# Homogenization of some evolutionary non-Newtonian flows in porous media

Yong Lu 111Department of Mathematics, Nanjing University, Nanjing 210093,
China<EMAIL_ADDRESS>Zhengmao Qian 222Department of Mathematics, Nanjing
University, Nanjing 210093, China<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider the homogenization of evolutionary incompressible
purely viscous non-Newtonian flows of Carreau-Yasuda type in porous media with
small perforation parameter $0<\varepsilon\ll 1$, where the small holes are
periodically distributed. Darcy’s law is recovered in the homogenization
limit. Applying Poincaré type inequality in porous media allows us to derive
the uniform estimates on velocity field, of which the gradient is small of
size $\varepsilon$ in $L^{2}$ space. This indicates the nonlinear part in the
viscosity coefficient does not contribute in the limit and a linear model
(Darcy’s law) is obtained. The estimates of the pressure rely on a proper
extension from the perforated domain to the homogeneous non-perforated domain.
By integrating the equations in time variable such that each term in the
resulting equations has certain continuity in time, we can establish the
extension of the pressure by applying the dual formula with the restriction
operator.

##  1 Introduction

In this paper we consider the homogenization of evolutionary incompressible
viscous non-Newtonian flows in porous media. Non-Newtonian fluids are
extensively involved in a number of applied problems involving the production
of oil and gas from underground reservoirs. There are at least two typical
situations: the flow of heavy oils and enhanced oil recovery. Therefore, it is
important to establish filtration laws governing non-Newtonian flows through
porous media. In this paper, we will consider only quasi-Newtonian fluids
where the viscosity can be expressed as a function of the shear rate. In
particular, we focus on the Carreau-Yasuda model in space-time cylinder
$(0,T)\times\Omega_{\varepsilon}$:

$\begin{cases}\varepsilon^{2}\partial_{t}\mathbf{u}_{\varepsilon}-{\rm
div}\,\big{(}\eta_{r}(D\mathbf{u}_{\varepsilon})D\mathbf{u}_{\varepsilon}\big{)}+(\mathbf{u}_{\varepsilon}\cdot\nabla)\mathbf{u}_{\varepsilon}+\nabla
p_{\varepsilon}=\mathbf{f},&{\rm in}\;(0,T)\times\Omega_{\varepsilon},\\\ {\rm
div}\,\mathbf{u}_{\varepsilon}=0,&{\rm
in}\;(0,T)\times\Omega_{\varepsilon},\\\ \mathbf{u}_{\varepsilon}=0,&{\rm
in}\;(0,T)\times\partial\Omega_{\varepsilon},\\\
\mathbf{u}_{\varepsilon}|_{t=0}=\mathbf{u}_{0},&{\rm
in}\;\Omega_{\varepsilon}.\end{cases}$ (1.1)

Here $\mathbf{u}_{\varepsilon}$ is the velocity,
$\nabla\mathbf{u}_{\varepsilon}$ is the gradient velocity tensor,
$D\mathbf{u}_{\varepsilon}=\frac{1}{2}(\nabla\mathbf{u}_{\varepsilon}+\nabla^{T}\mathbf{u}_{\varepsilon})$
denotes the rate-of-strain tensor, $p_{\varepsilon}$ denotes the pressure,
$\mathbf{f}$ is the density of the external force and ${\mathbf{u}}_{0}$ is
the initial velocity. In this paper, we assume $\mathbf{f}$ and
${\mathbf{u}}_{0}$ are independent of $\varepsilon$ and are both in
$L^{2}((0,T)\times\Omega;\mathbb{R}^{3})$. While, our main results still hold
if $\mathbf{f}$ and ${\mathbf{u}}_{0}$ depend on $\varepsilon$ and converge
strongly in $L^{2}((0,T)\times\Omega;\mathbb{R}^{3})$. The stress tensor
$\eta_{r}(D\mathbf{u}_{\varepsilon})$ is determined by the Carreau-Yasuda law:

$\eta_{r}(D\mathbf{u}_{\varepsilon})=(\eta_{0}-\eta_{\infty})(1+\lambda|D\mathbf{u}_{\varepsilon}|^{2})^{\frac{r}{2}-1}+\eta_{\infty},\quad\eta_{0}\geq\eta_{\infty},\
\lambda>0,\ r>1,$

where $\eta_{0}$ is the zero-shear-rate viscosity, $\lambda$ is a time
constant, $(r-1)$ is a dimensionless constant describing the slope in the
power law region of log $\eta_{r}$ versus log
$(|D({\mathbf{u}}_{\varepsilon})|)$.

The perforated domain $\Omega_{\varepsilon}$ under consideration is described
as follows. Let $\Omega$ be a bounded domain of class $C^{2,\mu},0<\mu<1$. The
holes in $\Omega$ are denoted by $T_{\varepsilon,k}$ which are assumed to
satisfy

$T_{\varepsilon,k}=\varepsilon x_{k}+\varepsilon T\subset\subset\varepsilon
Q_{k},$

where the cube $Q_{k}=(-\frac{1}{2},\frac{1}{2})^{3}+k$ and $x_{k}=x_{0}+k$
with $x_{0}\in(-\frac{1}{2},\frac{1}{2})^{3},\;k\in{\mathbb{Z}}^{3}$; $T$ is a
model hole which is assumed to be a closed domain contained in $Q_{0}$ with
$C^{2,\mu}$ boundary. The perforation parameter $\varepsilon$ is used to
measure the mutual distance and size of the holes, and $\varepsilon
x_{k}=\varepsilon x_{0}+\varepsilon k$ are the locations of the holes.

The perforated domain $\Omega_{\varepsilon}$ is then defined as:

$\Omega_{\varepsilon}=\Omega\backslash\bigcup_{k\in
K_{\varepsilon}}T_{\varepsilon,k},\quad{\rm where}\
K_{\varepsilon}=\\{k\in{\mathbb{Z}}^{3}:\varepsilon\overline{Q}_{k}\subset\Omega\\}.$
(1.2)

The study of homogenization problems in fluid mechanics have gained a lot
interest. In particular, the homogenization of Stokes system in perforated
domains has been systematically studied. In [24], Tartar considered the case
where the size of the holes is proportional to the mutual distance of the
holes and Darcy’s law was derived. Then Allaire [1, 2] considered general
cases and showed that the homogenized equation are determined by the ratio
$\sigma_{\varepsilon}$ between the size and the mutual distance of the holes:

$\sigma_{\varepsilon}=\big{(}\frac{\varepsilon^{d}}{a_{\varepsilon}^{d-2}}\big{)}^{\frac{1}{2}},\quad
d\geq 3;\qquad\sigma_{\varepsilon}=\varepsilon\big{|}{\rm
log}\frac{a_{\varepsilon}}{\varepsilon}\big{|}^{\frac{1}{2}},\quad d=2,$

where $\varepsilon$ and $a_{\varepsilon}$ are used to measure the mutual
distance of holes and the size of holes. Particularly, if
$\displaystyle\lim_{\varepsilon\rightarrow 0}\sigma_{\varepsilon}=0$
corresponding to the case of large holes, the homogenized system is the
Darcy’s law; if $\displaystyle\lim_{\varepsilon\rightarrow
0}\sigma_{\varepsilon}=\infty$ corresponding to the case of small holes, there
arise the same Stokes equations in homogeneous domains; if
$\displaystyle\lim_{\varepsilon\rightarrow
0}\sigma_{\varepsilon}=\sigma_{\ast}\in(0,+\infty)$ corresponding to the case
of critical size of holes, the homogenized equations are governed by the
Brinkman’s law–a combination of the Darcy’s law and the original Stokes
equations. Same results were shown in [18] by employing a generalized cell
problem inspired by Tartar [24].

Later, the homogenization study is extended to more complicated models
describing fluid flows: Mikelić [20] for the nonstationary incompressible
Navier-Stokes equations, Masmoudi [21] for the compressible Navier-Stokes
equations, Feireisl, Novotný and Takahashi [12] for the complete Navier-
Stokes-Fourier equations. In all these studies, only the case where the size
of holes is proportional to the mutual distance of the holes is considered and
the Darcy’s law is recovered in the limit.

Recently, cases with different sizes of holes are studied. Feireisl,
Namlyeyeva and Nečasová [10] studied the case with critical size of holes for
the incompressible Navier-Stokes equations and they derived Brinkman’s law;
Yang and the first author [19] studied the homogenization of evolutionary
incompressible Navier-Stokes system with large and small size of holes. In [8,
11, 16], with collaborators the first author considered the case of small
holes for the compressible Navier-Stokes equations and it is shown that the
homogenized equations remain the same as the original ones. Oschmann and
Pokorný [22] also considered the case of small holes for the unsteady
compressible Navier-Stokes equations for adiabatic exponent $\gamma>3$ which
improved the condition $\gamma>6$ of [16], and they showed that the
homogenized equations keep unchanged. Bella and Oschmann [6] considered the
homogenization of compressible Navier-Stokes equations for the case with
randomly perforated domains with small size of holes and they get the same
limiting equation. Höfer, Kowalczyk and Schwarzacher [13] studied the case of
large holes for the compressible Navier-Stokes equations at low Mach number
and derived the Darcy’s law; Bella and Oschmann [5] also studied the case with
critical size of holes for the compressible Navier-Stokes equations at low
Mach number and they derived incompressible Navier-Stokes equations with
Brinkman term. Bella, Feireisl and Oschmann [7] considered the case of
unsteady compressible Navier-Stokes equations at low Mach number under the
assumption $\Omega_{\varepsilon}\to\Omega$ in sense of Mosco’s convergence and
they derived the incompressible Navier-Stokes equations. Oschmann and Nečasová
[23] studied homogenization of the two-dimensional evolutionary compressible
Navier-Stokes equations with very small holes and limiting equations remain
unchanged.

There are not many mathematical studies concerning the homogenization of non-
Newtonian flows. Mikelić and Bourgeat [4] considered stationary case of
Carreau-Yasuda type flows under the assumption
$a_{\varepsilon}\sim\varepsilon$ and derived Darcy’s law. Mikelić summarized
some theory of stationary non-Newtonian flows in Chapter 4 of [14]. While for
evolutionary non-Newtonian fluid equations, according to the authors’
knowledge, there is no rigorous mathematical analysis results. In this paper,
we justify in porous media setting, the evolutionary Carreau-Yasuda model
converges to Darcy’s law.

###  1.1 Notations and weak solutions

We recall some notations of Sobolev spaces. Let $1\leq r\leq\infty$ and
$\Omega$ be a bounded domain. We use the notation $L_{0}^{r}(\Omega)$ to
denote the space of $L^{r}(\Omega)$ functions with zero mean value:

$L_{0}^{r}(\Omega)=\Big{\\{}f\in L^{r}(\Omega)\ :\ \int_{\Omega}f\,{\rm
d}x=0\Big{\\}}.$

We use $W^{1,r}(\Omega)$ to denote classical Sobolev space, and
$W^{1,r}_{0}(\Omega)$ denotes the completion of $C_{c}^{\infty}(\Omega)$ in
$W^{1,r}(\Omega)$. Here $C_{c}^{\infty}(\Omega)$ is the space of smooth
functions compactly supported in $\Omega$. We use $W^{-1,r^{\prime}}(\Omega)$
to denote the dual space of $W^{1,r}_{0}(\Omega)$. For $1\leq r<\infty$,
$W^{1,r}(\mathbb{R}^{3})=W^{1,r}_{0}(\mathbb{R}^{3})$. We introduce the
functional space $W_{0,\rm div}^{1,r}(\Omega),\ 1\leq r\leq\infty$ by

$W_{0,\rm div}^{1,r}(\Omega)=\left\\{u\in W_{0}^{1,r}(\Omega;\mathbb{R}^{3}):\
{\rm div}\,u=0\ {\rm in}\,\Omega\right\\}.$

Now we introduce the definition of finite energy weak solutions to (1.1):

######  Definition 1.1.

Let $T>0$. We say that $\mathbf{u}_{\varepsilon}$ is a finite energy weak
solution of (1.1) in $(0,T)\times\Omega_{\varepsilon}$ provided

  * •

$\mathbf{u}_{\varepsilon}\in C_{\rm
weak}([0,T);L^{2}(\Omega_{\varepsilon};\mathbb{R}^{3}))\cap L^{2}(0,T;W_{0,\rm
div}^{1,2}(\Omega_{\varepsilon}))\cap L^{r}(0,T;W_{0,\rm
div}^{1,r}(\Omega_{\varepsilon}))$.

  * •

The integral identity

$\displaystyle\int_{0}^{T}\int_{\Omega_{\varepsilon}}-\varepsilon^{2}\mathbf{u}_{\varepsilon}\cdot\partial_{t}\varphi-\mathbf{u}_{\varepsilon}\otimes\mathbf{u}_{\varepsilon}:\nabla\varphi+\eta_{r}(D\mathbf{u}_{\varepsilon})D\mathbf{u}_{\varepsilon}:D\varphi\,{\rm
d}x{\rm d}t$
$\displaystyle=\int_{0}^{T}\int_{\Omega_{\varepsilon}}\mathbf{f}\cdot\varphi\,{\rm
d}x{\rm
d}t+\varepsilon^{2}\int_{\Omega_{\varepsilon}}\mathbf{u}_{0}\cdot\varphi(0,\cdot)\,{\rm
d}x$

holds for any test function $\varphi\in
C_{c}^{\infty}([0,T)\times\Omega_{\varepsilon};\mathbb{R}^{3}),\ {\rm
div}_{x}\varphi=0$.

  * •

The energy inequality

$\displaystyle\int_{\Omega_{\varepsilon}}\frac{\varepsilon^{2}}{2}\mathbf{u}_{\varepsilon}^{2}\,{\rm
d}x+\int_{0}^{t}\int_{\Omega_{\varepsilon}}\eta_{r}(D\mathbf{u}_{\varepsilon})|D\mathbf{u}_{\varepsilon}|^{2}\,{\rm
d}x{\rm
d}t\leq\int_{\Omega_{\varepsilon}}\frac{\varepsilon^{2}}{2}\mathbf{u}_{0}^{2}\,{\rm
d}x+\int_{0}^{t}\int_{\Omega_{\varepsilon}}\mathbf{f}\cdot\mathbf{u}_{\varepsilon}\,{\rm
d}x{\rm d}t$ (1.3)

holds for a.a. $t\in(0,T)$.

The classical theory from Ladyzhenskaya [15], Theorem 1.1 in [3] and Theorem
1.3 in [26] gives the existence of at least one weak solution
$\mathbf{u}_{\varepsilon}\in C_{\rm
weak}([0,T);L^{2}(\Omega_{\varepsilon};\mathbb{R}^{3}))\cap L^{r}(0,T;W_{0,\rm
div}^{1,r}(\Omega_{\varepsilon}))$ for $r>2$ and $\mathbf{u}_{\varepsilon}\in
C_{\rm weak}([0,T);L^{2}(\Omega_{\varepsilon};\mathbb{R}^{3}))\cap
L^{2}(0,T;W_{0,\rm div}^{1,2}(\Omega_{\varepsilon}))$ for $1<r\leq 2$.

For brevity we use $C$ to denote a constant independent of $\varepsilon$
throughout the paper, while the value of $C$ may differ from line to line.

###  1.2 Restriction, extension, and some useful lemmas

Our goal is to obtain the limit system in homogeneous domains without holes,
so we need to extend $(\mathbf{u}_{\varepsilon},p_{\varepsilon})$ to the whole
of $\Omega$. Due to the zero boundary conditions on
${\mathbf{u}}_{\varepsilon}$, it is nature to extend
$\mathbf{u}_{\varepsilon}$ by zero to the holes. However, the extension of the
pressure is more delicate. It is defined by the restriction operator due to
Tartar [24] for the case where the size of the holes is proportional to their
mutual distance and extended to general sizes of holes by Allaire [1, 2]. For
$\Omega_{\varepsilon}$ defined in (1.2), there exists a linear operator, named
restriction operator,
$R_{\varepsilon}:W_{0}^{1,q}(\Omega;{\mathbb{R}}^{3})\rightarrow
W_{0}^{1,q}(\Omega_{\varepsilon};{\mathbb{R}}^{3})$ $(1<q<\infty)$ such that:

$\displaystyle u\in
W_{0}^{1,q}(\Omega_{\varepsilon};{\mathbb{R}}^{3})\Longrightarrow
R_{\varepsilon}(\tilde{u})=u\ {\rm in}\ \Omega_{\varepsilon},\ {\rm where}\
\tilde{u}\ {\rm is\ the\ zero\ extension\ of}\ u,$ (1.4) $\displaystyle{\rm
div}\,u=0{\rm\ in}\ \Omega\Longrightarrow{\rm div}\,R_{\varepsilon}(u)=0\ \rm
in\ \Omega_{\varepsilon},$ $\displaystyle\|\nabla
R_{\varepsilon}(u)\|_{L^{q}(\Omega_{\varepsilon})}\leq
C(\varepsilon^{-1}\|u\|_{L^{q}(\Omega)}+\|\nabla u\|_{L^{q}(\Omega)}).$

The construction of such a restriction operator can be found in [20]. Later
Allaire [1, 2] constructed such type of restriction operators for general
sizes of holes in $L^{2}$ framework. Recently, following the construction of
Allaire, the first author [17] gave a construction of restriction operators
for general sizes of holes in general $L^{q}$ framework.

The extension $\tilde{p}_{\varepsilon}$ of the pressure $p_{\varepsilon}$ with
$\nabla p_{\varepsilon}\in
W^{-1,q^{\prime}}(\Omega_{\varepsilon};\mathbb{R}^{3})$ is then defined
through the following dual formulation:

$\langle\nabla\tilde{p}_{\varepsilon},\varphi\rangle_{\Omega}=\langle\nabla
p_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{\Omega_{\varepsilon}},\qquad\forall\,\varphi\in
W^{1,q}_{0}(\Omega;{\mathbb{R}}^{3}).$

Such an extension $\tilde{p}_{\varepsilon}$ is well defined due to the three
properties in (LABEL:def-restriction) of the restriction operator.

Now we introduce several useful conclusions which will be frequently used
throughout this paper. Let’s first introduce Poincaré inequality in porous
media. One may find this proof in [24, 20].

######  Lemma 1.2.

For each $u\in W_{0}^{1,q}(\Omega_{\varepsilon};\mathbb{R}^{3}),\;1<q<\infty$,
where $\Omega_{\varepsilon}$ is defined in (1.2). Then there holds

$\|u\|_{L^{q}(\Omega_{\varepsilon})}\leq C\varepsilon\|\nabla
u\|_{L^{q}(\Omega_{\varepsilon})}.$ (1.5)

Next we introduce the restriction operator which concerns functions with time
variable. For each $\varphi\in L^{p}(0,T;W_{0}^{1,q}(\Omega))$ with $1\leq
p\leq\infty,\,1<q<\infty$, the restriction $R_{\varepsilon}(\varphi)$ is taken
only on spatial variable:

$R_{\varepsilon}(\varphi)(\cdot,t)=R_{\varepsilon}\big{(}\varphi(\cdot,t)\big{)}(\cdot),\quad\mbox{for\,
each}\ t\in(0,T).$ (1.6)

Then we can get the same properties as in (LABEL:def-restriction) for each
$t\in(0,T)$. Moreover, it is rather straightforward to deduce from (LABEL:def-
restriction) and (1.5) the following lemma:

######  Lemma 1.3.

Let $\Omega$ be a bounded domain of class $C^{1}$ and $\Omega_{\varepsilon}$
be defined in (1.2). Let $\varphi\in
L^{p}(0,T;W_{0}^{1,q}(\Omega;\mathbb{R}^{3})),\ 1\leq
p\leq\infty,\;1<q<\infty$. Then we have

$\|R_{\varepsilon}(\varphi)\|_{L^{p}(0,T;L^{q}(\Omega_{\varepsilon}))}+\varepsilon\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{p}(0,T;L^{q}(\Omega_{\varepsilon}))}\leq
C(\|\varphi\|_{L^{p}(0,T;L^{q}(\Omega))}+\varepsilon\|\nabla\varphi\|_{L^{p}(0,T;L^{q}(\Omega))}).$
(1.7)

We finally give the following Korn type inequality, see for example Chapter 10
in [9]:

######  Lemma 1.4.

(Korn inequality) Let $\Omega$ be a bounded domain of class $C^{1}$ and
$\Omega_{\varepsilon}$ be defined in (1.2). Let $1<q<\infty$. For arbitrary
$u\in W_{0}^{1,q}(\Omega_{\varepsilon};\mathbb{R}^{3})$, there holds

$\|\nabla u\|_{L^{q}(\Omega_{\varepsilon})}\leq
C(q)\|Du\|_{L^{q}(\Omega_{\varepsilon})},$ (1.8)

where $C(q)$ is independent of $\varepsilon$.

###  1.3 Main results

We now state our homogenization results, where the limits are taken up to
possible extractions of subsequences. We shall follow the idea of Mikelić [20]
and Teman [25] to consider new equations by integrating original equations
with respect to time variable. Let
$(\mathbf{u}_{\varepsilon},p_{\varepsilon})$ be a finite energy weak solution
of equations (1.1). Introduce

$U_{\varepsilon}=\int_{0}^{t}\mathbf{u}_{\varepsilon}\,{\rm
d}s,\;G_{\varepsilon}=\int_{0}^{t}(\mathbf{u}_{\varepsilon}\cdot\nabla)\mathbf{u}_{\varepsilon}\,{\rm
d}s,\;H_{\varepsilon}=\int_{0}^{t}(1+\lambda|D\mathbf{u}_{\varepsilon}|^{2})^{\frac{r}{2}-1}D\mathbf{u}_{\varepsilon}\,{\rm
d}s,\;F=\int_{0}^{t}\mathbf{f}\,{\rm d}s.$ (1.9)

Then we have $U_{\varepsilon}\in C([0,T];\;W^{1,2}_{0,\rm
div}(\Omega_{\varepsilon})),\;G_{\varepsilon}\in
C([0,T];L^{\frac{3}{2}}(\Omega_{\varepsilon})),\;F\in
C([0,T];L^{2}(\Omega_{\varepsilon}))$ and

$H_{\varepsilon}\in\begin{cases}C([0,T];L^{2}(\Omega_{\varepsilon}))&1<r\leq
2,\\\ C([0,T];L^{\frac{r}{r-1}}(\Omega_{\varepsilon}))&r>2.\end{cases}$

The classical theory of Stokes equations ensure the existence of

$P_{\varepsilon}\in\begin{cases}C_{\rm
weak}([0,T];L^{2}(\Omega_{\varepsilon}))&1<r\leq 2,\\\ C_{\rm
weak}([0,T];L^{\frac{r}{r-1}}(\Omega_{\varepsilon}))&r>2,\end{cases}$

such that for each $t\in[0,T]$,

$\nabla
P_{\varepsilon}=F-\varepsilon^{2}(\mathbf{u}_{\varepsilon}-\mathbf{u}_{0})+\frac{\eta_{\infty}}{2}\Delta
U_{\varepsilon}-G_{\varepsilon}+(\eta_{0}-\eta_{\infty}){\rm
div}\,H_{\varepsilon}\quad\mbox{in}\
\mathcal{D}^{\prime}(\Omega_{\varepsilon}).$ (1.10)

Now we are ready to state the main theorem:

######  Theorem 1.5.

Let $1<r<\infty$. Let $(\mathbf{u}_{\varepsilon},p_{\varepsilon})$ be a finite
energy weak solution of equations (1.1), and
$\tilde{\mathbf{u}}_{\varepsilon}$ is the zero extension of
$\mathbf{u}_{\varepsilon}$. The extension $\tilde{P}_{\varepsilon}$ is defined
through the restriction operator as follows,

$\langle\nabla\tilde{P}_{\varepsilon},\varphi\rangle_{(0,T)\times\Omega}=\langle\nabla
P_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}},\quad
for\ all\ \varphi\in C_{c}^{\infty}((0,T)\times\Omega),$

where $P_{\varepsilon}$ is defined in (1.10). Let
$\tilde{p}_{\varepsilon}=\partial_{t}\tilde{P}_{\varepsilon}$ be the extension
of $p_{\varepsilon}$. Then we can find $\mathbf{u}\in
L^{2}((0,T)\times\Omega)$ and

$p\in\begin{cases}W^{-1,2}(0,T;L^{2}(\Omega))&1<r\leq 2,\\\
W^{-1,\frac{r}{r-1}}(0,T;L^{\frac{r}{r-1}}(\Omega))&r>2,\end{cases}$

which satisfy

$\varepsilon^{-2}\tilde{\mathbf{u}}_{\varepsilon}\rightarrow\mathbf{u}\
weakly\ in\ L^{2}((0,T)\times\Omega),$ $\tilde{p}_{\varepsilon}\rightarrow p\
weakly\ in\begin{cases}W^{-1,2}(0,T;L^{2}(\Omega))&1<r\leq 2,\\\
W^{-1,\frac{r}{r-1}}(0,T;L^{\frac{r}{r-1}}(\Omega))&r>2.\end{cases}$

Moreover, the limit $(\mathbf{u},p)$ satisfies the Darcy’s law:

$\frac{1}{2}\eta_{0}\mathbf{u}=A(\mathbf{f}-\nabla p)\quad\mbox{in}\
\mathcal{D^{\prime}}((0,T)\times\Omega).$ (1.11)

We give several remarks concerning our main results and main ideas of proof:

######  Remark 1.6.

  * •

The permeability tensor $A$ which appears in (1.11) is a constant positive
definite matrix defined in (3.3).

  * •

Mikelić considered the homogenization of nonstationary Navier-Stokes equations
in [20], namely for the case $r=2$, and derived Darcy’s law. Observing the
strong convergence of the nonlinear viscosity coefficient
$\eta_{r}(D\mathbf{u}_{\varepsilon})$ to constant $\eta_{0}$, we derived the
Darcy’s law for arbitrary $r>1$.

  * •

The main difficulty compared to the steady case considered in [4] lies in
dealing with the estimates of the pressure $p_{\varepsilon}$, which lies in
some negative order Sobolev space with respect to time variable due to the
presence of the time derivative term $\partial_{t}{\mathbf{u}}_{\varepsilon}$.
We shall follow the idea of Mikelić [20] by integrating original equations
with respect to time variable, and this allows us to define the extension of
the pressure pointwisely in $t$.

The rest of the paper is devoted to the proof of Theorem 1.5. In Section 2, we
derive the uniform estimates of the velocity field and pressure. In Section 3,
we employ the cell problem to modify test functions, and then pass to limit in
the weak formulation of the new equations to derive the limit system—Darcy’s
law.

##  2 Uniform estimates

In this section, we derive the uniform estimates of the solutions. The
estimates of the velocity follows from the energy inequality by using the
Poincaré inequality and the Korn inequality (see Lemma 1.2 and 1.4).
Concerning the pressure $p_{\varepsilon}$, we will not deduce the estimates of
$p_{\varepsilon}$ directly. Instead, we will consider a proper extension of
its time integration $P_{\varepsilon}$ given in (1.10). Such an extension is
defined by a dual formula using restriction operator pointwisely in $t$. The
estimates of the extension of ${P_{\varepsilon}}$ follow from the estimates of
$U_{\varepsilon}$ and the estimates of the restriction operator.

###  2.1 Estimates of velocity field

Based on the energy inequality (1.3), we can derive the following estimates of
velocity field ${\mathbf{u}}_{\varepsilon}$:

######  Proposition 2.1.

Let $\mathbf{u}_{\varepsilon}$ be a weak solution of (1.1) in the sense of
Definition 1.1. There holds

$\displaystyle\|\nabla\mathbf{u}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon,\quad$
$\displaystyle\|\mathbf{u}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon^{2},\quad\|\mathbf{u}_{\varepsilon}\|_{L^{\infty}(0,T;L^{2}(\Omega_{\varepsilon}))}\leq
C,$ (2.1)
$\displaystyle\|\nabla\mathbf{u}_{\varepsilon}\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon^{\frac{2}{r}},\quad$
$\displaystyle\|\mathbf{u}_{\varepsilon}\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon^{\frac{2}{r}+1},\quad\mbox{if}\ r>2.$

###### Proof.

We first rewrite the energy inequality (1.3) as

$\displaystyle\frac{\varepsilon^{2}}{2}\int_{\Omega_{\varepsilon}}\mathbf{u}_{\varepsilon}^{2}\,{\rm
d}x+\int_{0}^{t}\int_{\Omega_{\varepsilon}}\eta_{\infty}|D\mathbf{u}_{\varepsilon}|^{2}+(\eta_{0}-\eta_{\infty})(1+\lambda|D\mathbf{u}_{\varepsilon}|^{2})^{\frac{r}{2}-1}|D\mathbf{u}_{\varepsilon}|^{2}\,{\rm
d}x{\rm d}t$ (2.2)
$\displaystyle\quad\leq\int_{0}^{t}\int_{\Omega_{\varepsilon}}\mathbf{f}\cdot\mathbf{u}_{\varepsilon}\,{\rm
d}x{\rm
d}t+\frac{\varepsilon^{2}}{2}\int_{\Omega_{\varepsilon}}\mathbf{u}_{0}^{2}\,{\rm
d}x,\quad\mbox{for a.a. $0<t\leq T$.}$

Applying the Poincaré inequality in porous media and the Korn inequality (see
Lemma 1.2 and 1.4) gives

$\displaystyle\int_{0}^{T}\int_{\Omega_{\varepsilon}}\mathbf{f}\cdot\mathbf{u}_{\varepsilon}\,{\rm
d}x{\rm d}t\leq
C\varepsilon\int_{0}^{T}\|\mathbf{f}\|_{L^{2}(\Omega)}\|D\mathbf{u}_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}{\rm
d}t.$ (2.3)

Then from (LABEL:est-u-1) and (2.3), and the assumptions that ${\mathbf{f}}$
and ${\mathbf{u}}_{0}$ are independent of $\varepsilon$ and are in
$L^{2}((0,T)\times\Omega)$, we deduce

$\displaystyle\|D\mathbf{u}_{\varepsilon}\|^{2}_{L^{2}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon\|D\mathbf{u}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\|{\mathbf{f}}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}+C\varepsilon^{2}.$

Again by the Poincaré inequality (1.5) and the Korn inequality (1.8), we
obtain the $L^{2}$ estimates in $\eqref{2.1.01}_{1}$.

If $r>2$, using the estimates in $\eqref{2.1.01}_{1}$, we deduce from
(LABEL:est-u-1) that

$\displaystyle\|D\mathbf{u}_{\varepsilon}\|^{r}_{L^{r}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon^{2},$

and the $L^{r}$ estimates in $\eqref{2.1.01}_{2}$ follow from the Poincaré
inequality and the Korn inequality.

∎

By the uniform estimates of velocity in Proposition 2.1, we have the following
uniform estimates:

######  Corollary 2.2.

Let $U_{\varepsilon}$, $G_{\varepsilon}$ and $H_{\varepsilon}$ be defined as
in (1.9). Then

$\displaystyle\|U_{\varepsilon}\|_{W^{1,2}(0,T;W^{1,2}_{0}(\Omega_{\varepsilon}))}\leq
C\varepsilon,\quad\|U_{\varepsilon}\|_{W^{1,2}(0,T;L^{2}(\Omega_{\varepsilon}))}\leq
C\varepsilon^{2},$ (2.4)
$\displaystyle\|U_{\varepsilon}\|_{W^{1,r}(0,T;W^{1,r}_{0}(\Omega_{\varepsilon}))}\leq
C\varepsilon^{\frac{2}{r}},\quad\|U_{\varepsilon}\|_{W^{1,r}(0,T;L^{r}(\Omega_{\varepsilon}))}\leq
C\varepsilon^{\frac{2}{r}+1},\quad\mbox{if}\ r>2,$
$\displaystyle\|G_{\varepsilon}\|_{W^{1,1}(0,T;L^{\frac{3}{2}}(\Omega_{\varepsilon}))}\leq
C\varepsilon^{2},$
$\displaystyle\|H_{\varepsilon}\|_{W^{1,2}(0,T;L^{2}(\Omega_{\varepsilon}))}\leq
C\varepsilon,\quad\mbox{if}\ 1<r\leq 2,$
$\displaystyle\|H_{\varepsilon}\|_{W^{1,\frac{r}{r-1}}(0,T;L^{\frac{r}{r-1}}(\Omega_{\varepsilon}))}\leq
C\varepsilon,\quad\mbox{if}\ r>2.$

###### Proof.

The estimates for $U_{\varepsilon}$ in $\eqref{est-U}_{1}$ and
$\eqref{est-U}_{2}$ follow immediately from its definition and the uniform
estimates of ${\mathbf{u}}_{\varepsilon}$ in (LABEL:2.1.01).

Using Sobolev embedding and Hölder’s inequality gives

$\displaystyle\|{\mathbf{u}}_{\varepsilon}\cdot\nabla{\mathbf{u}}_{\varepsilon}\|_{L^{\frac{3}{2}}(\Omega_{\varepsilon})}\leq\|{\mathbf{u}}_{\varepsilon}\|_{L^{6}(\Omega_{\varepsilon})}\|\nabla{\mathbf{u}}_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}\leq
C\|\nabla{\mathbf{u}}_{\varepsilon}\|_{L^{2}(\Omega_{\varepsilon})}^{2}.$

Thus

$\|{\mathbf{u}}_{\varepsilon}\cdot\nabla{\mathbf{u}}_{\varepsilon}\|_{L^{1}(0,T;L^{\frac{3}{2}}(\Omega_{\varepsilon}))}\leq
C\|\nabla{\mathbf{u}}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}^{2}\leq
C\varepsilon^{2},$

which gives the estimates of $G_{\varepsilon}$ in (LABEL:est-U).

We turn to the estimates of $H_{\varepsilon}$. If $1<r\leq 2$, there holds

$\displaystyle|(1+\lambda|D\mathbf{u}_{\varepsilon}|^{2})^{\frac{r}{2}-1}D\mathbf{u}_{\varepsilon}|\leq|D{\mathbf{u}}_{\varepsilon}|.$

Therefore

$\|H_{\varepsilon}\|_{W^{1,2}(0,T;L^{2}(\Omega_{\varepsilon}))}\leq
C\|(1+\lambda|D\mathbf{u}_{\varepsilon}|^{2})^{\frac{r}{2}-1}D\mathbf{u}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\leq
C\|D{\mathbf{u}}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon.$

If $r>2$, we have

$\displaystyle|(1+\lambda|D\mathbf{u}_{\varepsilon}|^{2})^{\frac{r}{2}-1}D\mathbf{u}_{\varepsilon}|\leq
C|D{\mathbf{u}}_{\varepsilon}|+C|D{\mathbf{u}}_{\varepsilon}|^{r-1}.$

Therefore

$\displaystyle\|H_{\varepsilon}\|_{W^{1,\frac{r}{r-1}}(0,T;L^{\frac{r}{r-1}}(\Omega_{\varepsilon}))}$
$\displaystyle\leq
C\|(1+\lambda|D\mathbf{u}_{\varepsilon}|^{2})^{\frac{r}{2}-1}D\mathbf{u}_{\varepsilon}\|_{L^{\frac{r}{r-1}}((0,T)\times\Omega_{\varepsilon})}$
$\displaystyle\leq
C\|D{\mathbf{u}}_{\varepsilon}\|_{L^{\frac{r}{r-1}}((0,T)\times\Omega_{\varepsilon})}+C\|D{\mathbf{u}}_{\varepsilon}\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}^{r-1}$
$\displaystyle\leq
C\|D{\mathbf{u}}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}+C\|D{\mathbf{u}}_{\varepsilon}\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}^{r-1}$
$\displaystyle\leq C\varepsilon+C\varepsilon^{\frac{2(r-1)}{r}}\leq
C\varepsilon,$

where we use the fact $\frac{r}{r-1}<2$ and $\frac{2(r-1)}{r}>1$ if $r>2$.

∎

###  2.2 Estimates of pressure: extension

Next we will define the extension of pressure and derive corresponding
estimates based on the uniform estimates in (LABEL:est-U).

######  Proposition 2.3.

Let $\tilde{P_{\varepsilon}}$ be the extension of $P_{\varepsilon}$ defined by
the restriction operator as follows

$\langle\nabla\tilde{P_{\varepsilon}},\varphi\rangle_{(0,T)\times\Omega}=\langle\nabla
P_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}},\quad\forall\varphi\in
C_{c}^{\infty}((0,T)\times\Omega),$ (2.5)

where $P_{\varepsilon}$ is defined in (1.10) and the restriction operator
$R_{\varepsilon}$ is given in (1.6) satisfying the estimates (1.7). Then we
have the following estimates:

$\big{|}\langle\nabla\tilde{P_{\varepsilon}},\varphi\rangle_{(0,T)\times\Omega}\big{|}\leq
C\begin{cases}\|\varphi\|_{L^{2}((0,T)\times\Omega)}+\varepsilon\|\nabla\varphi\|_{L^{2}((0,T)\times\Omega)}\quad
1<r\leq 2,\\\
\|\varphi\|_{L^{r}((0,T)\times\Omega)}+\varepsilon\|\nabla\varphi\|_{L^{r}((0,T)\times\Omega)}\quad
r>2.\end{cases}$ (2.6)

###### Proof.

By $(\ref{1.11.1})$, we have

$\displaystyle\langle\nabla\tilde{P_{\varepsilon}},\varphi\rangle_{(0,T)\times\Omega}$
$\displaystyle=\langle\nabla
P_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}}$
$\displaystyle=\langle
F-\varepsilon^{2}(\mathbf{u}_{\varepsilon}-\mathbf{u}_{0})+\frac{\eta_{\infty}}{2}\Delta
U_{\varepsilon}-G_{\varepsilon}+(\eta_{0}-\eta_{\infty}){\rm
div}\,H_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}}.$

Using the fact $F$ and ${\mathbf{u}}_{0}$ are both in
$L^{2}((0,T)\times\Omega;\mathbb{R}^{3})$ and $\eqref{2.1.01}_{1}$ implies

$\displaystyle\big{|}\langle
F-\varepsilon^{2}(\mathbf{u}_{\varepsilon}-\mathbf{u}_{0}),R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}}\big{|}\leq
C\|R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}.$

By the estimates in (LABEL:est-U) and Sobolev embedding inequality, we have:

$\displaystyle\big{|}\langle\Delta
U_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}}\big{|}$
$\displaystyle\leq\|\nabla
U_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\leq
C\varepsilon\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})},$
$\displaystyle\big{|}\langle
G_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}}\big{|}$
$\displaystyle\leq\|G_{\varepsilon}\|_{L^{2}(0,T;L^{\frac{3}{2}}(\Omega_{\varepsilon}))}\|R_{\varepsilon}(\varphi)\|_{L^{2}(0,T;L^{3}(\Omega_{\varepsilon}))}\leq
C\varepsilon^{2}\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})},$
$\displaystyle\big{|}\langle{\rm
div\,}H_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}}\big{|}$
$\displaystyle\leq\|H_{\varepsilon}\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}$
$\displaystyle\leq C\varepsilon\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\quad\mbox{if}\
1<r\leq 2,$ $\displaystyle\big{|}\langle{\rm
div\,}H_{\varepsilon},R_{\varepsilon}(\varphi)\rangle_{(0,T)\times\Omega_{\varepsilon}}\big{|}$
$\displaystyle\leq\|H_{\varepsilon}\|_{L^{\frac{r}{r-1}}((0,T)\times\Omega_{\varepsilon})}\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}$
$\displaystyle\leq C\varepsilon\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}\quad\mbox{if}\
r>2.$

Hence, by the estimates of restriction operator in (1.7), we have for $1<r\leq
2$,

$\displaystyle\big{|}\langle\nabla\tilde{P_{\varepsilon}},\varphi\rangle_{(0,T)\times\Omega}\big{|}$
$\displaystyle\leq
C\big{(}\|R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}+\varepsilon\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{2}((0,T)\times\Omega_{\varepsilon})}\big{)}$
$\displaystyle\leq
C\big{(}\|\varphi\|_{L^{2}((0,T)\times\Omega)}+\varepsilon\|\nabla\varphi\|_{L^{2}((0,T)\times\Omega)}\big{)}.$

For $r>2$,

$\displaystyle\big{|}\langle\nabla\tilde{P_{\varepsilon}},\varphi\rangle_{(0,T)\times\Omega}\big{|}$
$\displaystyle\leq
C\big{(}\|R_{\varepsilon}(\varphi)\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}+\varepsilon\|\nabla
R_{\varepsilon}(\varphi)\|_{L^{r}((0,T)\times\Omega_{\varepsilon})}\big{)}$
$\displaystyle\leq
C\big{(}\|\varphi\|_{L^{r}((0,T)\times\Omega)}+\varepsilon\|\nabla\varphi\|_{L^{r}((0,T)\times\Omega)}\big{)}.$

The proof is thus completed. ∎

##  3 Homogenization process

This section is devoted to the limit passage and derive the limit equations.
We first introduce the cell problem which is used to modify test functions.
Then by the estimates in Proposition 2.1 and Corollary 2.2, we can pass to the
limit term by term to get the limit system.

###  3.1 Cell Problem

To obtain the limit system, a natural way is to pass $\varepsilon\rightarrow
0$ in the weak formulation of (1.1), a proper surgery on
$C_{c}^{\infty}(\Omega)$ test functions needs to be done such that the test
functions vanish on the holes and then become good test functions for the
original equations in $\Omega_{\varepsilon}$. To this issue, Tartar [24]
considered the Stokes equations where the size of the holes is proportional to
the mutual distance of the holes. Then near each single hole in $\varepsilon
Q_{k}$ in the perforated domain $\Omega_{\varepsilon}$, after a scaling of
size $\varepsilon^{-1}$, there arises typically the following problem, named
cell problem:

Let $(w^{i},\pi^{i})(i=1,2,3)$ be a $Q_{0}$-periodic solution of the following
cell problem

$\begin{cases}-\Delta w^{i}+\nabla\pi^{i}=e^{i}&{\rm in}\ Q_{0}\setminus T,\\\
{\rm div}\,w^{i}=0&{\rm in}\ Q_{0}\setminus T,\\\ w^{i}=0&{\rm on}\
T.\end{cases}$ (3.1)

Here $\\{e^{i}\\}_{i=1,2,3}$ is the standard Euclidean coordinate of
$\mathbb{R}^{3}$. The cell problem (3.1) admits a unique weak solution
$(w^{i},\pi^{i})\in W^{1,2}(Q_{0}\setminus T;\mathbb{R}^{3})\times
L_{0}^{2}(Q_{0}\setminus T)$ with $(w^{i},\pi^{i})$ $Q_{0}-{\rm periodic}$.
Moreover, under the assumption $T$ is of class $C^{2,\mu}$, one has

$\displaystyle\|w^{i}\|_{W^{1,\infty}(Q_{0}\setminus
T)}+\|\pi^{i}\|_{L^{\infty}(Q_{0}\setminus T)}\leq C.$ (3.2)

The permeability tensor $A$ is defined as

$A_{i,j}=\int_{Q_{0}}w^{i}_{j}(y)\,{\rm d}y,\qquad A=(A_{i,j})_{1\leq i,j\leq
3},$ (3.3)

where $w^{i}_{j}$ denotes the $j$-th component of vector $w^{i}$. It is shown
in [24] that $A$ is symmetric and positive definite.

Next we set

$w^{i,\varepsilon}(x)=w^{i}(\frac{x}{\varepsilon}),\qquad\pi^{i,\varepsilon}(x)=\pi^{i}(\frac{x}{\varepsilon}).$

Then $(w^{i,\varepsilon}(x),\pi^{i,\varepsilon}(x))$ satisfies the following
equations

$\begin{cases}-\varepsilon\nabla\pi^{i,\varepsilon}+\varepsilon^{2}\Delta
w^{i,\varepsilon}+e^{i}=0&{\rm in}\;\varepsilon Q_{0}\setminus\varepsilon
T,\\\ {\rm div}\;w^{i,\varepsilon}=0&{\rm in}\;\varepsilon
Q_{0}\setminus\varepsilon T,\\\ w^{i,\varepsilon}=0&{\rm on}\;\varepsilon
T,\\\ (w^{i,\varepsilon},\pi^{i,\varepsilon})\ {\rm is}\ \varepsilon
Q_{0}-{\rm periodic}.\end{cases}$ (3.4)

Moreover, it follows from (3.2) that

$\|w^{i,\varepsilon}\|_{L^{\infty}(\Omega_{\varepsilon})}\leq C,\quad\|\nabla
w^{i,\varepsilon}\|_{L^{\infty}(\Omega_{\varepsilon})}\leq
C\varepsilon^{-1},\quad\|\pi^{i,\varepsilon}\|_{L^{\infty}(\Omega_{\varepsilon})}\leq
C.$ (3.5)

By the fact that $w^{i}$ is $Q_{0}-{\rm periodic}$, using (3.5) implies

$w^{i,\varepsilon}\rightarrow\bar{w}^{i}:=\int_{Q_{0}}w^{i}(y)\,{\rm
d}y\quad{\rm weakly\ in\ }L^{r}(\Omega),$ (3.6)

for each $1<r<\infty$.

###  3.2 Passing to the limit

Our main theorem actually follows from the following key proposition:

######  Proposition 3.1.

Let $(U_{\varepsilon},P_{\varepsilon})$ be the solutions of the equation
(1.10). Let $\tilde{P}_{\varepsilon}$ be the extension of $P_{\varepsilon}$
defined in (2.5) and $\tilde{U}_{\varepsilon}$ be the zero extension of
$U_{\varepsilon}$. Then we can find $U\in L^{2}((0,T)\times\Omega)$ and

$P\in\begin{cases}L^{2}((0,T)\times\Omega)&1<r\leq 2,\\\
L^{\frac{r}{r-1}}((0,T)\times\Omega)&r>2,\end{cases}$ (3.7)

such that

$\varepsilon^{-2}\tilde{U}_{\varepsilon}\rightarrow U\ weakly\ in\
L^{2}((0,T)\times\Omega),$ (3.8) $\tilde{P_{\varepsilon}}\rightarrow P\
weakly\ in\begin{cases}L^{2}((0,T)\times\Omega)&1<r\leq 2,\\\
L^{\frac{r}{r-1}}((0,T)\times\Omega)&r>2.\end{cases}$ (3.9)

Moreover, the limit $(U,P)$ satisfies the Darcy’s law:

$\displaystyle\frac{1}{2}\eta_{0}U=A(F-\nabla P)\quad{\rm
in}\,\mathcal{D^{\prime}}((0,T)\times\Omega).$ (3.10)

Here the permeability tensor $A$ is a constant positive definite matrix
determined in (3.3).

###### Proof.

The convergence in (3.8) follows directly from uniform estimates in
$\eqref{est-U}_{1}$.

From (2.6), we can obtain for $1<r\leq 2$,

$\|\tilde{P_{\varepsilon}}\|_{L^{2}((0,T)\times\Omega)}\leq
C\|\nabla\tilde{P_{\varepsilon}}\|_{L^{2}(0,T;W^{-1,2}(\Omega))}\leq C,$

and for $r>2$,

$\|\tilde{P_{\varepsilon}}\|_{L^{\frac{r}{r-1}}((0,T)\times\Omega)}\leq
C\|\nabla\tilde{P_{\varepsilon}}\|_{L^{\frac{r}{r-1}}(0,T;W^{-1,{\frac{r}{r-1}}}(\Omega))}\leq
C.$

Thus we can find

$P\in\begin{cases}L^{2}((0,T)\times\Omega)&1<r\leq 2,\\\
L^{\frac{r}{r-1}}((0,T)\times\Omega)&r>2,\end{cases}$

such that (3.9) holds.

Next we will use the cell problem to construct test functions. Clearly
$w^{i,\varepsilon}$ defined in (3.4) vanishes on the holes in
$\Omega_{\varepsilon}$. Given any scalar function $\phi\in
C_{c}^{\infty}((0,T)\times\Omega)$, taking $\phi w^{i,\varepsilon}$ as a test
function to (1.10) implies

$\displaystyle\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}\,{\rm
div}\,(\phi w^{i,\varepsilon})\,{\rm d}x{\rm
d}t-\int_{0}^{T}\int_{\Omega}\big{(}\frac{\eta_{\infty}}{2}\nabla\tilde{U}_{\varepsilon}+(\eta_{0}-\eta_{\infty})H_{\varepsilon}\big{)}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\big{(}-F+\varepsilon^{2}(\tilde{\mathbf{u}}_{\varepsilon}-\mathbf{u}_{0})+G_{\varepsilon}\big{)}\cdot\phi
w^{i,\varepsilon}\,{\rm d}x{\rm d}t.$ (3.11)

Then we will pass $\varepsilon\rightarrow 0$ term by term where the limits are
taken up to subsequences. It follows from (3.6) that

$\lim_{\varepsilon\rightarrow 0}\int_{0}^{T}\int_{\Omega}{F}\cdot
w^{i,\varepsilon}\phi\,{\rm d}x{\rm
d}t=\int_{0}^{T}\int_{\Omega}{F}\cdot\bar{w}^{i}\phi\,{\rm d}x{\rm d}t.$
(3.12)

The estimates of ${\mathbf{u}}_{\varepsilon}$ in (LABEL:2.1.01) ensure

$\big{|}\int_{0}^{T}\int_{\Omega}\varepsilon^{2}(\tilde{\mathbf{u}}_{\varepsilon}-\mathbf{u}_{0})\cdot\phi
w^{i,\varepsilon}\,{\rm d}x{\rm
d}t\big{|}\leq\varepsilon^{2}\|\tilde{\mathbf{u}}_{\varepsilon}-\mathbf{u}_{0}\|_{L^{2}((0,T)\times\Omega))}\|\phi
w^{i,\varepsilon}\|_{L^{2}((0,T)\times\Omega)}\leq C\varepsilon^{2}\to 0.$
(3.13)

For the term related to the pressure, using the divergence free condition
${\rm div}\,w^{i,\varepsilon}=0$ implies

$\displaystyle\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}\,{\rm
div}\,(\phi w^{i,\varepsilon})\,{\rm d}x{\rm
d}t=\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}w^{i,\varepsilon}\cdot\nabla\phi\,{\rm
d}x{\rm d}t$ (3.14)
$\displaystyle=\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\,{\rm
d}x{\rm
d}t+\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}\bar{w}^{i}\cdot\nabla\phi\,{\rm
d}x{\rm d}t.$

By the fact that

$\tilde{P_{\varepsilon}}\rightarrow P\ \rm weakly\
in\begin{cases}L^{2}((0,T)\times\Omega)&1<r\leq 2,\\\
L^{\frac{r}{r-1}}((0,T)\times\Omega)&r>2,\end{cases}$

we have

$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}\tilde{P}_{\varepsilon}\bar{w}^{i}\cdot\nabla\phi\,{\rm
d}x{\rm d}t=\int_{0}^{T}\int_{\Omega}P\bar{w}^{i}\cdot\nabla\phi\,{\rm d}x{\rm
d}t=\int_{0}^{T}\int_{\Omega}P\,{\rm div}\,(\bar{w}^{i}\phi)\,{\rm d}x{\rm
d}t.$ (3.15)

For each fixed $t\in(0,T)$, by the divergence free condition ${\rm
div}\,(w^{i,\varepsilon}-\bar{w}^{i})=0$, we have
$(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\in L_{0}^{r+2}(\Omega)$. By
employing the classical Bogovskii operator $\mathcal{B}$ in domain $\Omega$,
we can find
$\psi_{\varepsilon}=\mathcal{B}\big{(}(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\big{)}$
such that

${\rm
div}\,\psi_{\varepsilon}=(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi,\quad\mbox{for
each $t\in(0,T)$.}$ (3.16)

Moreover, we have the following estimate:

$\|\psi_{\varepsilon}\|_{L^{\infty}(0,T;W_{0}^{1,r+2}(\Omega))}\leq
C\|(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\|_{L^{\infty}(0,T;L^{r+2}(\Omega))}\leq
C.$ (3.17)

By the fact that
$\partial_{t}\psi_{\varepsilon}=\partial_{t}\mathcal{B}\big{(}(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\big{)}=\mathcal{B}\big{(}(w^{i,\varepsilon}-\bar{w}^{i})\cdot\partial_{t}\nabla\phi\big{)}$
we obtain

$\|\partial_{t}\psi_{\varepsilon}\|_{L^{\infty}(0,T;W_{0}^{1,r+2}(\Omega))}\leq
C\|(w^{i,\varepsilon}-\bar{w}^{i})\cdot\partial_{t}\nabla\phi\|_{L^{\infty}(0,T;L^{r+2}(\Omega))}\leq
C.$

By compact Sobolev embedding, we have, up to a subsequence that

$\psi_{\varepsilon}\rightarrow\psi\quad{\rm strongly\ in\
}L^{r+2}((0,T)\times\Omega)$ (3.18)

for some $\psi\in W^{1,r+2}(0,T;W_{0}^{1,r+2}(\Omega))$. Recall that
$w^{i,\varepsilon}\to\bar{w}^{i}$ weakly in $L^{r+2}((0,T)\times\Omega)$. Then

$(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\rightarrow 0\ {\rm weakly\
in}\ L^{r+2}((0,T)\times\Omega).$ (3.19)

By (3.16), (3.18) and (3.19), we can deduce that ${\rm div}\,\psi=0$. Then
using (2.6) implies

$\displaystyle\big{|}\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\,{\rm
d}x{\rm d}t\big{|}$
$\displaystyle=\big{|}\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}\,{\rm
div}\,\psi_{\varepsilon}\,{\rm d}x{\rm d}t\big{|}$
$\displaystyle=\big{|}\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}\,{\rm
div}\,(\psi_{\varepsilon}-\psi)\,{\rm d}x{\rm d}t\big{|}$
$\displaystyle=\big{|}\langle\nabla\tilde{P_{\varepsilon}},\psi_{\varepsilon}-\psi\rangle_{(0,T)\times\Omega}\big{|}$
$\displaystyle\leq
C\big{(}\|\psi_{\varepsilon}-\psi\|_{L^{r+2}((0,T)\times\Omega)}+\varepsilon\|\nabla(\psi_{\varepsilon}-\psi)\|_{L^{r+2}((0,T)\times\Omega)}\big{)}.$

Then, together with (3.17) and (3.18) we finally deduce

$\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}(w^{i,\varepsilon}-\bar{w}^{i})\cdot\nabla\phi\,{\rm
d}x{\rm d}t\to 0,$ (3.20)

and consequently, by (3.15) and (3.20), passing $\varepsilon\to 0$ in
$\eqref{limit-P-1}$ gives

$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}\tilde{P_{\varepsilon}}\,{\rm div}\,(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t=\int_{0}^{T}\int_{\Omega}P\,{\rm
div}\,(\phi\bar{w}^{i})\,{\rm d}x{\rm d}t.$ (3.21)

By $\eqref{est-U}_{3}$, we have

$\big{|}\int_{0}^{T}\int_{\Omega}G_{\varepsilon}\cdot\phi
w^{i,\varepsilon}\,{\rm d}x{\rm
d}t\big{|}\leq\|G_{\varepsilon}\|_{L^{\frac{3}{2}}((0,T)\times\Omega)}\|\phi\|_{L^{\infty}((0,T)\times\Omega)}\|w^{i,\varepsilon}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{2}\to 0.$ (3.22)

Next we calculate

$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}\nabla\tilde{U}_{\varepsilon}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t$

which equals to

$\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}\nabla\tilde{U}_{\varepsilon}:(\nabla\phi\otimes
w^{i,\varepsilon})\,{\rm d}x{\rm d}t+\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}\nabla\tilde{U}_{\varepsilon}:\nabla
w^{i,\varepsilon}\phi\,{\rm d}x{\rm d}t.$

By $\eqref{est-U}_{1}$ we obtain

$\big{|}\int_{0}^{T}\int_{\Omega}\nabla\tilde{U}_{\varepsilon}:\nabla\phi\otimes
w^{i,\varepsilon}\,{\rm d}x{\rm d}t\big{|}\leq
C\|w^{i,\varepsilon}\|_{L^{2}(\Omega)}\|\nabla\tilde{U}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega)}\leq
C\varepsilon\to 0.$ (3.23)

Taking $\phi\tilde{U}_{\varepsilon}$ as a test function to (3.4) gives

$\int_{0}^{T}\int_{\Omega}\varepsilon\pi^{i,\varepsilon}\nabla\phi\cdot\tilde{U}_{\varepsilon}-\varepsilon^{2}\nabla
w^{i,\varepsilon}:\nabla(\phi\tilde{U}_{\varepsilon})+e^{i}\phi\tilde{U}_{\varepsilon}\,{\rm
d}x{\rm d}t=0.$

It is equivalent to

$\int_{0}^{T}\int_{\Omega}\varepsilon^{-1}\pi^{i,\varepsilon}\nabla\phi\cdot\tilde{U}_{\varepsilon}-\nabla
w^{i,\varepsilon}:\nabla\tilde{U}_{\varepsilon}\phi-\nabla
w^{i,\varepsilon}:(\nabla\phi\otimes\tilde{U}_{\varepsilon})+\varepsilon^{-2}\phi\tilde{U}_{\varepsilon}\cdot
e^{i}\,{\rm d}x{\rm d}t=0.$ (3.24)

By $\eqref{est-U}_{1}$, (3.5), we have

$\displaystyle\big{|}\int_{0}^{T}\int_{\Omega}\varepsilon^{-1}\pi^{i,\varepsilon}\nabla\phi\cdot\tilde{U}_{\varepsilon}\,{\rm
d}x{\rm
d}t\big{|}\leq\varepsilon^{-1}\|\tilde{U}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega)}\|\pi^{i,\varepsilon}\|_{L^{2}(\Omega)}\|\nabla\phi\|_{L^{\infty}((0,T)\times\Omega)}\leq
C\varepsilon\to 0,$ $\displaystyle\big{|}\int_{0}^{T}\int_{\Omega}\nabla
w^{i,\varepsilon}:(\nabla\phi\otimes\tilde{U}_{\varepsilon})\,{\rm d}x{\rm
d}t\big{|}\leq\|\nabla
w^{i,\varepsilon}\|_{L^{2}(\Omega)}\|\tilde{U}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega)}\|\nabla\phi\|_{L^{\infty}((0,T)\times\Omega)}\leq
C\varepsilon\to 0.$

Then, passing $\varepsilon\rightarrow 0$ in (3.24) gives

$\displaystyle\lim_{\varepsilon\rightarrow 0}\int_{0}^{T}\int_{\Omega}\nabla
w^{i,\varepsilon}:\nabla\tilde{U}_{\varepsilon}\phi\,{\rm d}x{\rm
d}t=\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}\varepsilon^{-2}\phi\tilde{U}_{\varepsilon}\cdot
e^{i}\,{\rm d}x{\rm d}t=\int_{0}^{T}\int_{\Omega}\phi U_{i}\,{\rm d}x{\rm
d}t.$ (3.25)

Thus by (3.23) and (3.25) we have

$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}\nabla{\tilde{U}_{\varepsilon}}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t=\int_{0}^{T}\int_{\Omega}\phi U_{i}\,{\rm
d}x{\rm d}t.$ (3.26)

For the last term related to the nonlinear stress tensor $H_{\varepsilon}$,
due to the smallness of $D\tilde{\mathbf{u}}_{\varepsilon}$, we shall show its
contribution in the limit is nothing but a Newtonian stress tensor. Introduce
the decomposition

$\displaystyle H_{\varepsilon}$
$\displaystyle=\int_{0}^{t}(1+\lambda|D{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}D{\mathbf{u}}_{\varepsilon}\,{\rm
d}s=\int_{0}^{t}\big{(}(1+\lambda|D{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D{\mathbf{u}}_{\varepsilon}\,{\rm
d}s+\int_{0}^{t}D{\mathbf{u}}_{\varepsilon}\,{\rm d}s$
$\displaystyle=\int_{0}^{t}\big{(}(1+\lambda|D{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D{\mathbf{u}}_{\varepsilon}\,{\rm
d}s+D{U}_{\varepsilon}.$

Then

$\displaystyle\int_{0}^{T}\int_{\Omega}H_{\varepsilon}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t$ (3.27)
$\displaystyle=\int_{0}^{T}\int_{\Omega}D\tilde{U}_{\varepsilon}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm
d}t+\int_{0}^{T}\int_{\Omega}\left(\int_{0}^{t}\big{(}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D\tilde{\mathbf{u}}_{\varepsilon}{\rm
d}s\right):\nabla(\phi w^{i,\varepsilon})\,{\rm d}x{\rm d}t.$

Using the divergence free condition ${\rm div}\,\tilde{U}_{\varepsilon}=0$ and
(3.26) implies

$\lim_{\varepsilon\to
0}\int_{0}^{T}\int_{\Omega}D\tilde{U}_{\varepsilon}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t=\lim_{\varepsilon\to
0}\frac{1}{2}\int_{0}^{T}\int_{\Omega}\nabla\tilde{U}_{\varepsilon}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t=\frac{1}{2}\int_{0}^{T}\int_{\Omega}\phi
U_{i}\,{\rm d}x{\rm d}t.$ (3.28)

For the other term on the right-hand side of (LABEL:H_dec), based on different
values of $r$, we use different inequalities to show that its limit is
actually zero.

For $1<r<2$, by inequality $0\leq(1+s)^{\alpha}-s^{\alpha}\leq
1\,(0\leq\alpha\leq 1,\ s\geq 0)$, we have

$\big{|}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{|}=\big{|}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}\big{(}1-(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{1-\frac{r}{2}}\big{)}\big{|}\leq
C|D\tilde{\mathbf{u}}_{\varepsilon}|^{2-r}.$

Then, for $1<r<2$, there holds

$\begin{split}&\big{|}\int_{0}^{T}\int_{\Omega}\left(\int_{0}^{t}\big{(}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D\tilde{\mathbf{u}}_{\varepsilon}\,{\rm
d}s\right):\nabla w^{i,\varepsilon}\phi\,{\rm d}x{\rm d}t\,\big{|}\\\ &\leq
C\|\nabla w^{i,\varepsilon}\|_{L^{\infty}(\Omega)}\int_{0}^{T}{\rm
d}t\int_{0}^{t}{\rm
d}s\int_{\Omega}\mid(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\mid|D\tilde{\mathbf{u}}_{\varepsilon}|\,{\rm
d}x\\\ &\leq C\varepsilon^{-1}\int_{0}^{T}{\rm d}t\int_{0}^{t}{\rm
d}s\int_{\Omega}|D\tilde{\mathbf{u}}_{\varepsilon}|^{3-r}{\rm d}x\\\ &\leq
C\varepsilon^{-1}\|D\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{3-r}((0,T)\times\Omega)}^{3-r}\leq
C\varepsilon^{-1}\|D\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega)}^{3-r}\leq
C\varepsilon^{2-r}\to 0.\end{split}$

For $2<r\leq 4$, again by inequality $0\leq(1+s)^{\alpha}-s^{\alpha}\leq
1\,(0\leq\alpha\leq 1,\ s\geq 0)$, we have

$\big{|}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{|}\leq
C|D\tilde{\mathbf{u}}_{\varepsilon}|^{r-2}.$

Then using the estimate
$\|\nabla\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{r}((0,T)\times\Omega)}\leq
C\varepsilon^{\frac{2}{r}}$ in $\eqref{2.1.01}_{2}$ gives

$\displaystyle\big{|}\int_{0}^{T}\int_{\Omega}\left(\int_{0}^{t}\big{(}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D\tilde{\mathbf{u}}_{\varepsilon}\,{\rm
d}s\right):\nabla w^{i,\varepsilon}\phi\,{\rm d}x{\rm d}t\,\big{|}$
$\displaystyle\leq C\|\nabla
w^{i,\varepsilon}\|_{L^{\infty}(\Omega)}\int_{0}^{T}{\rm d}t\int_{0}^{t}{\rm
d}s\int_{\Omega}\mid(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\mid|D\tilde{\mathbf{u}}_{\varepsilon}|\,{\rm
d}x$ $\displaystyle\leq C\varepsilon^{-1}\int_{0}^{T}{\rm d}t\int_{0}^{t}{\rm
d}s\int_{\Omega}|D\tilde{\mathbf{u}}_{\varepsilon}|^{r-1}{\rm d}x$
$\displaystyle\leq
C\varepsilon^{-1}\|D\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{r-1}((0,T)\times\Omega)}^{r-1}\leq
C\varepsilon^{-1}\varepsilon^{\frac{2(r-1)}{r}}=C\varepsilon^{\frac{r-2}{r}}\to
0.$

For $r\geq 4$,

$\displaystyle\big{|}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{|}\leq\lambda(\frac{r}{2}-1)(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-2}|D\tilde{\mathbf{u}}_{\varepsilon}|^{2}\leq
C(|D\tilde{\mathbf{u}}_{\varepsilon}|^{2}+|D\tilde{\mathbf{u}}_{\varepsilon}|^{r-2}).$

Using the estimate
$\|\nabla\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{2}((0,T)\times\Omega)}^{2}+\|\nabla\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{r}((0,T)\times\Omega)}^{r}\leq
C\varepsilon^{2}$ and Hölder’s inequality implies

$\displaystyle\|\nabla\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{q}((0,T)\times\Omega)}^{q}\leq
C\varepsilon^{2},\quad\forall\,q\in[2,r].$

Thus,

$\displaystyle\big{|}\int_{0}^{T}\int_{\Omega}\left(\int_{0}^{t}\big{(}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D\tilde{\mathbf{u}}_{\varepsilon}\,{\rm
d}s\right):\nabla w^{i,\varepsilon}\phi\,{\rm d}x{\rm d}t\,\big{|}$
$\displaystyle\leq C\|\nabla
w^{i,\varepsilon}\|_{L^{\infty}(\Omega)}\int_{0}^{T}{\rm d}t\int_{0}^{t}{\rm
d}s\int_{\Omega}\mid(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\mid|D\tilde{\mathbf{u}}_{\varepsilon}|\,{\rm
d}x$ $\displaystyle\leq C\varepsilon^{-1}\int_{0}^{T}{\rm d}t\int_{0}^{t}{\rm
d}s\int_{\Omega}\big{(}|D\tilde{\mathbf{u}}_{\varepsilon}|^{r-1}+|D\tilde{\mathbf{u}}_{\varepsilon}|^{3}\big{)}{\rm
d}x$ $\displaystyle\leq
C\varepsilon^{-1}(\|D\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{r-1}((0,T)\times\Omega)}^{r-1}+\|D\tilde{\mathbf{u}}_{\varepsilon}\|_{L^{3}((0,T)\times\Omega)}^{3})\leq
C\varepsilon\to 0.$

To sum up, for $1<r<\infty$,

$\int_{0}^{T}\int_{\Omega}\left(\int_{0}^{t}\big{(}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D\tilde{\mathbf{u}}_{\varepsilon}\,{\rm
d}s\right):\nabla w^{i,\varepsilon}\phi\,{\rm d}x{\rm d}t\to 0.$ (3.29)

We can use same arguments to show that

$\left|\int_{0}^{T}\int_{\Omega}\left(\int_{0}^{t}\big{(}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D\tilde{\mathbf{u}}_{\varepsilon}\,{\rm
d}s\right):(w^{i,\varepsilon}\otimes\nabla\phi)\,{\rm d}x{\rm d}t\right|\leq
C\varepsilon\to 0.$ (3.30)

Thus by (3.29) and (3.30), we have

$\int_{0}^{T}\int_{\Omega}\left(\int_{0}^{t}\big{(}(1+\lambda|D\tilde{\mathbf{u}}_{\varepsilon}|^{2})^{\frac{r}{2}-1}-1\big{)}D\tilde{\mathbf{u}}_{\varepsilon}{\rm
d}s\right):\nabla(\phi w^{i,\varepsilon})\,{\rm d}x{\rm d}t\to 0.$ (3.31)

Then using (3.28), (3.31) and passing $\varepsilon\rightarrow 0$ in
(LABEL:H_dec) implies

$\displaystyle\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}H_{\varepsilon}:\nabla(\phi
w^{i,\varepsilon})\,{\rm d}x{\rm d}t=\frac{1}{2}\int_{0}^{T}\int_{\Omega}\phi
U_{i}\,{\rm d}x{\rm d}t.$ (3.32)

Finally using (3.12), (3.13), (3.21), (3.22), (3.26), (3.32) and passing
$\varepsilon\rightarrow 0$ in (3.1) gives

$\int_{0}^{T}\int_{\Omega}\frac{1}{2}\eta_{0}\phi U_{i}\,{\rm d}x{\rm
d}t=\int_{0}^{T}\int_{\Omega}F\cdot\phi\bar{w}^{i}\,{\rm d}x{\rm
d}t+\int_{0}^{T}\int_{\Omega}P\,{\rm div}\,(\phi\bar{w}^{i})\,{\rm d}x{\rm
d}t.$

This is nothing but the Darcy’s law (3.10) in the sense of distribution, with
permeability tensor $A=(\bar{w}_{j}^{i})$ determined in (3.3). Thus, we
completed the proof of Proposition 3.1. ∎

###  3.3 End of the proof

From Proposition 3.1, we can deduce the limit equation in ${\mathbf{u}}$. Set
$\tilde{p}_{\varepsilon}=\partial_{t}\tilde{P}_{\varepsilon}$ and
$p=\partial_{t}P$ in the weak sense. By (3.7) and (3.9) we have

$p\in\begin{cases}W^{-1,2}(0,T;L^{2}(\Omega))&1<r\leq 2,\\\
W^{-1,\frac{r}{r-1}}(0,T;L^{\frac{r}{r-1}}(\Omega))&r>2,\end{cases}$

and

$\tilde{p}_{\varepsilon}=\partial_{t}\tilde{P}_{\varepsilon}\rightarrow p\
{\rm weakly\ in}\begin{cases}W^{-1,2}(0,T;L^{2}(\Omega))&1<r\leq 2,\\\
W^{-1,\frac{r}{r-1}}(0,T;L^{\frac{r}{r-1}}(\Omega))&r>2.\end{cases}$

Thus, differentiating (3.10) with respect to time variable in the distribution
sense gives the limit equation (1.11). The proof of Theorem 1.5 is completed.

## Acknowledgements

Both authors are partially supported by the NSF of China under Grant 12171235.

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