\
c(\mathbf{x},0)=c_0(\mathbf{x}),
\quad \mathbf{x}\in \Omega.
\label{3.7}
\end{gather}
\end{lemma}

To prove this lemma we just fulfill the two-scale limit as
$\varepsilon\to 0$ in the integral identity \eqref{0.7}
for the functions $\tilde{c} ^{\varepsilon}$ in the form
\begin{equation}
\int_{\Omega_T}\chi^{\varepsilon}\Big(\tilde{c} ^{\varepsilon}
\frac{\partial\phi}{\partial
t}-\big(\nabla \tilde{c} ^{\varepsilon}
-\tilde{\mathbf{v}}^{\varepsilon} \tilde{c} ^{\varepsilon}\big)\cdot
\nabla \phi\Big)\,dx\,dt=-\int_{\Omega}\chi^{\varepsilon}
c_0(\mathbf{x})\phi(\mathbf{x},0)dx
\label{3.8}
\end{equation}
with the test functions $\phi=\phi(\mathbf{x},t)$.

\begin{lemma}\label{lem3.3}
Under conditions of Theorem \ref{thm1.2} the two-scale
limits $c(\mathbf{x},t)$ and $C(\mathbf{x},\mathbf{y},t)$
satisfy the microscopic integral identity
\begin{equation}
\int_{Y}\chi(\mathbf{y})\big(\nabla c +\nabla_{y}C
-c \mathbf{V}\big)\cdot
\nabla \phi_1\,dy=0 \label{3.9}
\end{equation}
for arbitrary 1-periodic in variable $\mathbf{y}$ smooth
functions $\phi_1(\mathbf{y})$.
\end{lemma}

The integral identity \eqref{3.9} follows from \eqref{3.8} after
fulfilling the two-scale limit as $\varepsilon\to 0$
with test functions $\phi=\varepsilon\phi_0(\mathbf{x},t)
\phi_1(\mathbf{x} /\varepsilon)$.

\begin{lemma}\label{lem3.4}
Let $C^{(i)}(\mathbf{y}), i=1,2,3$, be the solution to the
integral identity
\begin{equation}
\int_{Y}\chi(\mathbf{y})\big(\mathbf{e}_i+
\nabla_{y}C^{(i)}\big)\cdot \nabla \phi_1\,dy=0,
\label{3.10}
\end{equation}
and $C^{(0)}(\mathbf{y},\mathbf{x},t)$  be the solution to the
integral identity
\begin{equation}
\int_{Y}\chi(\mathbf{y})\big(\mathbf{V}+
\nabla_{y}C^{(0)}\big)\cdot \nabla \phi_1\,dy=0,
\label{3.11}
\end{equation}
with arbitrary 1-periodic in variable $\mathbf{y}$ smooth
functions $\phi_1(\mathbf{y})$.
Then the function
\begin{equation}
C(\mathbf{x},\mathbf{y},t)=\big(\sum_{i=1}^{3}
C^{(i)}(\mathbf{y})\otimes \mathbf{e}_i\big)
\cdot \nabla c(\mathbf{x},t)+C^{(0)}(\mathbf{y},\mathbf{x},t)
c(\mathbf{x},t)
\label{3.12}
\end{equation}
solves the integral identity \eqref{3.9}.

In \eqref{3.10}--\eqref{3.12} $\mathbf{e}_i$ is the
standard Cartesian basis vector and  the matrix
$\mathbf{a}\otimes \mathbf{b}$ is defined by the formula
$$
(\mathbf{a}\otimes \mathbf{b})\cdot
\mathbf{c}=\mathbf{a}(\mathbf{b}\cdot \mathbf{c})
$$
for any vectors  $\mathbf{a}, \mathbf{b}, \mathbf{c}$.
\end{lemma}

The proof of the lemma is straightforward. It is omitted.

Substitution \eqref{3.12} into \eqref{3.5} gives us desired
homogenized equation \eqref{1.3} with boundary and initial
conditions \eqref{1.4}--\eqref{1.5}.

The matrix $\mathbb{B}$ and the vector $\mathbf{v}_0(\mathbf{x},t)$
 are defined as
\begin{equation}
\mathbb{B}=m\mathbb{I}+\big(\sum_{i=1}^{3}
\langle \nabla_{y}C^{(i)}\rangle_{Y_f}\otimes \mathbf{e}_i\big),
\label{3.13}
\end{equation}
\begin{equation}
\mathbf{v}_0(\mathbf{x},t)=\langle \nabla_{y}C^{(0)}\rangle_{Y_f},
\label{3.14}
\end{equation}
where by definition $\langle
f\rangle_{Y_f}=\int_{Y_f}f(\mathbf{y})dy$.

\begin{lemma}\label{lem3.5}
The matrix $\mathbb{B}$ is symmetric and strictly positively defined.
\end{lemma}

The proof is well-known, see  \cite{BLP,JKO}.


\subsubsection*{Acknowledgments}
This research  is partially supported  by the Federal
Program ``Research and scientific-pedagogical brainpower of
Innovative Russia" for 2009-2013 (State Contract  02.740.11.0613).

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\end{document}