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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 14, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/14\hfil Short-time filtration]
{Homogenized models for a short-time  filtration  in
elastic porous media}

\author[A. M. Meirmanov\hfil EJDE-2007/14\hfilneg]
{Anvarbek M. Meirmanov}

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

\thanks{Submitted August 27, 2007. Published January 31, 2008.}
\subjclass[2000]{35M20, 74F10, 76S05}
\keywords{Stokes equations; Lam\'{e}'s equations;
hydraulic fracturing; \hfill\break\indent
two-scale convergence; homogenization of periodic structures}

\begin{abstract}
 We consider a linear system of differential equations describing a
 joint motion of elastic porous body and fluid occupying porous
 space.  The rigorous justification, under various conditions imposed
 on physical parameters, is fulfilled for homogenization procedures
 as the dimensionless size of the pores tends to zero, while the
 porous body is geometrically periodic and a characteristic time of
 processes is small enough. Such kind of models may describe, for
 example, hydraulic fracturing or  acoustic or seismic waves
 propagation. As the results, we derive homogenized equations
 involving non-isotropic Stokes system for fluid velocity coupled
 with   two different types of acoustic equations for the solid
 component, depending on ratios between physical parameters, or
 non-isotropic Stokes system for one-velocity continuum.
 The proofs are based on Nguetseng's two-scale convergence method
 of homogenization in periodic structures.
\end{abstract}

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


\section{Introduction} \label{Introduction}

In the present paper we consider a problem of  joint motion of a
deformable solid (elastic skeleton), perforated by  system of
 pores (pore space) and a fluid, occupying
pore space. In dimensionless variables (without primes)
$$
\mathbf x'=L \mathbf x, \quad t'=\tau t,\quad
\mathbf w'=\frac{L^{2}}{g\tau^{2}} \mathbf w, \quad \rho'_s= \rho_0
\rho_s,\quad \rho'_f =\rho_0 \rho_f,\quad \mathbf F'=g\mathbf F,
$$
differential equations of the problem in a domain $\Omega \in
\mathbb{R}^3$ for the dimensionless displacement vector
$\mathbf{w}$ of the continuum medium  have the form
\begin{gather}\label{0.1}
\bar{\rho} \frac{\partial^2\mathbf w}{\partial t^2}= \mathop{\rm div}
\mathbb{P} +\bar{\rho}\mathbf F, \\
\label{0.2}
\mathbb {P} =\bar{\chi}\mathbb {P}^{f}+(1-\bar{\chi})\mathbb {P}^{s} ,\\
\label{0.3}
\mathbb {P}^{f}=\alpha_\mu D\bigl(x,\frac{\partial\mathbf
w}{\partial t}\bigr)-p_{f}\mathbb I, \\
\label{0.4}
\mathbb {P}^{s}=\alpha_\lambda \mathbb {D}(x,\mathbf{w})+\alpha_\eta
(\mathop{\rm div}\mathbf{w})\mathbb I, \\
\label{0.5}
 p_{f}+\bar{\chi} \alpha_p \mathop{\rm div}\mathbf{w}=0.
\end{gather}
Hereafter we use  notation
 $$
\mathbb {D}(x,\mathbf u)=(1/2)\left(\nabla\mathbf u +
(\nabla\mathbf u)^T\right),\quad \bar{\rho}=\bar{\chi}\rho_f
+(1-\bar{\chi})\rho_s,
$$
The vector $\mathbb I$ is a unit tensor, the given
function $\bar{\chi}(\mathbf x)$ is a characteristic function of the
pore space, the given function
 $\mathbf F(\mathbf x,t)$ is a dimensionless vector  of distributed
mass forces,  $\mathbb {P}^{f}$ is a liquid stress tensor,
$\mathbb {P}^{s}$ is a stress tensor in a solid skeleton and $p_{f}$ is a
liquid pressure.

These differential equations (\ref{0.1})--(\ref{0.5}) mean that the
the displacement vector $\mathbf{w}$ satisfies the Stokes equations
 in the pore space $\Omega_{f}$  and  the Lame's equations
 in the solid skeleton  $\Omega_{s}$.

On the common boundary $\Gamma$ "solid skeleton-pore space" the
displacement vector $\mathbf{w}$ and the liquid pressure $p_{f}$
satisfy the usual continuity condition
\begin{equation} \label{0.6}
[\mathbf w](\mathbf x_0,t)=0,\quad \mathbf x_0\in \Gamma,\; t\geq
0
\end{equation}
and the momentum conservation law in the form
\begin{equation} \label{0.7}
[\mathbb {P}\cdot \mathbf{n}](\mathbf x_0,t)=0, \quad \mathbf x_0\in
\Gamma ,\; t\geq 0,
\end{equation}
 where $\mathbf{n}(\mathbf x_0)$ is a unit
normal to the boundary at the point $\mathbf x_0\in \Gamma$ and
\begin{gather*}
 [\varphi](\mathbf x_0,t)=\varphi_{(s)}(\mathbf x_0,t)
-\varphi_{(f)}(\mathbf x_0,t),\\
\varphi_{(s)}(\mathbf x_0,t) =\lim_{\mathbf x\to \mathbf x_0,\;
\mathbf x\in \Omega_s} \varphi(\mathbf x,t),\\
\varphi_{(f)}(\mathbf x_0,t) =\lim_{\mathbf x\to \mathbf x_0,\;
\mathbf x\in \Omega_f} \varphi(\mathbf x,t).
\end{gather*}


 The problem is endowed   with homogeneous initial and boundary conditions
\begin{gather} \label{0.8}
\mathbf w(\mathbf{x},0)=0,\quad
\frac{\partial \mathbf w}{\partial t}(\mathbf{x},0)=0,\quad
\mathbf x\in \Omega , \\
 \label{0.9}
\mathbf w(\mathbf{x},t)=0,\quad  \mathbf x \in S=\partial \Omega,
\quad t\geq 0.
\end{gather}

The dimensionless constants $\alpha_i$ $(i=\tau,\nu,\ldots)$ are defined
by the formulas
$$ \alpha_\mu =\frac{2\mu \tau}{L^{2}\rho_0},\quad
\alpha_\lambda =\frac{2\lambda \tau^{2}}{L^{2}\rho_0},\quad
 \alpha_p =\rho _{f}c^{2}\frac{\tau^{2}}{L^{2}},
 \quad \alpha_\eta =\frac{\eta\tau^{2}}{L^{2}\rho_0},
 $$
where $\mu$ is the  viscosity of fluid,  $\lambda$ and $\eta$ are
elastic Lam\'{e}'s constants, $c$ is a  speed of sound in fluid, $L$
is a characteristic size of the domain in consideration, $\tau$ is a
characteristic time of the process, $\rho_f$ and $\rho_s$ are
respectively mean dimensionless densities of liquid and rigid
phases, correlated with mean density of water and $g$ is the value
of acceleration of gravity.

 The corresponding mathematical model, describing by the system
\eqref{0.1}--\eqref{0.9} is commonly accepted (see \cite{B-K,S-P})
and contains a natural small parameter
 $\varepsilon$, which is  a characteristic size of pores $l$ divided by the
characteristic size $L$ of the entire porous body:
$$
\varepsilon =\frac{l}{L}.
$$
Our aim is to derive all possible limiting regimes (homogenized
equations)  as $\varepsilon\searrow 0$. Such an approximation
significantly simplifies the original problem and at the same time
preserves all of its main features. But even this approach is too
hard to work out, and some additional simplifying assumptions are
necessary. In terms of geometrical properties of the medium, the
most appropriate is to simplify the problem postulating that the
porous structure is periodic.

We accept the following constraints

\begin{assumption} \label{assumption1} \rm
The domain  $\Omega =(0,1)^3$ is a periodic repetition of an elementary
cell  $Y^\varepsilon =\varepsilon Y$, where $Y=(0,1)^3$ and quantity
$1/\varepsilon$ is integer, so that $\Omega$ always contains an
integer number of elementary cells $Y^\varepsilon$. Let $Y_s$  be a
"solid part" of $Y$, and the "liquid part"  $Y_f$ -- is its open
complement.  We denote as $\gamma =\partial Y_f \cap
\partial Y_s$ and $\gamma $ is a  Lipschitz continuous surface.
 A pore space  $\Omega ^{\varepsilon}_{f}$  is the periodic repetition of
the elementary cell $\varepsilon Y_f$, and a solid skeleton  $\Omega
^{\varepsilon}_{s}$  is the periodic repetition of the elementary
cell $\varepsilon Y_s$. A Lipschitz continuous boundary
$\Gamma^\varepsilon =\partial \Omega_s^\varepsilon \cap
\partial \Omega_f^\varepsilon$ is the periodic repetition in
$\Omega$ of the boundary $\varepsilon \gamma$.
The ``solid
skeleton" $\Omega _{s}^\varepsilon$ and the ``pore space" $\Omega
^{\varepsilon}_{f}$ are connected domains and an intersection
$\Omega ^{\varepsilon}_{f}$ with any plane
$\{x_{i}=\mbox{constant}, \;0<x_{i}<1, \, i=1,2,3\}$ is
an open (in plane topology) set.
\end{assumption}

In these assumptions
\begin{gather*}
\bar{\chi}(\mathbf x)=\chi^{\varepsilon}(\mathbf x)=\chi
 \left(\mathbf x / \varepsilon\right),\\
\bar{\rho}=\rho^{\varepsilon}(\mathbf x)
 =\chi^{\varepsilon}(\mathbf x)\rho _{f}+
(1-\chi^{\varepsilon}(\mathbf x))\rho_{s},
\end{gather*}
where $\chi (\mathbf y)$ is a characteristic function of  $Y_f$ in  $Y$.

Suppose that all dimensionless parameters depend on the small
parameter $\varepsilon$ and there exist limits (finite or infinite)
\begin{gather*}
\lim_{\varepsilon\searrow 0} \alpha_\mu(\varepsilon) =\mu_0, \quad
\lim_{\varepsilon\searrow 0} \alpha_\lambda(\varepsilon) =\lambda_0, \\
\lim_{\varepsilon\searrow 0} \alpha_\eta(\varepsilon) =\eta_0, \quad
\lim_{\varepsilon\searrow 0} \alpha_p(\varepsilon) =p_{*},\\
\lim_{\varepsilon\searrow 0}\frac{\alpha_\mu}{\varepsilon^{2}}=\mu_{1},\quad
\lim_{\varepsilon\searrow 0}\frac{\alpha_\lambda}{\varepsilon^{2}}
 =\lambda_{1}.
\end{gather*}

The first research with the aim of finding limiting regimes in the
case when the skeleton was assumed to be an absolutely rigid body
was carried out by  Sanchez-Palencia and  Tartar.
Sanchez-Palencia \cite[Sec. 7.2]{S-P} formally obtained Darcy's law
of filtration using the method of  two-scale asymptotic expansions,
and  Tartar \cite[Appendix]{S-P}  rigorously justified the
homogenization procedure. Using the same method of two-scale
expansions  Keller and Burridge \cite{B-K} derived formally the
system of Biot's equations from the problem
(\ref{0.1})--(\ref{0.9}) in the case when the   parameter
$\alpha _{\mu}$ was of
order $\varepsilon^2$, and the rest of the coefficients were fixed
independent of $\varepsilon$.
 Under the same assumptions as in the article \cite{B-K}, the rigorous
justification of Biot's model was given by  Nguetseng \cite{GNG}
and later by   Clopeaut \emph{et al}. \cite{G-M2}.  The most
general case of the problem \eqref{0.1}--\eqref{0.9} when
$$
\mu_{0} , \, \lambda_{0}^{-1}, \,
p_{*}^{-1} , \, \eta_0^{-1}<\infty
$$
has been studied in \cite{AM}.
All these authors have used Nguetseng's two-scale convergence method
\cite{NGU,LNW}.

In the present work by means of the same  method we investigate all
possible limiting regimes in the problem (\ref{0.1})--(\ref{0.9})
in the cases, when
\begin{equation}\label{0.10}
0<\mu_0 < \infty ; \quad \lambda_0=0; \quad 0<p_{*},\,\eta_0.
\end{equation}
These cases correspond, for example,  to the hydraulic fracturing,
when all processes  end during the seconds ($\tau\searrow 0$).

We show that for the case $\lambda_1<\infty $ the homogenized
equations describe two velocity continuum and consist of
non-isotropic Stokes equations for fluid velocity coupled with
acoustic equations for the solid component and for the case
$\lambda_1=\infty $ the homogenized equations describe one-velocity
continuum and consist of non-isotropic Stokes system for the
limiting displacements of the continuum.

This property of the mathematical model, which initially describes
one-velocity continuum and becomes a model describing  two-velocity
continuum after homogenization procedure, appears as a result of
different smoothness of the solution in the solid and in the liquid
components:
$$
\int_{\Omega}\alpha_{\mu}(\varepsilon)\chi^{\varepsilon}
|\nabla \mathbf{w}^{\varepsilon}|^{2}dx\leq C_{0},\quad
\int_{\Omega}\alpha_{\lambda}(\varepsilon)(1-\chi^{\varepsilon})
|\nabla \mathbf{w}^{\varepsilon}|^{2})dx\leq C_{0},
$$
where $C_{0}$
is a constant independent of the small parameter $\varepsilon $. To
preserve the best properties of the solution we must use the
well-known extension lemma \cite{ACE,JKO} and extend the solution
from the solid part to the liquid one and vice-versa. On this stage
criterion   $\lambda_{1}$ becomes crucial. Namely, let
$\mathbf{w}_{s}^{\varepsilon}$ be an extension of the solid
displacements to the liquid part and
 $\lambda_{1}=\infty$. Then the  limiting (homogenized)
 system describes  one-velocity continuum. It takes place
 because each of sequences $\{\mathbf{w}^{\varepsilon}\}$ and
 $\{\mathbf{w}_{s}^{\varepsilon}\}$ two --scale converges  to the
  function independent of the fast variable. This statement  easily
follows from Nguetseng's theorem.

\section{Formulation of the main results} \label{Main results}

There are various equivalent in the sense of distributions forms of
representation of equation (\ref{0.1}) in each domain
$\Omega^{\varepsilon}_{f}$ and $\Omega^{\varepsilon}_{s}$ and
boundary conditions \eqref{0.6}--\eqref{0.7} on the common boundary
$\Gamma^{\varepsilon}$ between pore space $\Omega^{\varepsilon}_{f}$
and solid skeleton $\Omega^{\varepsilon}_{s}$. In what follows, it
is convenient to write them in the form of the integral equalities.


 We say that  functions
$\big(\mathbf w^{\varepsilon},\,p_{f}^{\varepsilon},\,
p_{s}^{\varepsilon}\big)$ are called a generalized solution of the
problem \eqref{0.1}--\eqref{0.9}, if they satisfy the regularity
conditions
\begin{equation} \label{1.1}
\mathbf w^{\varepsilon},\, \mathbb {D}(x,\mathbf w^{\varepsilon}),\,
p_{f}^{\varepsilon},\, p_{s}^{\varepsilon} \in L^2(\Omega_{T})
\end{equation}
in the domain $\Omega_{T}=\Omega\times (0,T)$, boundary condition
(\ref{0.9}) on the outer boundary $S$ in the trace sense, equations
\begin{gather} \label{1.2}
 \frac{1}{\alpha_p}p_{f}^{\varepsilon}=
-\chi^{\varepsilon}\big(\mathop{\rm div} \mathbf w^{\varepsilon}-
\frac{\beta^{\varepsilon}}{m}\big), \\
\label{1.3}
 \frac{1}{\alpha_\eta}p_{s}^{\varepsilon}=
-(1-\chi^{\varepsilon})\big(\mathop{\rm div} \mathbf w^{\varepsilon}+
\frac{\beta^{\varepsilon}}{1-m}\big)
\end{gather}
a.e. in $\Omega_{T}$  and, finally,   integral identity
\begin{equation}\label{1.4}
 \begin{gathered}
\int_{\Omega_{T}} \Bigl(\rho
^{\varepsilon} \mathbf w^{\varepsilon}\cdot \frac{\partial
^{2}{\mathbf \varphi}}{\partial t^{2}} - \chi
^{\varepsilon}\alpha_\mu D(\mathbf x, \mathbf w^{\varepsilon}):
\mathbb {D}(x,\frac{\partial {\mathbf \varphi}}{\partial t})
-\rho ^{\varepsilon} \mathbf F\cdot {\mathbf \varphi}\\
+\{(1-\chi^{\varepsilon})\alpha_\lambda \mathbb {D}(x,\mathbf
w^{\varepsilon})- (p_{f}^{\varepsilon}+p_{s}^{\varepsilon})\mathbb
I\} : \mathbb {D}(x,{\mathbf \varphi})\Bigr) \,dx\,dt=0
\end{gathered}
\end{equation}
for all smooth  vector-functions  ${\mathbf \varphi}={\mathbf
\varphi}(\mathbf x,t)$ such that
$$
{\mathbf
\varphi}(\mathbf{x},t)=0,\;\mathbf{x}\in S,\; t>0;\quad
{\mathbf\varphi}(\mathbf{x},T)=\frac{\partial {\mathbf \varphi}}{
\partial t}(\mathbf{x},T)=0,\;\mathbf{x}\in \Omega .
$$

In this definition we changed the form of representation of the
stress tensor $\mathbb {P}$ in the integral identity (\ref{1.4}) by
introducing new unknown function $p_{s}^{\varepsilon}$, which in a
certain way has a sense of pressure. In what follows we call this
function $p_{s}^{\varepsilon}$ as a solid pressure and equations
(\ref{1.2}) and (\ref{1.3})-- as continuity equations. We also
introduced functionals
 $$
\beta^{\varepsilon}=
 \int_{\Omega}\chi^{\varepsilon}\mathop{\rm div}\mathbf w^{\varepsilon}dx
 \mbox{ if } p_*+\eta_{0}=\infty \quad\mbox{and}\quad
\beta^{\varepsilon}=0 \mbox{ if } p_*+\eta_{0}<\infty,
$$
 which have been chosen from the conditions
\begin{equation} \label{1.5}
\int_{\Omega}p_{f}^{\varepsilon}dx=\int_{\Omega}p_{s}^{\varepsilon}dx=0,
\end{equation}
if  $p_*+\eta_{0}=\infty$.
This special choice of continuity equations  permits to estimate
pressures, even if $p_*=\infty$ (incompressible liquid) or
$\eta_{0}=\infty$ (incompressible solid) and  simplifies the use of
homogenization procedure.

In (\ref{1.3})  by $A:B$ we denote the convolution (or,
equivalently, the inner tensor product) of two second-rank tensors
along the both indexes, i.e., $A:B=\mbox{tr\,} (B^*\circ
A)=\sum_{i,j=1}^3 A_{ij} B_{ji}$.


The following two theorems  are the main results of the paper.

\begin{theorem} \label{theorem1}
Let  $\mathbf F$  and  $\partial \mathbf F / \partial t $   are
bounded in $L^2(\Omega_{T})$.  Then for all $\varepsilon >0$
  on the arbitrary time interval
$[0,T]$ there exists a unique generalized solution of the problem
 \eqref{0.1}--\eqref{0.9} and
\begin{gather} \label{1.6}
  \max_{0\leq t\leq
T}\| \frac{\partial ^{2}\mathbf w^{\varepsilon}}{\partial t^{2}}(t)
\|_{2,\Omega}+\|\chi^{\varepsilon}\sqrt{\alpha_\mu} \nabla
\frac{\partial ^{2}\mathbf
w^\varepsilon}{\partial t^{2}}|\|_{2,\Omega _{T}} \leq C_{0} , \\
 \label{1.7}
  \max_{0\leq t\leq
T}\|\chi^\varepsilon \sqrt{\alpha_\mu}|\nabla_x \frac{\partial
\mathbf w^{\varepsilon}}{\partial t}(t)|+(1-\chi^\varepsilon)
 \sqrt{\alpha_\lambda}|\nabla_x\frac{\partial \mathbf w^{\varepsilon}}{\partial
t}(t)| \|_{2,\Omega} \leq C_{0} , \\
\label{1.8}
\max_{0\leq t\leq T}\||p_{f}^{\varepsilon}(t)|+
|p_{s}^{\varepsilon}(t)|\|_{2,\Omega}\leq C_{0},
\end{gather}
where $C_{0}$ does not depend on the small parameter $\varepsilon $.
\end{theorem}

\begin{theorem} \label{theorem2}
Assume that the hypotheses in theorem \ref{theorem1}  and
restrictions \eqref{0.10}  hold. Then functions $\partial \mathbf
w^{\varepsilon} /\partial t$  admit an extension $\mathbf
v^{\varepsilon}$ from $\Omega_f^\varepsilon \times (0,T)$
 into $\Omega_{T}$ such that sequence $\{\mathbf v^{\varepsilon}\}$
 converges strongly in $L^{2}(\Omega_{T})$ and weakly in
 $L^{2}((0,T);W^1_2(\Omega))$ to the function $\mathbf v$. At the same time,
 sequences $\{\mathbf w^\varepsilon\}$,  $\{(1-\chi^\varepsilon)\mathbf w^\varepsilon\}$,
 $\{p_{f}^{\varepsilon}\}$  and
 $\{p_{s}^{\varepsilon}\}$ converge weakly in $L^{2}(\Omega_{T})$
 to $\mathbf w$, $\mathbf u_{s}$,  $p_{f}$  and $p_{s}$, respectively.

(I) If $\lambda _{1}=\infty $, then $\partial\mathbf
u_{s}/\partial t=(1-m)\mathbf v=(1-m)\partial\mathbf w/\partial t$
and weak and strong limits  $p_{f}$,  $p_{s}$ and $\mathbf v$
satisfy in $\Omega_{T}$ the initial-boundary value problem
\begin{gather}
\label{1.9} \begin{aligned}
&\hat{\rho}\frac{\partial \mathbf v}{\partial
t}+\nabla(p_{f}+p_{s})-\hat{\rho}\mathbf F\\
&= \mathop{\rm div}\{\mu_{0}\mathbb A^{f}_{0} :\mathbb {D}(x,\mathbf
v)+\mathbb B^{f}_{0}p_{s} +\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v+
\int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau\},
\end{aligned}
\\
\label{1.10} \begin{aligned}
&p_{*}^{-1}\partial p_{f} /\partial t+\mathbb C^{f}_{0}:\mathbb{D}(x,\mathbf v)
+ a^{f}_{0}p_{s} \\
&+ (a^{f}_{1}+m)\mathop{\rm div}\mathbf v
+\int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau=0,
\end{aligned}\\
\label{1.11}
\frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t}
+\frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial
t}+\mathop{\rm div}\mathbf v=0,
\end{gather}
where $\hat{\rho}=m \rho_{f} + (1-m)\rho_{s}$ is the average density
of the mixture, $m=\int_{Y}\chi dy$ is a porosity and the symmetric
strictly positively defined constant fourth-rank tensor $\mathbb
A^{f}_{0}$, matrices $\mathbb C^{f}_{0}, \mathbb B^{f}_{0}$,
$\mathbb B^{f}_{1}$ and $\mathbb B^{f}_{2}(t)$ and scalars
$a^{f}_{0}$, $a^{f}_{1}$ and $a^{f}_{2}(t)$ are defined below by
formulas \eqref{0.10}, \eqref{4.32} and \eqref{4.34}, where
$\mathbb B^{f}_{1}=0$, $a^{f}_{1}=0$ if $p_{*}<\infty$, and
$\mathbb B^{f}_{2}=0$, $a^{f}_{2}=0$ if $p_{*}=\infty$.

Differential equations \eqref{1.9} are endowed with homogeneous
initial and boundary conditions
 \begin{equation}\label{1.12}
 \mathbf v(\mathbf x,0)=0,\quad \mathbf x\in \Omega;
 \quad \mathbf v(\mathbf x,t)=0, \quad \mathbf x\in S, \quad t>0.
\end{equation}

(II) If $\lambda _{1}<\infty $, then  weak and strong limits
$\mathbf u_{s}$,  $p_{f}$, $p_{s}$ and $\mathbf v$ satisfy in
$\Omega_{T}$ the initial-boundary value problem, which consists of
Stokes like system
\begin{gather}\label{1.13}
\begin{aligned}
& \rho_{f}m\frac{\partial
\mathbf v}{\partial t}+\rho_{s}\frac{\partial ^2\mathbf u_{s}}{
\partial t^2} + \nabla (p_{f}+p_{s})-\hat{\rho}\mathbf F\\
&=\mathop{\rm div}\{\mathbb B^{f}_{0}p_{s}+\mu_{0}\mathbb A^{f}_{0}:\mathbb
{D}(x,\mathbf v)+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v
+\int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau\},
\end{aligned}
\\
\label{1.14} \begin{aligned}
&p_{*}^{-1}\partial p_{f} /\partial t+\mathbb C^{f}_{0}:\mathbb
{D}(x,\mathbf v)+ a^{f}_{0}p_{s}\\
&+ (a^{f}_{1}+m)\mathop{\rm div}\mathbf v
+\int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau=0,
\end{aligned}
\end{gather}
for the liquid component, coupled with the continuity equation
\begin{equation}\label{1.15}
\frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t}
+\frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial t}+\mathop{\rm div}
\frac{\partial\mathbf u_{s}}{\partial t} +m\mathop{\rm div}\mathbf v=0,
\end{equation}
 the relation
\begin{equation}\label{1.16}
\begin{gathered}
\frac{\partial \mathbf u_{s}}{\partial t}=(1-m)\mathbf v(\mathbf
x,t)+\int_{0}^{t}\mathbb B^{s}_{1}(t-\tau)\cdot \mathbf z(\mathbf
x,\tau )d\tau, \\
\mathbf z(\mathbf x,t)=-\frac{1}{1-m}\nabla p_{s}(\mathbf x,t)
+\rho_{s}\mathbf F(\mathbf x,t)-\rho_{s}\frac{\partial \mathbf
v}{\partial t}(\mathbf x,t)
\end{gathered}
\end{equation}
in the case of $\lambda_{1}>0$, or the
balance of momentum equation in the form
\begin{equation}\label{1.17}
\rho_{s}\frac{\partial^{2}\mathbf u_{s}}{\partial
t^{2}}=\rho_{s}\mathbb B^{s}_{2}\cdot \frac{\partial \mathbf
v}{\partial t}+((1-m)I-\mathbb B^{s}_{2})\cdot(-\frac{1}{1-m}\nabla
p_{s}+\rho_{s}\mathbf F)
\end{equation}
in the case of $\lambda_{1}=0$ for the solid component. The problem
is supplemented by boundary and initial conditions (\ref{1.12}) for
the velocity $\mathbf v$ of the liquid component and by the
homogeneous initial conditions
\begin{equation}\label{1.18}
\mathbf u_{s}(\mathbf x,0)=\frac{\partial\mathbf u_{s}}{\partial
t}(\mathbf x,0)=0, \quad (\mathbf x,t) \in \Omega
\end{equation}
and homogeneous boundary condition
\begin{equation}\label{1.19}
\mathbf u_{s}(\mathbf x,t)\cdot \mathbf n(\mathbf x)=0,
     \quad (\mathbf x,t) \in S, \quad t>0,
\end{equation}
for the displacements $\mathbf u_{s}$ of the solid component. In
\eqref{1.16}--\eqref{1.19} $\mathbf n(\mathbf x)$ is the unit
normal vector to $S$ at a point $\mathbf x \in S$, and matrices
$\mathbb B^{s}_{1}(t)$ and $\mathbb B^{s}_{2}$ are given below by
\eqref{4.38} and \eqref{4.40}, where the matrix
$((1-m)\mathbb I - \mathbb B^{s}_{2})$ is symmetric and strictly positively
definite.
\end{theorem}

\section{Preliminaries} \label{Preliminaries}

\subsection{Nguetseng's theorem}
Justification of theorem  \ref{theorem2}
relies on systematic use of the method of two-scale convergence,
which had been proposed by G. Nguetseng \cite{NGU} and has been
applied recently to a wide range of homogenization problems (see,
for example, the survey \cite{LNW}).

\begin{definition} \label{TS} \rm
A sequence $\{w^\varepsilon\}\subset L^2(\Omega_{T})$ is said to be
\textit{two-scale convergent} to a 1- periodic in $\mathbf{y}$
function $W (\mathbf{x},\mathbf{y},t)\in L^2(\Omega_{T}\times Y)$,
if and only if for any 1-periodic in $\mathbf y$ function
$\sigma=\sigma(\mathbf x,t,\mathbf y)$
\begin{equation}\label{2.1}
\int_{\Omega_{T}} w^\varepsilon(\mathbf x,t) \sigma\big(\mathbf
x,t,\frac{\mathbf x}{\varepsilon}\big)\,dx\,dt\to\int
_{\Omega_{T}}\int_Y W(\mathbf x,t,\mathbf y)\sigma(\mathbf
x,t,\mathbf y)dy \,dx\,dt
\end{equation}
as $\varepsilon\to 0$.
\end{definition}

Existence and main properties of weakly convergent sequences are
established by the following fundamental theorem \cite{NGU,LNW}.

\begin{theorem}[Nguetseng's theorem] \label{theorem3}
\begin{enumerate}
\item Any bounded in $L^2(\Omega_{T})$ sequence contains a
subsequence, two-scale convergent to some limit
$W\in L^2(\Omega_{T}\times Y)$.

\item  Let sequences $\{w^\varepsilon\}$ and $\{\varepsilon
\nabla_x w^\varepsilon\}$ be uniformly bounded in $L^2(\Omega_{T})$.
Then there exist a 1-periodic in $\mathbf y$ function $W=W(\mathbf
x,t,\mathbf y)$ and a subsequence $\{w^\varepsilon\}$ such that
$W,\, \nabla_y W\in L^2(\Omega_{T}\times Y)$, and the subsequences
$\{w^\varepsilon\}$ and $\{\varepsilon \nabla w^\varepsilon\}$
two-scale converge to $W$ and $\nabla_y W$,
respectively.

\item  Let sequences $\{w^\varepsilon\}$ and $\{\nabla
w^\varepsilon\}$ be bounded in $L^2(Q)$. Then there exist functions
$w\in L^2(\Omega_{T})$ and $W \in L^2(\Omega_{T}\times Y)$ and a
subsequence from $\{\nabla w^\varepsilon\}$ such that the function
$W$ is 1-periodic in $\mathbf y$, $\nabla w \in L^2(\Omega_{T})$,
$\nabla_y W \in L^2(\Omega_{T}\times Y)$, and the subsequence
$\{\nabla w^\varepsilon\}$ two-scale converge to the function
$(\nabla w(\mathbf x,t)+\nabla_y W(\mathbf x,t,\mathbf y))$.
\end{enumerate}
\end{theorem}

\begin{corollary} \label{corollary2.1}
Let $\sigma\in L^2(Y)$ and $\sigma^\varepsilon(\mathbf
x)=\sigma(\mathbf x/\varepsilon)$. Assume that a sequence
$\{w^\varepsilon\}\subset L^2(\Omega_{T})$ two-scale converges to $W
\in L^2(\Omega_{T}\times Y)$. Then the sequence
$\{\sigma^\varepsilon w^\varepsilon\}$ two-scale converges to the
function $\sigma W$.
\end{corollary}

\subsection{An extension lemma}
 The typical difficulty in
homogenization problems, like problem \eqref{0.1}--\eqref{0.9},
while passing to a limit   as $\varepsilon \searrow 0$ arises
because of the fact that the bounds on the gradient of displacement
$\nabla_x \mathbf w^\varepsilon$ may be distinct in liquid and rigid
components. The classical approach in overcoming this difficulty
consists of constructing of extension to the whole $\Omega$ of the
displacement field defined merely on $\Omega_s$ or $\Omega_f$. The
following lemma is valid due to the well-known results from
\cite{ACE,JKO}. We formulate it in appropriate for us form:

\begin{lemma} \label{lemma2.1}
Suppose that assumption \ref{assumption1} on geometry of periodic
structure holds, $w^\varepsilon\in W^1_2(\Omega^\varepsilon_f)$ and
$w^\varepsilon =0$ on $S_{f}^{\varepsilon}=\partial\Omega
^\varepsilon_f \cap
\partial \Omega$ in the trace sense.  Then there exists a function
$w_{f}^\varepsilon \in
 W^1_2(\Omega)$ such that its restriction on the sub-domain
$\Omega^\varepsilon_f$ coincide with $w^\varepsilon$, i.e.,
\begin{equation} \label{2.2}
\chi^\varepsilon(\mathbf x)(w_{f}^\varepsilon(\mathbf
x)-w^\varepsilon (\mathbf x))=0,\quad \mathbf x\in\Omega,
\end{equation}
and, moreover, the estimate
\begin{equation} \label{2.3}
\|w_{f}^\varepsilon\|_{2,\Omega}\leq C\|w^\varepsilon\|_{2,\Omega
^{\varepsilon}_{f}}  , \quad \|\nabla w_{f}^\varepsilon\|_{2,\Omega}
\leq  C \|\nabla w^\varepsilon\|_{2,\Omega ^{\varepsilon}_{f}}
\end{equation}
hold true, where the constant $C$ depends only on geometry $Y$ and
does not depend on $\varepsilon$.
\end{lemma}

\subsection{Friedrichs--Poincar\'{e}'s inequality in periodic
structure} The following lemma was proved by  Tartar in
\cite[Appendix]{S-P}. It specifies Friedrichs--Poincar\'{e}'s
inequality for $\varepsilon$-periodic structure. We formulate this
lemma for our particular case just to estimate  functions in the
$\varepsilon$--layer $Q^{\varepsilon}$ of the boundary $S$. This
domain $Q^{\varepsilon}$ consists of all elementary cells
$\varepsilon Y$  touching the boundary $\partial\Omega$.  We
consider special class of functions $w_{f}^\varepsilon$, which are
extensions of functions $w^{\varepsilon}\in
W^1_2(\Omega^\varepsilon_f)$, vanishing on the part
$S_{f}^{\varepsilon}
 =\partial\Omega^\varepsilon_f \cap\partial \Omega$  of the boundary
 $S=\partial\Omega$,  from  subdomain
$\Omega^\varepsilon_f$ onto whole domain $\Omega$ (see lemma \ref{lemma2.1}).
Due to supposition on the structure of the pore space, the
intersection of the boundary of the  "liquid part"  $Y_f$ with each
sides of the boundary $\partial Y$ is a set with nonempty interior
and strictly positive measure. Therefore on the each side of the
boundary $S$ the function $w_{f}^\varepsilon$ is equal to zero on
some  set with nonempty interior, periodic structure and strictly
positive measure, independent of $\varepsilon$.

\begin{lemma} \label{lemma2.2}
Suppose that assumptions on the geometry of $\Omega^\varepsilon_f$
hold true. Then for any function $w_{f}^\varepsilon\in W^1_2(\Omega)$
 such that $w_{f}^\varepsilon=0$ on the part $S_{f}^{\varepsilon}
 =\partial\Omega^\varepsilon_f \cap\partial \Omega$  of the boundary
 $S$, the inequality
\begin{equation} \label{2.4}
\int_{Q^{\varepsilon}} |w_{f}^\varepsilon|^2 dx \leq C \varepsilon^2
\int_{Q^{\varepsilon}} |\nabla w_{f}^\varepsilon|^2 dx
\end{equation}
holds true with some constant $C$ independent of the small parameter
$\varepsilon$.
\end{lemma}

\subsection{Some notation}
Further we denote (1)
\begin{gather*}
 \langle\Phi \rangle_{Y} =\int_Y \Phi  dy, \quad
 \langle\Phi \rangle_{Y_{f}} =\int_Y \chi \Phi  dy,
 \quad
 \langle\Phi \rangle_{Y_{s}} =\int_Y (1-\chi )\Phi  dy, \\
\langle\varphi  \rangle_{\Omega } =\int_{\Omega } \varphi  dx,
\quad
  \langle\varphi  \rangle_{\Omega_{T}} =\int_{\Omega_{T}} \varphi
  \,dx\,dt.
\end{gather*}

(2) If $\mathbf{a}$ and $\mathbf{b}$ are two vectors then 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 vector
$\mathbf{c}$.

(3) If $\mathbb B$ and $\mathbb C$ are two matrices, then $\mathbb
B\otimes \mathbb C$ is a forth-rank tensor such that its convolution
with any matrix $\mathbb A$ is defined by the formula
$$
(\mathbb B\otimes \mathbb C):\mathbb A=\mathbb B (\mathbb C:\mathbb A).
$$

(4) By $\mathbb I^{ij}={\mathbf e}_i \otimes {\mathbf e}_j$ we denote
the $3\times 3$-matrix with just one non-vanishing entry, which is
equal to one and stands in the $i$-th row and the $j$-th column.

(5) We  also  introduce
$$
\mathbb J^{ij}=\frac{1}{2}(\mathbb I^{ij}+\mathbb I^{ji})
=\frac{1}{2} ({\mathbf e}_i \otimes {\mathbf e}_j + {\mathbf e}_j
\otimes {\mathbf e}_i),\quad \mathbb J=\sum_{i,j=1}^{3}\mathbb
J^{ij}\otimes \mathbb J^{ij},
$$
where $({\mathbf e}_1, {\mathbf e}_2, {\mathbf e}_3)$ are the
standard Cartesian basis vectors.


\section{Proof of theorem \ref{theorem2}}

Estimates (\ref{1.6})-(\ref{1.7})  follow from the energy equality in
 the form
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\{\int_{\Omega}\rho^{\varepsilon}(\frac{\partial
^{2}\mathbf w^\varepsilon}{\partial t^{2}})^{2}dx+
\alpha_\lambda\int_{\Omega}(1-\chi^{\varepsilon})\mathbb
D(x,\frac{\partial\mathbf w^{\varepsilon}}{\partial t}):\mathbb
D(x,\frac{\partial\mathbf w^{\varepsilon}}{\partial t})dx
\\
&+\alpha_p\int_{\Omega}\chi^{\varepsilon}(\mathop{\rm div}\frac{\partial\mathbf
w^{\varepsilon}}{\partial t})^{2}dx+\alpha_\eta\int_{\Omega}
(1-\chi^{\varepsilon})(\mathop{\rm div}\frac{\partial\mathbf
w^{\varepsilon}}{\partial t})^{2}dx\}
\\
&+ \alpha_\mu\int_{\Omega}\chi^{\varepsilon}\mathbb
D(x,\frac{\partial^{2}\mathbf w^{\varepsilon}}{\partial
t^{2}}):\mathbb D(x,\frac{\partial^{2}\mathbf
w^{\varepsilon}}{\partial t^{2}})dx\\
&=\int_{\Omega} \frac{\partial
\mathbf{F}}{\partial t}\cdot\frac{\partial ^{2}\mathbf
w^\varepsilon}{\partial t^{2}}dx\\
&\quad +
\frac{\partial \beta^{\varepsilon}}{\partial
t}\big(\frac{\alpha_p}{m}\int_{\Omega}\chi^{\varepsilon}
\mathop{\rm div}\frac{\partial^{2}\mathbf w^{\varepsilon}}{\partial
t^{2}}dx+\frac{\alpha_\eta}{(1-m)}\int_{\Omega}
(1-\chi^{\varepsilon})\mathop{\rm div}\frac{\partial^{2}\mathbf
w^{\varepsilon}}{\partial t^{2}}dx\big).
\end{aligned} \label{3.1}
\end{equation}
We obtain this  equality if we  differentiate  equation  for $\mathbf
w^{\varepsilon}$ with respect to time, multiply the result  by
$\partial ^{2} \mathbf w^{\varepsilon}/\partial t^{2}$ and integrate
the product by parts using continuity equations (\ref{1.2}) and
(\ref{1.3}). Note, that all terms on the common interface
$\Gamma^{\varepsilon}$ "solid skeleton--pore space" disappear due to
boundary conditions \eqref{0.6}--\eqref{0.7}.

In fact, if $p_{*}+\eta_0<\infty $  ($\beta^{\varepsilon}=0$), then
 we just use H\"{o}lder and Gronwall inequalities in (\ref{3.1})  and get
\begin{equation} \label{3.2}
\begin{aligned}
&\max_{0<t<T}\big(\| |\frac{\partial ^{2}\mathbf
w^\varepsilon}{\partial t^{2}}(t)|+(1-\chi^{\varepsilon})(
\sqrt{\alpha_\lambda} |\nabla \frac{\partial\mathbf
w^{\varepsilon}}{\partial t}(t)| +\sqrt{\alpha_\eta}|\mathop{\rm div}
\frac{\partial\mathbf w^{\varepsilon}}{\partial t}(t)|)
\|_{2,\Omega} \\
&+\sqrt{\alpha_p}\|\chi^{\varepsilon}\mathop{\rm div}\frac{\partial\mathbf
w^{\varepsilon}}{\partial t}(t) \|_{2,\Omega}
\big)+\|\chi^{\varepsilon} \nabla \frac{\partial ^{2}\mathbf
w^\varepsilon}{\partial t^{2}}|\|_{2,\Omega _{T}} \leq C_{0},
\end{aligned}
\end{equation}
where  $C_{0}$ is independent of the small parameter  $\varepsilon$.

 Estimates (\ref{1.8}) for
pressures follow from estimates (\ref{3.2})  and continuity
equations.

For the case $p_{*}+\eta_0=\infty $ estimates  (\ref{1.6}) and
(\ref{1.7}) follow again  from energy  identity (\ref{3.1}) in the
same way as before, if we additionally use inequalities
\begin{gather*}
\frac{1}{m}(\int_{\Omega}\chi^{\varepsilon}
\mathop{\rm div}\frac{\partial\mathbf w^{\varepsilon}}{\partial
t}dx)^{2}\leq\int_{\Omega}\chi^{\varepsilon}
(\mathop{\rm div}\frac{\partial\mathbf w^{\varepsilon}}{\partial
t})^{2}dx,\\
\frac{1}{(1-m)}(\int_{\Omega}(1-\chi^{\varepsilon})
\mathop{\rm div}\frac{\partial\mathbf w^{\varepsilon}}{\partial
t}dx)^{2}\leq\int_{\Omega}(1-\chi^{\varepsilon})
(\mathop{\rm div}\frac{\partial\mathbf w^{\varepsilon}}{\partial
t})^{2}dx.
\end{gather*}

To estimate pressures we use  estimates (\ref{1.6}) and (\ref{1.7})
and integral identity (\ref{1.4})  in the form
\begin{align*}
&\int_{\Omega}\big(p_{f}^{\varepsilon}+p_{s}^{\varepsilon}\big)
\mathop{\rm div}{\mathbf\psi}dx\\
&=\int_{\Omega} \Bigl(\rho ^{\varepsilon}\big
(\frac{\partial ^{2}\mathbf w^{\varepsilon}}{\partial t^{2}}-\mathbf
F\big)\cdot {\mathbf \psi} +\{\chi ^{\varepsilon}\alpha_\mu \mathbb
D(x,\frac{\partial\mathbf w^{\varepsilon}}{\partial t})+(1-\chi
^{\varepsilon})\alpha_\lambda \mathbb D(x,\mathbf w^{\varepsilon})\}
:\mathbb D(x,{\mathbf\psi})\Bigr)dx,
\end{align*}
 Considering
the sum of pressures
$q=p_{f}+p_{s}$
as a linear functional on the space
 $\stackrel{\!\!\circ}{W^1_2}(\Omega)$  we get
$$
|\int_{\Omega}q\mathop{\rm div}\mathbf{\psi}dx|
\leq C_{0}\max_{0\leq t\leq
T}\|\mathbf{\psi}(t)\|_{W^{1}_{2}(\Omega)},
$$ where  $C_{0}$ is
independent of the small parameter  $\varepsilon$.

Choosing now ${\mathbf{\psi}}$ such that $\mathop{\rm div}
{\mathbf{\psi}}=q$ we arrive at
\begin{equation}\label{3.3}
\max_{0\leq t\leq
T}\|\mathop{\rm div}\mathbf{\psi}(t)\|_{\Omega}^{2}= \max_{0\leq
t\leq T}\|q(t)\|_{\Omega}^{2}\leq C_{0}\max_{0\leq t\leq
T}\|\mathbf{\psi}(t)\|_{W^{1}_{2}(\Omega)}.
\end{equation}
Such a choice of the function ${\mathbf{\psi}}$ is always possible
(see \cite{LAD}), if we put
$$
{\mathbf{\psi}}=\nabla \varphi + {\mathbf{\psi_{0}}},
$$
where
\begin{gather}\label{3.4}
\Delta \varphi=q,\quad \mathbf{x} \in \Omega,\quad
\varphi=0,\quad \mathbf{x} \in \partial\Omega, \\
\label{3.5}
\mathop{\rm div} {\mathbf\psi_{0}}=0, \quad \mathbf{x} \in \Omega,\quad
\mathbf{\psi_{0}}=-\nabla \varphi,\quad \mathbf{x} \in
\partial\Omega.
\end{gather}
In fact, extending the solution $\varphi$ of the problem (\ref{3.4})
as  odd function over boundaries $\{x_{i}=0,1;\;i=1,2,3\}$ we
conclude that
$$
\varphi\in \stackrel{\!\!\circ}{W^2_2}(\Omega), \quad \mbox{and}
\quad \max_{0\leq t\leq
T}\|\nabla\varphi(t)\|_{W^{1}_{2}(\Omega)}\leq C\max_{0\leq
t\leq T}\|q(t)\|_{\Omega}.
$$
Now we look for the solution $\mathbf \psi_{0}$ of the problem
(\ref{3.5}) as a solution of the Stokes system
$$
\Delta \mathbf \psi_{0}+\nabla p=0, \;
\mathop{\rm div}{\mathbf{\psi_{0}}}=0, \quad \mathbf{x}\in \Omega
$$
with non-homogeneous boundary condition
$$
{\mathbf{\psi_{0}}}=-\nabla \varphi,\quad \mathbf{x}\in \partial\Omega.
$$
The above problem has  unique solution, such that
$$
\max_{0\leq t\leq T}\|\mathbf{\psi}_{0}(t)\|_{W^{1}_{2}(\Omega)}\leq
C\max_{0\leq t\leq T}\|\nabla
\varphi(t)\|_{W^{1}_{2}(\Omega)},
$$
 if and only if
$$
\int_{\Omega}\mathop{\rm div}(\nabla \varphi )dx\equiv
\int_{\Omega}\Delta\varphi dx=\int_{\Omega}q dx=0.
$$
This solvability condition follows from  conditions (\ref{1.5}).
Thus, gathering all estimates together we obtain  the desired
estimates for the sum of pressures
$(p_{f}^\varepsilon +p_{s}^{\varepsilon})$. Finally, thanks
to the property that the product
of these two functions is equal to zero, we get bounds for each of
pressures  $p_{f}^\varepsilon $ and $p_{s }^{\varepsilon}$.


\section{Proof of theorem \ref{theorem2}} %\label{Theorem 2.2}

\subsection*{Weak and two-scale limits of sequences of displacement
and pressures} On the strength of theorem \ref{theorem1}, the
sequences $\{p_{f}^\varepsilon\}$, $\{p_{s}^\varepsilon\}$  and
$\{\mathbf w^\varepsilon \}$   are uniformly in $\varepsilon$
bounded in $L^2(\Omega_{T})$. Hence there exist a subsequence of
small parameters $\{\varepsilon>0\}$ and functions $p_{f}$, $p_{s}$
and $\mathbf w$  such that
$$
p_{f}^\varepsilon \to p_{f}, \quad
p_{s}^\varepsilon \to p_{s},\quad
\mathbf w^\varepsilon \to \mathbf w
$$
weakly in  $L^2(\Omega_T)$ as $\varepsilon\searrow 0$.
Relabeling if necessary, we assume that the sequences converge
themselves.
At the same time
\begin{equation} \label{4.1}
(1-\chi^\varepsilon )\alpha_\lambda \mathbb {D}(x,\mathbf
w^\varepsilon) \to 0.
\end{equation}
strongly in  $L^2(\Omega_T)$ and  the sequence
$\{\mathop{\rm div}\mathbf w^\varepsilon \}$ converges weakly in
$L^2(\Omega_T)$ to
$\mathop{\rm div}\mathbf w $ as  $\varepsilon\searrow 0$.

Moreover, due to extension lemma \ref{lemma2.1} there are functions
$$
\mathbf v^\varepsilon \in L^\infty (0,T;W^1_2(\Omega))
$$
such that $\mathbf v^\varepsilon =\partial \mathbf w^\varepsilon /
\partial t$ in $\Omega_{f}\times (0,T)$,  $v^{\varepsilon}=0$
on the part $S^{\varepsilon}_{f}$ of the boundary $S$ and
\begin{gather} \label{4.2}
\|\frac{\partial\mathbf v^{\varepsilon}} {\partial
t}\|_{2,\Omega_{T}}+\|\nabla \frac{\partial\mathbf
v^{\varepsilon}}{\partial t}\|_{2,\Omega_{T}}\leq C_{0},
\\
 \label{4.3}
\max_{0\leq t\leq T}\big(\|\mathbf
v^{\varepsilon}(t)\|_{2,\Omega}+\|\nabla \mathbf
v^{\varepsilon}(t)\|_{2,\Omega}\big)\leq C_{0},
\end{gather}
where $C_{0}$ does not depend on the small parameter $\varepsilon $.

\begin{lemma} \label{lemma4.1}
 There exist a subsequence of $\{\varepsilon>0\}$
and function
$$
\mathbf v\in L^{\infty}\big(0,T;W^1_2(\Omega)\big),
$$
such that
\begin{enumerate}
\item $\mathbf v^\varepsilon (\,,t)\to \mathbf v(\,,t)$
weakly in $ W^1_2(\Omega)$ as
$\varepsilon \searrow 0$ for all $t\in [0,T]$, and

\item $\mathbf v(\,,t)\in {\mathaccent"7017 W}_2^1 (\Omega)$
for all $t\in [0,T]$.
\end{enumerate}
\end{lemma}

\begin{proof}
First of all note, that there are a subsequence of small parameters
$\{\varepsilon>0\}$ and function $\mathbf v$,  such that
$$
 \mathbf v,\frac{\partial\mathbf v}{\partial t}\in
L^{2}\big(0,T;W^1_2(\Omega)\big),
$$
and
$\mathbf v^\varepsilon (\,,t)\to \mathbf v(\,,t)$
weakly in $ L^{2}\big(0,T;W^1_2(\Omega)\big)$ as
$\varepsilon \searrow 0$.

Now, let $\mathbf{\varphi}(\mathbf{x})$ be an arbitrary smooth
function, and
$$
J^{\varepsilon}_{\varphi}(t)=
\int_{\Omega}\Big(\big(\mathbf v^\varepsilon(\mathbf{x},t) -\mathbf
v(\mathbf{x},t)\big)\cdot\mathbf{\varphi}(\mathbf{x})
+\nabla\big(\mathbf v^\varepsilon(\mathbf{x},t) -\mathbf
v(\mathbf{x},t)\big)\cdot
\nabla\mathbf{\varphi}(\mathbf{x})\Big)dx.
$$
By construction
$$
\int_{0}^{T}J^{\varepsilon}_{\varphi}(t)\psi(t)dt\to 0
$$
as $\varepsilon \searrow 0$ for any $\psi\in L^{2}(0,T)$.
The first statement of the lemma means that
$$
J^{\varepsilon}_{\varphi}(t)\to 0
$$
as $\varepsilon \searrow 0$ for all $t\in [0,T]$.
Estimates (\ref{4.2}) and (\ref{4.3}) imply
$$
\int_{0}^{T}|\frac{dJ^{\varepsilon}_{\varphi}}{dt}(t)|^{2}dt
\leq C_{0}^{2}.
$$
Using this  estimate, the initial condition
$J^{\varepsilon}_{\varphi}(0)=0$, and the weak convergence in
$L^{2}(0,T)$  of the sequence $\{J^{\varepsilon}_{\varphi}\}$ to
zero, one may easily prove that
$$
J^{\varepsilon}_{\varphi}(t)\to 0 \quad \mbox{in } C[0,T],
$$
which proves the first part of the lemma.

To prove the second part of the lemma note, that
$$
\mathbf v^\varepsilon (\,,t)\to \mathbf v(\,,t)\quad  \mbox{
strongly in } \quad L^2(S)\quad \mbox{as} \quad \varepsilon \searrow
0 \quad \mbox{for all} \quad t\in [0,T].
$$
This fact follows from the well-known imbedding theorem, which
states that any weakly convergent sequence in $W^{1}_{2}(\Omega)$
converges strongly in $L^2(S)$.

Now we use lemma \ref{lemma2.2} and estimate (\ref{2.4}) to conclude that
\begin{equation} \label{4.4}
\max_{0\leq t\leq T}\|\mathbf
v^{\varepsilon}(t)\|^{2}_{2,S}\leq \varepsilon C_{0}.
\end{equation}
In fact, we may prove it for each facet separately.
 Considering, for example, the facet
$S_{3,0}=\{x_{3}=0$, $x^{'}= (x_{1},x_{2})\in (0,1)\times(0,1)\}$
one has
\begin{align*}
&|\mathbf v^{\varepsilon}(x^{'},0,t)|^{2}\\
&= |\mathbf v^{\varepsilon}(x^{'},x_{3},t)|^{2}+2\int_{0}^{x_{3}}
\mathbf v^{\varepsilon}(x^{'},y_{3},t)\frac{\partial\mathbf
v^{\varepsilon}}{\partial y_{3}}(x^{'},y_{3},t)dy_{3}\\
&\leq |\mathbf v^{\varepsilon}(x^{'},x_{3},t)|^{2}+
2\Big(\int_{0}^{\varepsilon}|\mathbf
v^{\varepsilon}(x^{'},y_{3},t)|^{2}dy_{3}\Big)^{1/2}
\Big(\int_{0}^{\varepsilon}|\frac{\partial\mathbf
v^{\varepsilon}}{\partial
y_{3}}(x^{'},y_{3},t)|^{2}dy_{3}\Big)^{1/2}
\end{align*}
and consequently, after integration over $S_{3,0}$  and interval
$x_{3}\in (0,\varepsilon)$,
$$
\varepsilon \int_{S_{3,0}}|\mathbf v^{\varepsilon}|^{2}dx^{'}
\leq \int_{Q^{\varepsilon}}|\mathbf
v^{\varepsilon}|^{2}dx+2\varepsilon
\Big(\int_{Q^{\varepsilon}}|\mathbf
v^{\varepsilon}|^{2}dx\Big)^{1/2}
\Big(\int_{\Omega}|\nabla\mathbf
v^{\varepsilon}|^{2}dx\Big)^{1/2}.
$$
Using estimates (\ref{2.4}) and (\ref{4.3}) we finally get
 estimate (\ref{4.4}), which means that
$$
\mathbf v^\varepsilon (\,,t)\to 0 \quad \mbox{ strongly in }
\ L^2(S)
$$
as $\varepsilon \searrow 0$ for all $t\in [0,T]$  and
that $\mathbf v=0$ on the boundary $S$.
\end{proof}


On the strength of Nguetseng's theorem, there exist 1-periodic in
$\mathbf y$ functions $P_{f}(\mathbf x,t,\mathbf y)$, $P_{s}(\mathbf
x,t,\mathbf y)$,  $\mathbf W(\mathbf x,t,\mathbf y)$ and $\mathbf
V(\mathbf x,t,\mathbf y)$ such that the sequences
$\{p_{f}^\varepsilon\}$, $\{p_{s}^\varepsilon\}$,  $\{\mathbf
w^\varepsilon \}$ and $\{\nabla\mathbf v^\varepsilon \}$ two-scale
converge to $P_{f}(\mathbf x,t,\mathbf y)$, $P_{s}(\mathbf
x,t,\mathbf y)$,  $\mathbf W(\mathbf x,t,\mathbf y)$ and $\nabla
\mathbf v(\mathbf x,t) +\nabla_{y}\mathbf V(\mathbf x,t,\mathbf
y)$, respectively.


\subsection{Micro- and macroscopic equations I}

\begin{lemma} \label{lemma4.2}
For all $ \mathbf x \in \Omega$ and $\mathbf y\in Y$ weak  and
two-scale limits of the sequences $\{p_{f}^\varepsilon\}$,
$\{p_{s}^\varepsilon\}$,  $\{\mathbf w^\varepsilon\}$, and
$\{\mathbf v^\varepsilon\}$ satisfy the relations
\begin{gather}\label{4.5}
P_{s}=p_{s}\frac{(1-\chi)}{(1-m)}, \quad P_{f}=\chi P_{f}, \\
\label{4.6}
\frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t}+m\mathop{\rm div}\mathbf
v+\langle \mathop{\rm div}{}_y\mathbf V\rangle_{Y_{f}}=\frac{\partial \beta
}{\partial t}, \\
\label{4.7}
\frac{1}{p_{*}}\frac{\partial P_{f}}{\partial
t}+\chi(\mathop{\rm div}\mathbf v+ \mathop{\rm div}{}_y\mathbf V)=\frac{\chi
}{m}\frac{\partial \beta}{\partial t}, \\
\label{4.8}
\frac{1}{p_{*}}p_{f}+\frac{1}{\eta_{0}}p_{s}+\mathop{\rm div}\mathbf w=0, \\
\label{4.9}
\mathbf w(\mathbf x,t)\cdot \mathbf n(\mathbf x)=0, \quad \mathbf x\in S,\\
\label{4.10}
\mathop{\rm div}{}_y \mathbf W=0, \\
\label{4.11}
\frac{\partial \mathbf W}{\partial t}=\chi \mathbf
v+(1-\chi)\frac{\partial \mathbf W}{\partial t},
\end{gather}
where $\partial \beta /\partial t =\langle\langle
\mathop{\rm div}{}_y\mathbf V\rangle_{Y_{f}}\rangle_{\Omega}$, if
$p_{*}+\eta_{0}=\infty$  and $\beta=0$, if $p_{*}+\eta_{0}<\infty$
and  $\mathbf n(\mathbf x)$ is the unit normal vector to $S$ at a
point $\mathbf x \in S$.
\end{lemma}

\begin{proof}
To prove (\ref{4.5}), into (\ref{1.4}) we insert a test
function ${\mathbf \psi}^\varepsilon =\varepsilon {\mathbf
\psi}\left(\mathbf x,t,\mathbf x / \varepsilon\right)$, where
${\mathbf \psi}(\mathbf x,t,\mathbf y)$ is an arbitrary 1-periodic
and finite on $Y_s$ function in $\mathbf y$. Passing to the limit as
$\varepsilon \searrow 0$, we get
\begin{equation} \label{4.12}
\nabla_y P_{s}(\mathbf x,t,\mathbf y)=0, \quad \mathbf y\in Y_{s}.
\end{equation}
Next, fulfilling the two-scale limiting passage in equality
$$\chi^{\varepsilon}p_{s}^{\varepsilon}=0$$
we arrive at
$\chi P_{s}=0$
which along with (\ref{4.12}) justifies  (\ref{4.5}).

Equations (\ref{4.6})--(\ref{4.9}) appear as the results of two-scale
limiting passages in   (\ref{1.2})--(\ref{1.3}) with the proper
test functions being involved. Thus, for example,  (\ref{4.8})
and (\ref{4.9}) arise, if we consider the sum of (\ref{1.2}) and
(\ref{1.3}),
\begin{equation}\label{4.13}
\frac{1}{\alpha_{p}}p_{f}^\varepsilon
+\frac{1}{\alpha_{\eta}}p_{s}^\varepsilon +\mathop{\rm div}\mathbf
w^\varepsilon =\frac{1}{m(1-m)}\beta ^{\varepsilon}(\chi^\varepsilon
-m);
\end{equation}
multiply by an arbitrary function, independent of the ``fast''
variable $\mathbf x/\varepsilon$, and then pass to the limit as
$\varepsilon\searrow 0$. In order to prove  (\ref{4.10}), it is
sufficient to consider the two-scale limiting relations in
(\ref{4.13}) as $\varepsilon \searrow 0$ with the test functions
$\varepsilon \psi \left(\mathbf x / \varepsilon\right) h(\mathbf
x,t)$, where $\psi$ and $h$ are arbitrary smooth  functions. In
order to prove  (\ref{4.11}) it is sufficient to consider the
two-scale limiting relations in
$$
\chi ^{\varepsilon}(\frac{\partial \mathbf w^{\varepsilon}}{\partial
t}-\mathbf v^{\varepsilon})=0.
$$
\end{proof}

\begin{corollary}\label{corollary4.1}
If $p_{*}+\eta_{0}=\infty$, then weak limits  $p_{f}$  and $p_{s}$
satisfy relations
\begin{equation}\label{4.14}
\langle p_{f} \rangle _{\Omega}=\langle p_{s}\rangle _{\Omega}=0.
\end{equation}
\end{corollary}

\begin{lemma} \label{lemma4.3}
For all $(\mathbf x,t) \in \Omega_{T}$ the relations
\begin{equation} \label{4.15}
\mathop{\rm div}{}_y \{\mu_0\chi (\mathbb {D}(y,\mathbf V)+\mathbb
{D}(x,\mathbf v))- (P_{f} +\frac{(1-\chi)}{(1-m)}p_{s})\cdot\mathbb
{I}\}=0,
\end{equation}
holds true.
\end{lemma}

\begin{proof}
Substituting a test function of the form ${\mathbf \psi}^\varepsilon
=\varepsilon {\mathbf \psi}\left(\mathbf x,t,\mathbf x / \varepsilon
\right)$, where ${\mathbf \psi}(\mathbf x,t,\mathbf y)$ is an
arbitrary 1-periodic in $\mathbf y$ function vanishing on the
boundary $S$, into integral identity (\ref{1.4}), and passing to the
limit as $\varepsilon \searrow 0$, we arrive at  (\ref{4.15}).
\end{proof}

\begin{lemma} \label{lemma4.4}
Let $\hat{\rho}=m \rho_{f} + (1-m)\rho_{s}$. Then functions $\mathbf
u_{s}=\langle \mathbf W\rangle _{Y_{s}}$, $\mathbf v$, $p_{f}$ and
$p_{s}$ satisfy in $\Omega_{T}$ the system of macroscopic equations
\begin{equation}\label{4.16}
\rho_{f}m\frac{\partial \mathbf v}{\partial t}+
\rho_{s}\frac{\partial ^2\mathbf u_{s}}{\partial t^2}-\hat{\rho}\mathbf F
=\mathop{\rm div}\{\mu _{0}(m\mathbb {D}(x,\mathbf v)+\langle \mathbb
{D}(y,\mathbf V)\rangle _{Y_{f}})-(p_{f}+p_{s})\cdot\mathbb {I}\},
\end{equation}
and the homogeneous initial conditions
\begin{equation} \label{4.17}
\mathbf u_{s}(\mathbf{x},0)=\rho_{f}m\mathbf
v(\mathbf{x},0)+\rho_{s}\frac{\partial \mathbf u_{s}}{\partial
t}(\mathbf{x},0)=0, \quad \mathbf{x}\in \Omega .
\end{equation}
\end{lemma}

\begin{proof}
Equations (\ref{4.16}) and initial conditions (\ref{4.17}) arise as the
limit of  (\ref{1.4}) with test functions being independent of
$\varepsilon$ in $\Omega_T$.
\end{proof}

\subsection*{ Micro- and macroscopic equations II}

\begin{lemma}\label{lemma4.5}
If $\lambda_{1}=\infty$, then the weak  limits of
$\{\mathbf v^\varepsilon\}$ and
$\{\partial \mathbf w^\varepsilon /\partial t\}$ coincide and
$$
(1-m)\mathbf v=\frac{\partial\mathbf u_{s}}{\partial t}.
$$
\end{lemma}

\begin{proof}
Let  $\Psi(\mathbf x,t,\mathbf y)$   be an arbitrary smooth function
periodic in $\mathbf y$. The sequence
$\{\sigma_{ij}^{\varepsilon}\}$, where
$$
\sigma_{ij}^{\varepsilon}=\int_{\Omega}\sqrt{\alpha_{\lambda}}
\frac{\partial w_{i}^\varepsilon}{\partial x_{j}} (\mathbf
x,t)\Psi(\mathbf x,t,\mathbf x /\varepsilon )dx, \quad \mathbf
w^\varepsilon=(w_{1}^\varepsilon, w_{2}^\varepsilon,
w_{3}^\varepsilon )
$$
is uniformly bounded in $\varepsilon$.  Therefore,
$$
\int_{\Omega}\varepsilon \frac{\partial w_{i}^\varepsilon}
{\partial x_{j}} (\mathbf x,t)\Psi(\mathbf x,t,\mathbf x
/\varepsilon )dx=\frac{\varepsilon}{\sqrt{\alpha_{\lambda}}}
\sigma_{ij}^{\varepsilon}\to 0
$$
as  $\varepsilon\searrow 0$, which is equivalent to
 $$
\int_{\Omega}\int_{Y} W_{i}(\mathbf x,t,\mathbf y)
\frac{\partial\Psi}{\partial y_{j}}(\mathbf x,t,\mathbf y)dxdy=0,
\quad \mathbf W=(W_{1}, W_{2}, W_{3}),
$$
 or  $\mathbf W(\mathbf x,t,\mathbf y)=\mathbf w(\mathbf x,t)$.
Therefore, taking the
two-scale limit as $\varepsilon\searrow 0$ in the
 equality
$$
\chi^{\varepsilon}(\mathbf v^\varepsilon-
\frac{\partial\mathbf w^\varepsilon}{\partial t})=0
$$
we arrive at the first statement of the lemma.  The last statement
follows from the definition of $\mathbf u_{s}$.
\end{proof}

\begin{lemma} \label{lemma4.6}
Let $\lambda_1 <\infty$. Then the weak and two-scale limits $p_{s}$
and $\mathbf W$  satisfy the microscopic relations
\begin{gather}\label{4.18}
\rho_{s}\frac{\partial ^{2}\mathbf W}{\partial t^{2}}=
\lambda_{1}\Delta_y \mathbf W -\nabla_y R -\frac{1}{1-m}\nabla
p_{s} +\rho_{s}\mathbf F, \quad \mathbf y \in Y_{s}, \\
\label{4.19}
\frac{\partial \mathbf W}{\partial t}=\mathbf v, \quad \mathbf y \in
\gamma
\end{gather}
in the case $\lambda_{1}>0$, and relations
\begin{gather}\label{4.20}
\rho_{s}\frac{\partial ^{2}\mathbf W}{\partial t^{2}}= -\nabla_y R
-\frac{1}{1-m}\nabla p_{s}+\rho_{s}\mathbf F, \quad \mathbf y \in
Y_{s}, \\
\label{4.21}
(\frac{\partial \mathbf W}{\partial t}-\mathbf v)\cdot{\mathbf n}=0,
\quad \mathbf y \in \gamma
\end{gather}
in the case $\lambda_{1}=0$.

Differential equations (\ref{4.18}) and (\ref{4.20}) are endowed
with homogeneous initial conditions
\begin{equation}\label{4.22}
\mathbf W(\mathbf y,0)=\frac{\partial \mathbf W}{\partial t}(\mathbf
y,0)=0, \quad \mathbf y \in Y_{s}.
\end{equation}
In  (\ref{4.21}), ${\mathbf n}$ is the unit normal to $\gamma$.
\end{lemma}

\begin{proof}
 Differential equations (\ref{4.18}), (\ref{4.20}) and initial conditions
 (\ref{4.22})  follow as $\varepsilon\searrow 0$
 from integral equality (\ref{1.4}) with the test function ${\mathbf
\psi}={\mathbf \varphi}(x\varepsilon^{-1})\cdot h({\mathbf x},t)$,
where ${\mathbf \varphi}$ is solenoidal and finite in $Y_{s}$.

Boundary condition (\ref{4.19}) is a  consequence of the two-scale
convergence of $\{\sqrt{\alpha_{\lambda}}\nabla\mathbf
w^{\varepsilon}\}$ to the function
 $\sqrt{\lambda_{1}}\nabla_y\mathbf W(\mathbf x,t,\mathbf y)$.
On the strength of this convergence, the function $\nabla_y \mathbf
W (\mathbf x,t,\mathbf y)$ is $L^2$-integrable in $Y$.  The boundary
condition (\ref{4.21}) follows from Eqs. (\ref{4.10})-(\ref{4.11}).
\end{proof}

\subsection{Homogenized equations I}
In this section we derive homogenized equations for the liquid
component.

 \begin{lemma} \label{lemma4.7}
If $\lambda_1 =\infty$  then $\partial \mathbf w / \partial t=\mathbf v$
and  the weak limits $\mathbf v$, $p_{f}$  and $p_{s}$ satisfy in
$\Omega_{T}$ the initial-boundary value problem
 \begin{gather}\label{4.23}
\begin{aligned}
&\hat{\rho}\frac{\partial \mathbf v}{\partial
t}+\nabla(p_{f}+p_{s})-\hat{\rho}\mathbf F\\
&= \mathop{\rm div}\{\mu _{0}\mathbb A^{f}_{0}:\mathbb {D}(x,\mathbf v)
+\mathbb B^{f}_{0}p_{s}+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v+
\int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau\},
\end{aligned}\\
\label{4.24} \begin{aligned}
&p_{*}^{-1}\partial p_{f} /\partial t+\mathbb C^{f}_{0}:\mathbb
{D}(x,\mathbf v)+ a^{f}_{0}p_{s}
+(a^{f}_{1}+m)\mathop{\rm div}\mathbf v\\
&+\int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau=0,
\end{aligned}\\
\label{4.25}
\frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t}
+\frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial
t}+\mathop{\rm div}\mathbf v=0,
\end{gather}
where the symmetric strictly  positively defined constant
fourth-rank tensor $\mathbb A^{f}_{0}$, matrices  $\mathbb
C^{f}_{0}, \mathbb B^{f}_{0}$, $\mathbb B^{f}_{1}$ and $\mathbb
B^{f}_{2}(t)$ and scalars $a^{f}_{0}$, $a^{f}_{1}$ and
$a^{f}_{2}(t)$ are defined below by formulas \eqref{0.10},
\eqref{4.32} and \eqref{4.34}, where $\mathbb B^{f}_{1}=0$,
$a^{f}_{1}=0$ if $p_{*}<\infty$, and $\mathbb B^{f}_{2}=0$,
$a^{f}_{2}=0$ if $p_{*}=\infty$.

Differential equations (\ref{4.23}) are endowed with homogeneous
initial and boundary conditions
\begin{equation}\label{4.26}
\mathbf v(\mathbf x,0)=0,\quad \mathbf x\in \Omega, \quad \mathbf
v(\mathbf x,t)=0, \quad \mathbf x\in S, \quad t>0.
\end{equation}
\end{lemma}

\begin{proof}
First note that $\mathbf v =\partial \mathbf w /
\partial t$ due to lemma \ref{lemma4.5}.

The homogenized equations (\ref{4.23}) follow from the macroscopic
equations (\ref{4.16}), after we insert in them the expression
$$\mu_{0}\langle \mathbb {D}(y,\mathbf V)\rangle _{Y_{f}}=
\mu_{0}\mathbb A^{f}_{1}:\mathbb {D}(x,\mathbf v)+\mathbb
B^{f}_{0}p_{s}+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf v+
\int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau+\mathbb{A}(t).$$ In turn, this expression follows by
virtue of solutions of (\ref{4.7})  and (\ref{4.15}) on the
pattern cell $Y_{f}$.
In fact, if  $ p_{*}<\infty $,   then setting
\begin{gather*}
\mathbf V=\sum_{i,j=1}^{3}\mathbf V^{(ij)}(\mathbf y)D_{ij}
+\mathbf V^{(0)}(\mathbf y)p_{s}+\int_{0}^{t}\mathbf V^{(2)}(\mathbf
y,t-\tau)\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau,
\\
P_{f} =\mu_{0}\sum_{i,j=1}^{3}P^{ij}(\mathbf y)D_{ij}
+P^{0}(\mathbf y)p_{s}+\int_{0}^{t}P^{(2)}(\mathbf y,t-\tau)
\mathop{\rm div}\mathbf v(\mathbf x,\tau)d\tau,
\end{gather*}
where
$$
D_{ij}(\mathbf x,t)=\frac{1}{2}(\frac{\partial v_{i}}
{\partial x_{j}}(\mathbf x,t)+ \frac{\partial v_{j}}{\partial
x_{i}}(\mathbf x,t)),
$$
we arrive at the following periodic-boundary
value problems in $Y$:
\begin{gather}\label{4.27}
\mathop{\rm div}{}_y\{\chi \mathbb D(y,\mathbf V^{(ij)})-\chi
P^{(ij)}\mathbb I+\chi J^{ij}\}=0, \quad \chi\mathop{\rm div}{}_y \mathbf
V^{(ij)} =0; \\
\label{4.28}
\mathop{\rm div}{}_y \{\mu_{0}\chi \mathbb D(y,\mathbf V^{(0)}) -\big(\chi
P^{(0)}+\frac{1-\chi}{1-m}\big)\mathbb I \}=0, \quad
\chi\mathop{\rm div}{}_y \mathbf V^{(0)} =0;
\\
\mathop{\rm div}{}_y \{\mu_{0}\chi \mathbb D(y,\mathbf V^{(2)}) -\chi
P^{(2)}\mathbb  I \}=0,
\\
\label{4.29}
\frac{1}{p_{*}}\frac{\partial P^{(2)}}{\partial t} +\chi\mathop{\rm div}{}_y
\mathbf V^{(2)} =0, \, \frac{1}{p_{*}}P^{(2)}(\mathbf y,0)=-\chi(\mathbf y).
\end{gather}
For the case  $p_{*}=\infty$  we put
\begin{gather*}
 \mathbf V=\sum_{i,j=1}^{3}\mathbf V^{(ij)}(\mathbf y)D_{ij}
+\mathbf V^{(0)}(\mathbf y)p_{s}+\mathbf V^{(1)}(\mathbf
y)\mathop{\rm div}\mathbf v,\\
P_{f} =\sum_{i,j=1}^{3}P^{ij}(\mathbf y)D_{ij}
+P^{0}(\mathbf y)p_{s}+P^{(1)}(\mathbf y)\mathop{\rm div} \mathbf v,
\end{gather*}
where functions $\mathbf V^{(1)}$ and $P^{(1)}$ satisfy in $Y$ the
following periodic-boundary value problem in $Y$:
\begin{equation}\label{4.30}
\mathop{\rm div}{}_y \{\mu_{0}\chi \mathbb D(y,\mathbf V^{(1)}) -\chi
P^{(1)}\mathbb I \}=0,\, \chi(\mathop{\rm div}{}_y \mathbf V^{(1)}+1) =0.
\end{equation}


Note, that for all cases the functional $\beta$ is equal to zero due
to the special choice of the function $\mathbf{V}$, boundary
condition (\ref{4.26}) for the function $\mathbf{v}$ and conditions
(\ref{4.14}).

On the strength of the assumptions on the geometry of the pattern
``liquid'' cell $Y_{f}$, problems (\ref{4.27})--(\ref{4.30}) have
unique solution, up to an arbitrary constant vector. In order to
discard the arbitrary constant vectors we demand
$$
\langle\mathbf V^{(ij)}\rangle _{Y_{f}}=\langle\mathbf
V^{(0)}\rangle_{Y_{f}} =\langle\mathbf
V^{(1)}\rangle_{Y_{f}}=\langle\mathbf V^{(2)}\rangle_{Y_{f}}=0.
$$
Thus
\begin{gather}\label{4.31}
\mathbb A^{f}_{0}=m\mathbb J+\mathbb A^{f}_{1}, \quad \mathbb
A^{f}_{1}=\sum_{i,j=1}^{3}\langle \mathbb {D}(y,\mathbf
V^{(ij)})\rangle _{Y_{f}}\otimes \mathbb J^{ij}, \\
\label{4.32}
\mathbb B^{f}_{i}=\mu_{0}\langle \mathbb {D}(y,\mathbf
V^{(i)})\rangle _{Y_{f}}, \quad i=0,1,2.
\end{gather}
 Symmetry of the tensor $\mathbb A^{f}_{0}$ follows from symmetry of
 the tensor $\mathbb A^{f}_{1}$. And symmetry of the latter one follows
 from the equality
\begin{equation}\label{4.33}
\langle \mathbb {D}(y,\mathbf V^{(ij)})\rangle _{Y_{f}} : J^{kl}
=-\langle \mathbb {D}(y,\mathbf V^{(ij)}) : \mathbb {D}(y,\mathbf
V^{(kl)})\rangle_{Y_{f}}
\end{equation}
which appears by means of multiplication of  (\ref{4.27}) for
$\mathbf V^{(ij)}$ by $\mathbf V^{(kl)}$ and by integration by
parts.

This equality also implies positive definiteness of the tensor
$\mathbb A^{f}_{0}$. Indeed, let $\mathbb Z=(Z_{ij})$ be an
arbitrary symmetric matrix. Setting
 $
\mathbf{Z}=\sum_{i,j=1}^{3}\mathbf V^{(ij)}Z_{ij}
$
and taking into account (\ref{4.33}) we get
$$
\langle \mathbb {D}(y,\mathbf{Z})\rangle _{Y_{f}}:\mathbb Z
=-\langle \mathbb {D}(y,\mathbf{Z}): \mathbb
{D}(y,\mathbf{Z})\rangle_{Y_{f}}.
$$
This equality and the definition of the tensor $A_{0}^f$ give
$$
(\mathbb A_{0}^f:\mathbb Z):\mathbb Z=
\langle(\mathbb {D}(y,\mathbf{Z})+\mathbb Z): (\mathbb
{D}(y,\mathbf{Z})+\mathbb Z)\rangle_{Y_{f}}.
$$
Now the strict positive definiteness of the tensor $\mathbb
A_{0}^{f}$ follows immediately  from the equality  above and the
geometry of the elementary cell $Y_{f}$. Namely, suppose that
$(\mathbb A_{0}^{s}:\mathbb Z):\mathbb Z=0$ for some matrix $\mathbb
Z$, such that $\mathbb Z:\mathbb Z=1$. Then $(\mathbb
{D}(y,\mathbf{Z})+\mathbb Z)=0$, which is possible if and
only if $\mathbf{Z}$
is a linear function in $\mathbf y$. On the other hand, all linear
periodic functions on $Y_{f}$ are constant. Finally, the
normalization condition $\langle\mathbf V^{(ij)}\rangle_{Y_{f}} =0$
yields that $\mathbf{Z}=0$. However, this is impossible because the
functions $\mathbf V^{(ij)}$ are linearly independent.

Equations (\ref{4.24}) and (\ref{4.25}) for the pressures follow
from  (\ref{4.6}),  (\ref{4.8}) and equality
$$
\langle \mathop{\rm div}{}_y\mathbf V\rangle_{Y_{f}}
=\mathbb{C}^{f}_{0}:\mathbb {D}(x,\mathbf v)+ a^{f}_{0}p_{s}
+a^{f}_{1}\mathop{\rm div}\mathbf v+
\int_{0}^{t}a^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau
$$
 with
\begin{equation}\label{4.34}
\mathbb C^{f}_{0}=\sum_{i,j=1}^{3}\langle \mathop{\rm div}{}_y\mathbf
V^{(ij)}\rangle _{Y_{f}}\mathbb{J}^{ij}, \quad
a^{f}_{i}=\langle\mathop{\rm div}{}_y\mathbf V^{(i)}\rangle _{Y_{f}},\quad
i=0,1,2.
\end{equation}
Finally note, that  initial conditions (\ref{4.26}) follow from
initial conditions (\ref{4.17}) and lemma \ref{lemma4.5}.
\end{proof}

\subsection{Homogenized equations II}
We complete  the proof of theorem \ref{theorem2} with homogenized
equations for the solid component.

Let $\lambda_{1}<\infty$. In the same manner as above, we verify
that the limit $\mathbf v$ of the sequence $\{\mathbf
v^\varepsilon\}$ satisfies the initial-boundary value problem likes
(\ref{4.23})-- (\ref{4.25}). The main difference here that, in
general, the weak limit $\partial\mathbf w / \partial t$ of the
sequence $\{\partial\mathbf w^\varepsilon /\partial t\}$ differs
from $\mathbf v$. More precisely, the following statement is true.

\begin{lemma} \label{lemma4.8}
Let $\lambda_{1}<\infty$. Then the weak limits $\mathbf v$, $\mathbf
u_{s}$, $p_{f}$,  and $p_{s}$ of the sequences
 $\{\mathbf v^\varepsilon\}$, $\{(1-\chi^{\varepsilon})\mathbf w^\varepsilon\}$,
  $\{p_{f}^\varepsilon\}$,   and $\{p_{s}^\varepsilon\}$
satisfy the initial-boundary value problem in $\Omega_T$, consisting
of the balance of momentum equation
\begin{equation}\label{4.35}
\begin{aligned}
 &\rho_{f}m\frac{\partial
\mathbf v}{\partial t}+\rho_{s} \frac{\partial ^2\mathbf
u_{s}}{\partial t^2} + \nabla(p_{f}+p_{s})
-\hat{\rho}\mathbf F\\
&= \mathop{\rm div}\{\mu_{0}\mathbb A^{f}_{0}:\mathbb {D}(x,\mathbf v) +
\mathbb B^{f}_{0}p_{s}+\mathbb B^{f}_{1}\mathop{\rm div}\mathbf
v+\int_{0}^{t}\mathbb B^{f}_{2}(t-\tau)\mathop{\rm div}\mathbf v(\mathbf
x,\tau)d\tau\},
\end{aligned}
\end{equation}
 and  the continuity equation (\ref{4.24})  for the liquid component,
where $\mathbb A^{f}_{0}$, $\mathbb B^{f}_{0}$-- $\mathbb B^{f}_{2}$
are the same as in (\ref{4.23}),  the continuity equation
\begin{equation} \label{4.36}
\frac{1}{p_{*}}\frac{\partial p_{f}}{\partial t}+
\frac{1}{\eta_{0}}\frac{\partial p_{s}}{\partial t}+\mathop{\rm div}
\frac{\partial\mathbf u_{s}}{\partial t} +m \mbox {div}\mathbf v=0,
\end{equation}
 the relation
\begin{gather}\label{4.37}
\frac{\partial \mathbf u_{s}}{\partial t}=(1-m)\mathbf v(\mathbf
x,t)+\int_{0}^{t}\mathbb B^{s}_{1}(t-\tau)\cdot \mathbf z(\mathbf
x,\tau )d\tau , \\
\mathbf z(\mathbf x,t)=-\frac{1}{1-m}\nabla_x p_{s}
(\mathbf x,t)+\rho_{s}\mathbf F(\mathbf x,t)- \rho_{s}\frac{\partial
\mathbf v}{\partial t}(\mathbf x,t)
\end{gather}
in the case $\lambda_{1}>0$, or the balance of momentum equation
in the form
\begin{equation}\label{4.38}
\rho_{s}\frac{\partial^{2}\mathbf u_{s}}{\partial
t^{2}}=\rho_{s}\mathbb B^{s}_{2}\cdot \frac{\partial \mathbf
v}{\partial t}+((1-m)I-\mathbb B^{s}_{2})\cdot(-\frac{1}{1-m}\nabla
p_{s}+\rho_{s}\mathbf F)
\end{equation}
in the case of $\lambda_{1}=0$ for the solid component. The problem
is supplemented by boundary and initial conditions (\ref{4.26}) for
the velocity $\mathbf v$ of the liquid component and by homogeneous
initial conditions and the boundary condition
\begin{equation}\label{4.39}
\mathbf u_{s}(\mathbf x,t)\cdot \mathbf n(\mathbf x)=0, \quad
(\mathbf x,t) \in S, \quad t>0,
\end{equation}
for the displacement $\mathbf u_{s}$ of the solid component. In Eqs.
(\ref{4.37})--(\ref{4.39}) $\mathbf n(\mathbf x)$ is the unit normal
vector to $S$ at a point $\mathbf x \in S$, and matrices $\mathbb
B^{s}_{1}(t)$ and $\mathbb B^{s}_{2}$ are given below by Eqs.
(\ref{4.41}) and (\ref{4.43}).
\end{lemma}

\begin{proof}
The boundary condition (\ref{4.39}) follows from (\ref{4.9}), the
equality
$$
\frac{\partial\mathbf w}{\partial t}
=\frac{\partial\mathbf u_{s}}{\partial t}+m\mathbf v,
$$
and the homogeneous boundary condition for $\mathbf v$.

 The same equality and (\ref{4.8}) imply (\ref{4.36}).
 The homogenized equations of balance of momentum (\ref{4.35})
  derives exactly as before.  Therefore we
omit the relevant proofs now and focus ourself only on derivation of
homogenized equation of the balance of momentum for the solid
displacements $\mathbf u_{s}$.

(a) If  $\lambda_{1}>0$, then the solution of the system of
microscopic equations (\ref{4.10}), (\ref{4.18}), and (\ref{4.19}),
provided with the homogeneous initial data  (\ref{4.22}), is given
by formula
\begin{gather*}
\mathbf W=\int_{0}^{t}( \mathbf v(\mathbf x,\tau )+
\sum_{i=1}^{3}\mathbf W^{i}(\mathbf y,t-\tau)\otimes {\mathbf
e}_{i}\cdot\mathbf z(\mathbf x,\tau ))d\tau ,
\\
 R=\int_{0}^{t}\sum_{i=1}^{3}R^{i}(\mathbf
y,t-\tau){\mathbf e}_{i}\cdot\mathbf z(\mathbf x,\tau )d\tau ,
\end{gather*}
in which  functions $\mathbf W^{i}(\mathbf y,t)$ and
$R^{i}(\mathbf y,t)$ are defined by virtue of the periodic
initial-boundary value problem
\begin{equation}\label{4.40}
\begin{gathered}
\rho_{s}\frac{\partial ^{2} \mathbf W^{i}}{\partial
t^{2}}-\lambda_{1}\Delta \mathbf W^{i} +\nabla R^{i} =0, \quad
\mathop{\rm div}{}_y \mathbf W^{i} =0, \quad \mathbf y \in Y_{s},\;t>0,
\\
\mathbf W^{i}=0, \quad \mathbf y \in \gamma ,\;  t>0,
\\
\mathbf W^{i}(y,0)=0, \quad \rho_{s}\frac{\partial\mathbf
W^{i}}{\partial t}(y,0)={\mathbf e}_{i},\quad \mathbf y \in Y_{s}.
\end{gathered}
\end{equation}
In  \eqref{4.40}, ${\mathbf e}_{i}$ is the standard Cartesian
 basis vector.
Therefore,
\begin{equation}\label{4.41}
B^{s}_{1}(t)= \sum_{i=1}^{3}\langle \frac{\partial \mathbf
W^{i}}{\partial t}\rangle _{Y_{s}}\otimes {\mathbf e}_{i}(t).
\end{equation}
Note, that differential equations in  \eqref{4.40} are understood in
the sense of distributions (the compatibility conditions on the
boundary $\gamma$ at $t=0$ have no place) and therefore the
functions $\partial\mathbf W^{i}/\partial t$  have  no time
derivative at $t=0$.


 (b) If  $\lambda_{1}=0$ then in the
process of solving the system (\ref{4.10}), (\ref{4.18}), and
(\ref{4.19}) we firstly find the pressure $R(\mathbf x,t,\mathbf y)$
by virtue of solving the Neumann problem for Laplace's equation in
 $Y_{s}$ in the form
 $$
R(\mathbf x,t,\mathbf y)=\sum_{i=1}^{3} R_{i}(\mathbf y)
 {\mathbf e}_{i}\cdot \mathbf z(\mathbf x,t),
$$
 where $R^{i}(\mathbf y)$ is the solution of the problem
 \begin{equation}\label{4.42}
\Delta_y R_{i}=0,\quad \mathbf y \in Y_{s}; \quad \nabla_y
R_{i}\cdot \mathbf n =\mathbf n\cdot{\mathbf e}_{i}, \quad \mathbf y
\in \gamma .
\end{equation}
Formula (\ref{4.35}) appears as the result of  homogenization of
(\ref{4.18}) and
 \begin{equation}\label{4.43}
B^{s}_{2}=\sum_{i=1}^{3}\langle \nabla R_{i}(\mathbf y)\rangle
_{Y_{s}}\otimes {\mathbf e}_{i},
\end{equation}
where the matrix $((1-m)I - B^{s}_{2})$ is symmetric and positively
definite.  In fact, let $\tilde{R}=\sum_{i=1}^{3}R_{i}\mathbf
xi_{i}$ for any unit vector $\xi$. Then
 $$
(B\cdot\xi)\cdot\xi=\langle(\xi-\nabla\tilde{R})^{2}\rangle_{Y_{f}}>0
$$
due to the same reasons as in lemma \ref{lemma4.6}.
On the strength of the assumptions on the geometry of the pattern
``solid'' cell $Y_{s}$, problem  \eqref{4.40} has unique solution
and problem (\ref{4.42}) has unique solution up to an arbitrary
constant.
\end{proof}

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\end{thebibliography}


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