# A poroelastic plate model obtained by simultaneous homogenization and
dimension reduction

Marin Bužančić 111Faculty of Chemical Engineering and Technology, University
of Zagreb, Marulićev trg 19, 10000 Zagreb, Croatia. Email<EMAIL_ADDRESS>Pedro Hernández-Llanos222Instituto de Ciencias de la Ingeniería, Universidad
de O’Higgins, Avenida Libertador Bernardo O’Higgins 611, 2841935 Rancagua,
Chile. E-mail<EMAIL_ADDRESS>Igor Velčić333Faculty of Electrical
Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb,
Croatia. E-mail<EMAIL_ADDRESS>Josip Žubrinić444Faculty of Electrical
Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb,
Croatia. E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, the starting point of our analysis is coupled system of
elasticity and Stokes equation. We consider two small parameters: the
thickness $h$ of the thin plate and the pore scale $\varepsilon(h)$ which
depend on $h$. We will focus specifically on the case when the pore size is
small relative to the thickness of the plate. The main goal here is derive a
model for a poroelastic plate from the $3D$ problem as $h$ goes to zero using
simultaneous homogenization and dimension reduction techniques. The obtained
model generalizes the poroelastic plate model derived by A. Mikelić et. al. in
2015 by dimension reduction techniques from $3D$ Biot’s equations in the sense
that it also covers the case of contact of poroelastic and elastic plate as
well as the evolution equation with inertia term.

Key words Homogenization; dimension reduction; poroelastic plate; two-scale
convergence

2020 AMS Subject Classification: 35B27; 74F10; 74K20; 74Q15; 76M50

Dedicated to the memory of Andro Mikelić.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
  2. 2 Setting of the problem
    1. 2.1 Geometry and material assumptions
    2. 2.2 The microscopic equations
    3. 2.3  A priori estimates
      1. 2.3.1 A priori estimates for displacement
      2. 2.3.2 A priori estimate for pressure
  3. 3 Quasi-static case
    1. 3.1 Compactness result
    2. 3.2 Effective tensors
    3. 3.3 Limit equations
    4. 3.4 Analysis of the limit equations
    5. 3.5 Strong convergence
    6. 3.6 Further discussions on the limit model
      1. 3.6.1 Surface loads
      2. 3.6.2 Fluid part intersects the longitudinal boundary $\partial\omega\times\\{-1/2,1/2\\}\\}$
  4. 4 The problem with inertial term
    1. 4.1 Compactness and limit equations
    2. 4.2 Analysis of the limit equations
    3. 4.3 Strong convergence
  5. A Appendix
    1. A.1 Griso’s decomposition and its consequences
    2. A.2 Auxiliary claims on two-scale convergence

##  1 Introduction

Derivation of Biot’s equations by homogenization approach is relatively old
topic (see [8],[9]). Recently there is a growing attention in derivation and
studying poroelastic equations in different context. We will just mention a
few of them. In [14] the authors study a thin layer of elastic body in a fluid
modeled by Stokes equations, in a recent work [15] the author derives the
poroelastic plate model in the regime when the size of the pores is on the
same squale as the thickness of the body, in [17] the authors analyze the
quasistatic and evolution equation for poroelastic plate, in [3] the authors
discuss the existence result for the interaction between fluid and poroelastic
structure, in [5] the authors analyze quasi-static Biot system of
poroelasticity for both compressible and incompressible constituents, in [19]
the authors derive the model of the contact between poroelastic and elastic
body and in [18] the authors derive the model of quasi-static Biot’s plate by
doing dimension reduction of $3d$ Biot’s equations.

The current paper deals with derivation of Biot’s plate equations in the
regime the size of the pores is much less than the thickness of the plate by
doing simultaneous homogenization and dimension reduction. The starting point
are the coupled evolution Stokes equation for incompressible fluid and the
linearized equations of elasticity. We assume that the size of the pores
$\varepsilon(h)$ is much less than the thickness of the body $h$,
$\varepsilon(h)\ll h$. By using some modifications Griso’s decomposition (see
[16], [7])) we do simultaneous homogenization and dimension reduction. Griso’s
decomposition has already been used for obtaining the model of homogenized
plate in the context of linear and non-linear elasticity (see [6] and [7] for
the case of the elastic plate in high-contrast). In the limit (in the quasi-
static state) we obtain the generalization of the model of Biot’s plate
obtained in [18]. Our derivation covers the situations of the contact between
elastic and poroelastic plate, as well as the case of the contact between two
different poroelastic plates. The consequence of this is that the limit
equations are more complex and do not decouple as the equations obtained in
[18] and also analyzed in [17]. Derivation of the model of poroelastic plate
by doing simultaneous homogenization and dimension reduction provides also the
additional information on the limit Darcy’s law and appropriate boundary
conditions, i.e. their connection with the microscopic model (see Section 3.6
below). The difference between quasi-static model obtained in this paper and
the one obtained in [15] is significant. Here (like in [18]) in the limit
equations only appears the derivative of the pressure in the vertical
direction (in the effective Darcy’s law the microscopic fluid velocity is
driven only by the derivative of the pressure in the vertical direction),
while in [15] the macroscopic pressure depends only on (time and) macroscopic
in-plane variables. Consequently, the limit equations are different.

One of the difficulties for the derivation of the model was to obtain the
correct scalings of the constants (i.e. the relation of the elasticity
constants with respect to viscosity) which gives Biot’s plate model in the
limit, since we have to scale them with respect to two small parameters
$\varepsilon$ and $h$.

In the derivation of the model we also cover the case of the plate model with
inertia term and thus justify the model analyzed in [17]. Considering the
models with inertia term, it is well known that full $3d$ macroscopic Biot’s
equations have memory effects (see [9]). In that sense there is a certain
similarity between poroelastic equations and linear elastic equations with
high-contrast inclusions (to obtain Biot’s equations one has to put viscosity
in contrast to elasticity constants). However, the limit plate equations
obtained here doesn’t have memory effect. The reason for that is the fact that
we derive the model in the bending regime which is valid for long times (we
implicitly scale the time also for the derivation of evolution equations with
inertia term). The techniques presented here would allow allow us to obtain
the models in the membrane regime (see [7] for the detailed discussion on
this), when it is expected that the memory effects for the macroscopic
equations are present. These equations are, however, degenerate in the
vertical component of the deformation and thus of less interest.

The derivation of the limit equations in this paper is done under minimal
regularity assumption on the loads. On the microscopic level we assume that
the forces have $H^{1}$ regularity in time which, according to [12],
guarantees the existence of the pressure with $L^{2}$ regularity in time and
space. In the quasi-static case we impose the boundedness in $H^{1}(0,T)$ of
the loads, while in the evolution case the boundedness in $H^{1}(0,T)$ is only
imposed for the in-plane components of the loads, while for the vertical
component we only impose $L^{2}(0,T)$ boundedness (note that the derivation in
[19] and [15] require higher regularity). This is also minimal requirement for
the existence result of the limit equations. This has the consequence that the
pressure converges only weakly in $H^{-1}(0,T)$ and the additional regularity
of the limit pressure is obtained from the limit equations.

For the analysis of the limit equations we use the approach of [17].
Additional effort needs to be done since the equations are more general than
the ones analyzed in [17]. This is spetially valid for the equations with
inertia term. In this case we use the approach via Galerkin approximation to
obtain existence result. However, in order to derive the certain energy type
identity we need to approximate the solution and this is done via the
semigroup approach (which alone doesn’t provide the existence of regular
enough solution). This energy type identity is of independent interest, but
here is used to obtain the strong convergence result.

Next we briefly outline the structure of the paper. In Section 2 we discuss
the setting of the problem and give some a priori estimates for the fluid-
elastic structure interaction. In Section 3 we derive and analyze the quasi-
static Biot’s plate model while in Section 4 we derive and analyze the Biot’s
plate model with inertia term in the bending regime. Appendix gives us some
auxiliary claims useful for dimension reduction (Griso’s decomposition and its
consequences) as well as auxiliary definitions and claims about two-scale
convergence. The main results of the paper are the following: Theorem 3.1
gives us the compactness result for the regime of quasi-static case, Theorem
3.2 provides us the convergence result, i.e. identify the limit model, Theorem
3.4 gives us uniqueness and existence result for the general quasi-static
Biot’s plate model, Theorem 3.5 and Theorem 3.6 provide us the strong
convergence result. Similarly, Theorem 4.1 provide us compactness result as
well as the limit Biot’s equations with inertia term, Theorem 4.2 gives us the
uniqueness result and Theorem 4.3 gives us existence result for Biot’s
equations with inertia term, while Theorem 4.4 gives us strong convergence
result.

###  1.1 Notation

Throughout this paper, we denote by
$\mathbb{N},\mathbb{Z},\mathbb{R},\mathbb{R}^{+},\mathbb{R}^{m\times n}$ and
$\mathbb{R}^{n\times n}_{sym}$ the sets of natural, integer, real, nonnegative
real numbers, the space of real $m\times n$ matrices and the space of real
$n\times n$ symmetric matrices respectively, this two last spaces endowed with
the usual Euclidean norm
$\|\boldsymbol{F}\|=\sqrt{\text{tr}\,\boldsymbol{F}^{T}\boldsymbol{F}}$. If
$\mathbf{A},\mathbf{B}\in\mathbb{R}^{n\times m}$, we denote by
$\mathbf{A}:\mathbf{B}$ the scalar product

$\mathbf{A}:\mathbf{B}=\text{tr}(\mathbf{A}^{T}\mathbf{B}).$

For $\mathbf{A}\in\mathbb{R}^{n\times n}$, $\operatorname{sym}\mathbf{A}$
denotes the symmetric part of $\mathbf{A}$. Set $I=(-\frac{1}{2},\frac{1}{2})$
and the unit $Y=[0,1]^{3}$ and we denote by
$\mathcal{Y}=\mathbb{R}^{3}/\mathbb{Z}^{3}$ the flat torus with quotient
topology. For the open set $A$ and $k\in\mathbb{N}_{0}\cup{\infty}$,
$C^{k}(A)$ denotes the set of $k$-differentiable functions on $A$.
$C_{c}^{k}(A)$ denotes the same set of functions with compact support in $A$.
$C^{k}(\mathcal{Y})$ denotes the set of $k$-differentiable functions on torus,
while $H^{k}(\mathcal{Y})$ denotes the closure of $C^{k}(\mathcal{Y})$ in
$H^{k}$ Sobolev norm. If $\omega\subset\mathbb{R}^{2}$ is a rectangle and
$\Omega^{h}=\omega\times hI$, the set $C^{k}_{\\#}(\Omega^{h})$ denotes the
set of $k$ differentiable functions on $\Omega$ that can be extended to
$C^{k}(\mathbb{R}^{2}\times hI)$ periodic functions in $x_{1},x_{2}$
variables. The set $C^{k}_{\\#}(\omega)$ is the set of $k$\- differentiable
functions that can be extended to $C^{k}(\mathbb{R}^{2})$ periodic functions.
$H^{k}_{\\#}(\Omega^{h})$, i.e. $H^{k}_{\\#}(\omega)$ denotes the closure of
the set $C^{k}_{\\#}(\Omega^{h})$, i.e. $C^{k}_{\\#}(\omega)$ in $H^{k}$ norm.
The sets $\dot{C}^{k}(A)$, denotes the subspace of functions in $C^{k}(A)$
with zero mean. In the analogous way we define $\dot{C}_{c}^{k}(A)$,
$\dot{H}^{k}(A)$, $\dot{H}^{k}_{\\#}(\Omega^{h})$,
$\dot{H}_{\\#}^{k}(\omega)$.

For $f\in L^{1}(A\times I)$ we denote by $\bar{f}\in L^{1}(A)$ the function

$\bar{f}=\int_{I}f\,dx_{3}.$

For $f\in L^{1}(Y)$ we denote by

$\langle f\rangle_{Y}:=\int_{Y}f\,dy.$

These definitions of functional spaces are naturally extended for spaces of
functions taking values in $\mathbb{R}+$. By $\boldsymbol{e}^{i}$ we denote
coordinate vectors for $i=1,2,3$. Unless otherwise stated the greek indices
$\alpha,\beta$ take values in the set $\\{1,2\\}$. For the coordinates
$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$, we denote by
$\hat{x}=(x_{1},x_{2})$. For the vector valued function $\boldsymbol{u}$
taking values in $\mathbb{R}^{3}$ by ${\boldsymbol{u}}_{*}$ we denote the
first two components $\boldsymbol{u}_{*}=(u_{1},u_{2})$. For
$\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$, we denote by
$\mathbf{a}\odot\mathbf{b}$ the symmetric part of tensor product
$\mathbf{a}\otimes\mathbf{b}$. $\iota:\mathbb{R}^{2\times
2}\to\mathbb{R}^{3\times 3}$ denotes the natural inclusion

$\iota(\mathbf{A})_{\alpha\beta}=\mathbf{A}_{\alpha\beta},\quad\iota(\mathbf{A})_{i3}=\iota(\mathbf{A})_{3i}=0,\quad\alpha,\beta\in\\{1,2\\},\
i\in\\{1,2,3\\}.$

For $V,W$ vector spaces $L(V,W)$ is the set of continuous linear operators
from $V$ to $W$. For the operator $\mathcal{A}$ we denote by
$\mathcal{D}(\mathcal{A})$ the domain of the operator $\mathcal{A}$. For $h>0$
we denote by $\pi_{h}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ the mapping

$\pi_{h}(\mathbf{a})=(ha_{1},ha_{2},a_{3}).$

The definition naturally extends to $\mathbb{R}^{3}$ valued functions. For a
mapping $\boldsymbol{u}:\Omega\subset\mathbb{R}^{3}\to\mathbb{R}^{3}$, we
denote by $e(\boldsymbol{u})=\operatorname{sym}\nabla\boldsymbol{u}$. In the
analogous way we define $e_{\hat{x}}\boldsymbol{u}$, $e_{y}(\boldsymbol{u})$
for the mapping $\boldsymbol{u}:\omega\subset\mathbb{R}^{2}\to\mathbb{R}^{2}$,
i.e. $\boldsymbol{u}:Y(\mathcal{Y})\to\mathbb{R}^{3}$. For a mapping
$\boldsymbol{u}:\Omega\subset\mathbb{R}^{3}\to\mathbb{R}^{k}$,
$\Omega\subset\mathbb{R}^{3}$ we denote by $\nabla_{h}$ the scaled gradient:

$\nabla_{h}\boldsymbol{u}:=(\partial_{1}\boldsymbol{u},\partial_{2}\boldsymbol{u},\frac{1}{h}\partial_{3}\boldsymbol{u}).$

For $\boldsymbol{u}:\Omega\subset\mathbb{R}^{3}\to\mathbb{R}^{3}$ we denote by

$\operatorname{div}_{h}\boldsymbol{u}:=\text{tr}\,\nabla_{h}\boldsymbol{u},\quad
e_{h}(\boldsymbol{u}):=\operatorname{sym}\nabla_{h}\boldsymbol{u}.$

For $A\subset\mathbb{R}^{n}$, $\chi_{A}$ denotes the characteristic function.
For $x\in\mathbb{R}.$ and $r>0$, B(x,r) denotes the ball of radius $r$
centered at $x$. In general, $\rightarrow$ denotes the strong convergence,
$\rightharpoonup$ denotes the weak (or weak *) convergence.

##  2 Setting of the problem

In this section we pose the microscopic problem. In Section 2.1 we discuss
about the geometry of the microscopic problem, in Section 2.2 we give the
equations of the microscopic problem and give the existence result, while in
Section 2.3 we obtain some a priori estimates necessary to obtain the limit
equations.

We take $\omega\subset\mathbb{R}^{2}$ a rectangle with vertices with integer
coordinates and take $\varepsilon=\frac{1}{k}$, $k\in\mathbb{N}$,
$h=\frac{1}{kl}$, $l\in\mathbb{N}$. We will write $\varepsilon=\varepsilon(h)$
and assume $\lim_{h\to 0}\frac{\varepsilon(h)}{h}=0$. We define
$\Omega^{h}=\omega\times hI$, $\Omega=\omega\times I$.

###  2.1 Geometry and material assumptions

We assume the existence of finite number of typical cells consisting of fluid
and solid part, i.e. there exists $m>0$ and pairs $(Y_{f}^{i},Y_{s}^{i})$,
i.e. $(\mathcal{Y}_{f}^{i},\mathcal{Y}_{s}^{i})$ (we demand periodicity) for
$i=1,\dots,m$, open sets with Lipschitz boundary such that (see Figure 1)

$Y_{f}^{i}\cap
Y_{s}^{i}=\emptyset,\quad\overline{Y_{f}^{i}}\cup\overline{Y_{s}^{i}}=Y,\quad
i=1,\dots,m,$

i.e.,

$\mathcal{Y}_{f}^{i}\cap\mathcal{Y}_{s}^{i}=\emptyset,\quad\overline{\mathcal{Y}_{f}^{i}}\cup\overline{\mathcal{Y}_{s}^{i}}=\mathcal{Y},\quad
i=1,\dots,m,$

We also assume that both $Y_{f}^{i},Y_{s}^{i}$, i.e.,
$\mathcal{Y}_{f}^{i},\mathcal{Y}_{s}^{i}$, for $i=1,\dots,m$ are connected
sets.

Figure 1: The fluid and solid phase

The set $\Omega^{h}$ is divided into $\tilde{\Omega}_{s}^{h}$ and
$\tilde{\Omega}_{f}^{h}$, the regions occupied by solid and fluid. For each
$h$ we define

$\tilde{\Omega}^{h}_{f/s}=\cup_{i=1}^{m}\cup_{k}\varepsilon(h)(k+Y_{f/s}^{i_{h}(k)}),$

where the union is taken over all $k\in\mathbb{Z}^{3}$ such that
$k+Y\subset\Omega$ and $i_{h}:\mathbb{Z}^{3}\to\\{1,\dots,m\\}$ is a given
map. We assume that $\tilde{\Omega}^{h}_{f/s}$ are Lipschitz and that
$\tilde{\Omega}_{s}^{h}$ is connected. We define
$\tilde{\Gamma}^{h}=\partial\tilde{\Omega}^{h}_{f}\cap\partial\tilde{\Omega}^{h}_{s}$.
For simplicity we will assume that

$\partial\tilde{\Omega}^{h}_{f}\cap\left(\\{x_{3}=-h/2\\}\cup\\{x_{3}=h/2\\}\right)=\emptyset.$
(2.1)

Additionally we assume that

$\left|\chi_{\Omega^{h}_{f/s}}(x)-\sum_{i=1}^{m}\chi_{U_{i}}(x)\chi_{\mathcal{Y}^{i}_{f/s}}(\hat{x}/\varepsilon,x_{3}/(\varepsilon/h))\right|\to
0,\textrm{ almost everywhere in }\Omega.$ (2.2)

Here $U_{1},\dots,U_{m}\subset\Omega$ are disjoint, open sets with Lipschitz
boundary such that the union of its closures is whole $\Omega$.
$\Omega^{h}_{f/s}\subset\Omega$ are rescaled $\tilde{\Omega}^{h}_{f/s}$, i.e.
the images of $\tilde{\Omega}^{h}_{f/s}$ through rescaling

$(x_{1},x_{2},x_{3})=(x_{1}^{h},x_{2}^{h},h^{-1}x_{3}^{h}),\quad(x^{h}_{1},x^{h}_{2},x^{h}_{3})\in\Omega^{h}.$
(2.3)

Actually, for the derivation of the model we will need more restrictive
assumption than (2.2) (see Remark 3.3). We assume the following:

$\displaystyle\forall\delta>0,\ \exists h_{0}>0,\textrm{ such that for all
}h<h_{0},\forall x\in U_{i},i=1,\dots,m,$
$\displaystyle\quad\operatorname{dist}(x,\partial U_{i})>\delta\implies
B(x^{h},2\varepsilon(h))\cap\tilde{\Omega}^{h}_{f}=B(x^{h},2\varepsilon(h))\cap\mathcal{Y}^{i}_{f}.$
(2.4)

For $i=1,\dots,m$ we take elasticity tensors
$\mathbb{A}^{1},\dots,\mathbb{A}^{m}$ that satisfy: $\exists\nu>0$ such that

$\nu|\boldsymbol{\xi}|^{2}\leq\mathbb{A}^{i}\boldsymbol{\xi}:\boldsymbol{\xi}\leq\nu^{-1}|\boldsymbol{\xi}|^{2},\quad\forall\boldsymbol{\xi}\in\mathbb{R}^{3\times
3}_{\text{sym}},\quad\textrm{ for }i=1,\dots,m,$ (2.5)

and further assume that the following symmetries hold:

$\mathbb{A}^{s}_{ijkl}=\mathbb{A}^{s}_{jikl}=\mathbb{A}^{s}_{klij},\quad
i,j,k,l\in\\{1,2,3\\},\quad s=1,\dots,m.$

Define

$\tilde{\mathbb{A}}^{h}(x^{h})=\sum_{k\in\mathbb{Z}^{3},k+Y\subset\Omega}\mathbb{A}^{i_{h}(k)}\chi_{k+Y_{s}^{i_{h}(k)}}(x^{h}),\quad
x^{h}\in\Omega^{h}.$

As a consequence of (2.2) we have

$\left|\mathbb{A}^{h}(x)-\mathbb{A}\left(x,\hat{x}/\varepsilon,x_{3}/(\varepsilon/h)\right)\right|\to
0,\textrm{ almost everywhere in }\Omega,$ (2.6)

where

$\mathbb{A}^{h}(x)=\tilde{\mathbb{A}}^{h}(x_{1},x_{2},hx_{3}),\quad
x\in\Omega,$

and

$\mathbb{A}(x,y)=\chi_{U_{1}}(x)\mathbb{A}^{1}\chi_{\mathcal{Y}_{s}^{1}}(y)+\cdots+\chi_{U_{m}}(x)\mathbb{A}^{m}\chi_{\mathcal{Y}_{s}^{m}}(y).$

We denote by

$\mathcal{I}=\cup_{i=1}^{m}\partial U_{i}.$

The reason behind these assumptions is to cover with our analysis also the
contacts between different types of poroelastic plates as well as contacts
between elastic and poroelastic plate (see Figure 2 and Figure 2). We define
for $x\in U_{i}$, $\mathcal{Y}_{f/s}(x):=\mathcal{Y}_{f/s}^{i}$ and for
$x\in\mathcal{I}$, $\mathcal{Y}_{f}(x):=\emptyset$,
$\mathcal{Y}_{s}(x):=\mathcal{Y}$. We also define the space

$\Omega\times\mathcal{Y}_{f/s}^{x}:=\\{(x,y)\in\Omega\times\mathcal{Y}:y\in\mathcal{Y}_{f/s}(x)\\}.$

Figure 2: Contacts between elastic and poroelastic region, $\eta\ll h$ or
$\eta\sim h$

###  2.2 The microscopic equations

We consider the following fluid-solid-structure interaction problem on
$\Omega^{h}$ which couples Stokes equation with linearized elasticity:

$\displaystyle\eta\tilde{\kappa}^{h}_{f}\frac{\partial^{2}\tilde{\boldsymbol{u}}^{h}}{\partial
t^{2}}+\frac{\nabla\tilde{p}^{h}}{h^{2}}=\frac{\varepsilon^{2}}{h^{4}}\Delta\frac{\partial\tilde{\boldsymbol{u}}^{h}}{\partial
t}+\tilde{\boldsymbol{F}}^{h}\quad\text{in}\,\,\Omega^{h}_{f},$ (2.7)
$\displaystyle\text{div}\,\frac{\partial\tilde{\boldsymbol{u}}^{h}}{\partial
t}=0\quad\text{in}\,\,\Omega^{h}_{f},$ (2.8)
$\displaystyle\eta\tilde{\kappa}_{s}^{h}\frac{\partial^{2}\tilde{\boldsymbol{u}}^{h}}{\partial
t^{2}}=\frac{1}{h^{2}}\text{div}\,(\tilde{\mathbb{A}}^{h}e(\tilde{\boldsymbol{u}}^{h}))+\tilde{\boldsymbol{F}}^{h}\quad\text{in}\,\Omega^{h}_{s},$
(2.9) $\displaystyle[\tilde{\boldsymbol{u}}^{h}]=0\quad\text{on}\,\Gamma^{h},$
(2.10)
$\tilde{\sigma}^{f,h}:=-\frac{1}{h^{2}}\tilde{p}^{h}I+2\frac{\varepsilon^{2}}{h^{4}}e\left(\displaystyle\frac{\partial\tilde{\boldsymbol{u}}^{h}}{\partial
t}\right)\quad\text{in}\,\,\tilde{\Omega}^{h}_{f},$ (2.11)
$\tilde{\sigma}^{s,h}:=\frac{1}{h^{2}}\tilde{\mathbb{A}}^{h}e(\tilde{\boldsymbol{u}}^{h})\quad\text{in}\,\,\tilde{\Omega}_{s}^{h},$
(2.12)
$\tilde{\sigma}^{f,h}\boldsymbol{n}_{f}=\tilde{\sigma}^{s,h}\boldsymbol{n}_{s}\quad\text{on}\,\,\tilde{\Gamma}^{h},$
(2.13)
$\tilde{\sigma}^{s,h}\boldsymbol{n}=0\quad\text{on}\,\,\\{x_{3}=h/2\\}\cup\\{x_{3}=-h/2\\},$
(2.14)

$\left.\tilde{\boldsymbol{u}}^{h}\right|_{\\{t=0\\}}=\left.\partial_{t}\tilde{\boldsymbol{u}}^{h}\right|_{\\{t=0\\}}=0\quad\text{on}\,\,\Omega^{h},$
(2.15) $\left\\{\tilde{\boldsymbol{u}}^{h},\tilde{p}^{h}\right\\}\quad\text{is
periodic in }\,(x_{1},x_{2}).$ (2.16)

Here $\tilde{\boldsymbol{u}}^{h}$ is the deformation in the solid, i.e.
$\frac{\partial\tilde{\boldsymbol{u}}^{h}}{\partial t}$ is the fluid velocity.
$\tilde{p}^{h}$ is a pressure, $\tilde{\boldsymbol{F}}^{h}$ are volume forces,
$\tilde{\kappa}_{f}^{h}$ and $\tilde{\kappa}_{s}^{h}$ is fluid and solid
density (here bounded from below and above positive functions), $\eta=\eta(h)$
is additional small parameter that corresponds to scaling the time (we will
analyze the case when $\lim_{h\to 0}\eta(h)=0$, which corresponds to quasi-
static case and $\eta(h)=1$ which corresponds to the bending regime with
inertal term). $\tilde{\sigma}^{f,h}$ and $\tilde{\sigma}^{s,h}$ are stresses
in fluid and solid respectively, $\boldsymbol{n}$ denotes the unit normal at
the point of interface in (2.13) from different side of interface, i.e. at the
point of boundary in (2.14). (2.7)-(2.8) are Stokes equations for
incompressible fluid, while (2.9) is the equation of linearized elasticity (to
al of them we apply appropriate scaling of the constants). (2.10) is the
continuity of the deformation at the interface ($[\tilde{\boldsymbol{u}}^{h}]$
denotes the jump of the function, i.e. difference of traces from two sides of
interface), while (2.13) is the continuity of stresses in the normal
direction. (2.14) is the Neumann boundary condition at the transverzal
boundary {colorred (this can be changed by adding surface loads)}, while
(2.15) is the zero initial condition (our analysis also allows different
initial conditions). Periodic boundary conditions are imposed on longitudinal
boundary by (2.16) for simplicity.

Considering the scaling it is not obvious at all why to impose this scaling
(we will additionally later scale the loads). These scalings can always be
interpreted as appropriate scalings of viscosity and time, i.e. density (to
see that one has to multiply the equations (2.7) and (2.9) with $h^{2}$).
These scalings give the limit Biot’s plate equation.

The functional space for (2.7)-(2.16) is the following:

$\displaystyle\tilde{V}^{h}=$ $\displaystyle\left\\{\psi\in
H^{1}_{\\#}(\Omega^{h};\mathbb{R}^{3})\left.\right|\,\operatorname{div}\psi=0\textrm{
in }\Omega_{f}^{h}\right\\}.$ (2.17)

The variational formulation which corresponds to (2.7)-(2.16) is given by:
Find $\tilde{\boldsymbol{u}}^{h}\in H^{1}(0,T;\tilde{V}^{h})$ with
$\frac{d^{2}\tilde{\boldsymbol{u}}^{h}}{dt^{2}}\in
L^{2}(0,T;L^{2}(\Omega^{h};\mathbb{R}^{3}))$ such that

$\displaystyle\displaystyle\eta\frac{d^{2}}{dt^{2}}\int_{\Omega^{h}}\tilde{\kappa}^{h}\tilde{\boldsymbol{u}}^{h}(t)\varphi\,dx^{h}+\displaystyle\frac{d}{dt}\frac{\varepsilon^{2}}{h^{4}}\int_{\tilde{\Omega}^{h}_{f}}2e(\tilde{\boldsymbol{u}}^{h}(t)):e(\varphi)\,dx^{h}$
$\displaystyle+\frac{1}{h^{2}}\int_{\tilde{\Omega}_{s}^{h}}\tilde{\mathbb{A}}^{h}\left(x^{h}\right)e(\tilde{\boldsymbol{u}}^{h}(t)):e(\varphi)\,dx^{h}-\displaystyle\frac{1}{h^{2}}\int_{\tilde{\Omega}^{h}_{f}}\tilde{p}^{h}\,\text{div}\,\varphi\,dx^{h}$
$\displaystyle=\int_{\Omega^{h}}\frac{1}{h}\tilde{\boldsymbol{F}}^{h}\varphi\,dx^{h},\quad\forall\varphi\in
H^{1}_{\\#}(\Omega^{h};\mathbb{R}^{3}),\quad\text{a.e.}\,\,\text{in}\,\,(0,T),$
(2.18)

where

$\tilde{\kappa}^{h}=\tilde{\kappa}^{h}_{f}\chi_{\Omega^{h}_{f}}+\tilde{\kappa}^{h}_{s}\chi_{\Omega^{h}_{s}}.$
(2.19)

and with initial conditions (2.15).

In order to work in a fixed domain $\Omega$, we apply the change of variables
(2.3) and we define
$\boldsymbol{u}^{h}(x):=\tilde{\boldsymbol{u}}^{h}(x^{h})$. In the analogous
way we define $p^{h}$, $\boldsymbol{F}^{h}$, $\kappa^{h}$. After doing this
transformation we obtain the following problem: Find $\boldsymbol{u}^{h}\in
H^{1}(0,T;V^{h})$ with $\frac{d^{2}\boldsymbol{u}^{h}}{dt^{2}}\in
L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))$ and $p^{h}\in
L^{2}(0,T;L^{2}(\Omega^{h}_{f}))$ such that

$\displaystyle\displaystyle\frac{d^{2}}{dt^{2}}\int_{\Omega}\eta\kappa^{h}$
$\displaystyle\boldsymbol{u}^{h}(t)\boldsymbol{v}\,dx+\displaystyle\frac{d}{dt}\frac{\varepsilon^{2}}{h^{4}}\int_{\Omega^{h}_{f}}2e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{v})\,dx$
$\displaystyle+\frac{1}{h^{2}}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}(x)e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{v})\,dx-\displaystyle\frac{1}{h^{2}}\int_{\Omega^{h}_{f}}p^{h}\text{div}_{h}\,\boldsymbol{v}\,dx$
$\displaystyle=\int_{\Omega}\boldsymbol{F}^{h}\boldsymbol{v}\,dx,\quad\forall\boldsymbol{v}\in
H^{1}_{\\#}(\Omega;\mathbb{R}^{3}),\quad\text{a.e.}\,\,\text{in}\,\,(0,T),$
(2.20)

with initial conditions

$\left.\boldsymbol{u}^{h}\right|_{\\{t=0\\}}=\left.\partial_{t}\boldsymbol{u}^{h}\right|_{\\{t=0\\}}=0\quad\text{on}\,\,\Omega.$
(2.21)

Here we denoted by $\Omega^{h}_{f/s}$ the rescaled $\Omega^{h}_{f/s}$. We also
define $\Gamma^{h}$ as rescaled $\tilde{\Gamma}^{h}$. The space $V^{h}$ is
defined in the following way

$\displaystyle V^{h}=$ $\displaystyle\left\\{\psi\in
H^{1}_{\\#}(\Omega;\mathbb{R}^{3})\left.\right|\,\operatorname{div}_{h}\psi=0\textrm{
in }\Omega_{f}^{h}\right\\}.$ (2.22)

We define $\kappa\in L^{\infty}(\Omega)$ as weak star limit of $\kappa^{h}$.

Next we discuss the existence result for the problem (2.20) -(2.21). The
following result is proven in [12, Theorem 3.2, Theorem 3.4].

######  Proposition 2.1.

Let us suppose that $\boldsymbol{F}\in
H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{)})$. Then, $3D$ problem given in (2.20)
has a unique solution $(\boldsymbol{u}^{h},p^{h})\in\left(H^{1}(0,T;V^{h})\cap
H^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))\right)\times
L^{2}(0,T;L^{2}(\Omega^{h}_{f}))$. It additionally satisfies:

$\displaystyle\|\partial_{tt}\boldsymbol{u}^{h}(t)\|^{2}_{L^{2}(\Omega;\mathbb{R}^{3})}+\|\partial_{tt}\boldsymbol{u}^{h}\|_{L^{2}(0,T;H^{1}(\Omega_{f}^{h};\mathbb{R}^{3}))}+\|\partial_{t}\boldsymbol{u}(t)\|_{H^{1}(\Omega_{s}^{h};\mathbb{R}^{3})}+\|p^{h}\|_{L^{2}(0,T;L^{2}(\Omega_{f}^{h}))}\leq$
$\displaystyle\hskip
43.05542ptC(h,\Omega)e^{C(h,\Omega)T}\|\boldsymbol{F}\|_{H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))},$

for some $C(h,\Omega)>0$.

######  Remark 2.1.

The existence and uniqueness of the weak solution for (2.20) can be obtained
for the forces in $L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))$ (see [12, Theorem
2,4]). However, then the pressure is not in $L^{2}$ neither in space nor in
time variable and under this condition we cannot continue the analysis.

###  2.3  A priori estimates

####  2.3.1 A priori estimates for displacement

In order to prove the a priori estimates, we need the following lemmas and
propositions. Firstly we introduce the following spaces:

$\displaystyle V(\Omega^{h}_{f})$ $\displaystyle=$
$\displaystyle\left\\{\boldsymbol{\varphi}\in
H^{1}(\Omega_{f}^{h};\mathbb{R}^{3})|\,\boldsymbol{\varphi}=0\,\text{on}\,\,\Gamma^{h}\,\text{and}\,\boldsymbol{\varphi}\,\text{is
periodic in}\,(x_{1},x_{2})\right\\},$ $\displaystyle V(\Omega^{h}_{s})$
$\displaystyle=$ $\displaystyle\left\\{\boldsymbol{\varphi}\in
H^{1}(\Omega_{s}^{h};\mathbb{R}^{3})|\,\boldsymbol{\varphi}\,\text{is periodic
in}\,(x_{1},x_{2})\right\\}.$

######  Lemma 2.1.

There exists $C>0$ such that for every $h>0$ there exists a linear extension
operator from $V(\Omega^{h}_{s})$ to $H^{1}_{\\#}(\Omega;\mathbb{R}^{3})$
($\boldsymbol{\varphi}\mapsto\hat{\boldsymbol{\varphi}}$) that satisfies

$\displaystyle\|\hat{\boldsymbol{\varphi}}\|_{H^{1}_{\\#}(\Omega;\mathbb{R}^{3})}$
$\displaystyle\leq$ $\displaystyle
C\|\hat{\boldsymbol{\varphi}}\|_{V(\Omega^{h}_{s})},$
$\displaystyle\|e_{h}(\hat{\boldsymbol{\varphi}})\|_{L^{2}(\Omega;\mathbb{R}^{3\times
3})}$ $\displaystyle\leq$ $\displaystyle
C\|e_{h}(\hat{\boldsymbol{\varphi}})\|_{L^{2}(\Omega_{s}^{h};\mathbb{R}^{3\times
3})},\quad\forall\boldsymbol{\varphi}\in V(\Omega^{h}_{s}).$

###### Proof.

See [20, Chapter 4]. ∎

######  Remark 2.2.

Notice that constant functions are necessarilly extended by the (same)
constant function.

######  Lemma 2.2.

Let $\boldsymbol{\varphi}\in V(\Omega^{h}_{f})$. Then we have

$\|\boldsymbol{\varphi}\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3})}\leq
C\varepsilon\|\nabla_{h}\boldsymbol{\varphi}\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\leq
C\varepsilon\|e_{h}(\boldsymbol{\varphi})\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}.$ (2.23)

###### Proof.

The estimate can be established on each small cube contained in $\Omega$ by
rescaling it on the physical domain. ∎

######  Proposition 2.2.

Let $\boldsymbol{\xi}\in H^{1}\left(0,T;H^{1}(\Omega;\mathbb{R}^{3})\right)$
such that $\boldsymbol{\xi}(0)=0$ and
$\int_{\Omega}\kappa^{h}\boldsymbol{\xi}(t)\,dx=0$, for all $t\in[0,T]$ and
some $0<h<1$. Assume that for some $M>0$ we have $\varepsilon\leq Mh$. Then
the following estimate holds for all $t\in[0,T]$, with a constant $C$
independent of $h$,

$\|\pi_{1/h}\boldsymbol{\xi}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}\leq\displaystyle
C\left\\{\frac{1}{h}\|e_{h}(\boldsymbol{\xi}(t))\|_{L^{2}(\Omega^{h}_{s};\mathbb{R}^{3\times
3})}+\frac{\varepsilon}{h}\int_{0}^{t}\|e_{h}(\partial_{\tau}\boldsymbol{\xi}(\tau))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\,d\tau\right\\}.$ (2.24)

###### Proof.

For every $t\in[0,T]$, let $\hat{\boldsymbol{\xi}}(t)=P_{h}\boldsymbol{\xi}$
be the extension of $\left.\boldsymbol{\xi}\right|_{\Omega^{h}_{s}}$, given by
Lemma 2.1. Let
$\boldsymbol{\omega}(t)=\hat{\boldsymbol{\xi}}(t)-\boldsymbol{\xi}(t)$ on
$\Omega^{h}_{f}$ and zero elsewhere. Then for every $t\in(0,T)$ we have
$\boldsymbol{\omega}(t)\in V(\Omega^{h}_{f})$ and $\omega(t)$ periodic in
$(x_{1},x_{2})$. As a consequence of Lemma 2.2 we have

$\|\boldsymbol{\omega}(t)\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3})}\leq
C\varepsilon\|\nabla_{h}\boldsymbol{\omega}(t)\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\leq
C\varepsilon\|e_{h}(\boldsymbol{\omega}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})},$ (2.25)

for all $t\in(0,T)$. From (2.25) we have

$\displaystyle\|\hat{\boldsymbol{\xi}}(t)-\boldsymbol{\xi}(t)\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3})}$
$\displaystyle\leq
C\varepsilon\left\\{\|e_{h}(\hat{\boldsymbol{\xi}}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}+\|e_{h}(\boldsymbol{\xi}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\right\\}.$ (2.26)

Next , from (2.26) and using the fact that $\varepsilon,h\leq 1$ and
Proposition A.1 we obtain

$\displaystyle\|\pi_{1/h}\boldsymbol{\xi}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}$
$\displaystyle\leq\|\pi_{1/h}\hat{\boldsymbol{\xi}}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}+\|\pi_{1/h}(\hat{\boldsymbol{\xi}}(t)-\boldsymbol{\xi}(t))\|_{L^{2}(\Omega;\mathbb{R}^{3})}$
$\displaystyle\leq\|\pi_{1/h}\hat{\boldsymbol{\xi}}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}+\frac{1}{h}\|\hat{\boldsymbol{\xi}}(t)-\boldsymbol{\xi}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}$
$\displaystyle\leq\|\pi_{1/h}\hat{\boldsymbol{\xi}}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}+\frac{C\varepsilon}{h}\left\\{\|e_{h}(\hat{\boldsymbol{\xi}}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}+\|e_{h}(\boldsymbol{\xi}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\right\\}$ $\displaystyle\leq
C\Bigg{\\{}\frac{1}{h}\|e_{h}(\hat{\boldsymbol{\xi}}(t))\|_{L^{2}(\Omega;\mathbb{R}^{3\times
3})}+\left|\int_{\Omega}\pi_{1/h}(\kappa^{h}\hat{\boldsymbol{\xi}}(t))\right|+\frac{\varepsilon}{h}\|e_{h}(\hat{\boldsymbol{\xi}}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}$ $\displaystyle\hskip
21.52771pt+\frac{\varepsilon}{h}\|e_{h}(\boldsymbol{\xi}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\Bigg{\\}}$ (2.27)

Note that since $\int_{\Omega}\kappa^{h}\boldsymbol{\xi}(t)\,dx=0$ we have
that

$\int_{\Omega}\pi_{1/h}(\kappa^{h}\hat{\boldsymbol{\xi}}(t)))\,dx=\int_{\Omega}\pi_{1/h}(\kappa^{h}\hat{\boldsymbol{\xi}}(t)-\kappa^{h}\boldsymbol{\xi}(t)))\,dx.$
(2.28)

From (2.26) and (2.28) we conclude

$\left|\int_{\Omega}\pi_{1/h}(\kappa^{h}\hat{\boldsymbol{\xi}}(t)))\,dx\right|\leq\frac{C\varepsilon}{h}\left\\{\|e_{h}(\hat{\boldsymbol{\xi}}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}+\|e_{h}(\boldsymbol{\xi}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\right\\}.$ (2.29)

Next, we remark that using $\boldsymbol{\xi}(0)=0$, we have

$\displaystyle\|e_{h}(\boldsymbol{\xi}(t))\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}$
$\displaystyle\displaystyle\leq\left\|\int_{0}^{t}e_{h}\left(\partial_{\tau}\boldsymbol{\xi}(t)\right)d\tau\right\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}\leq\displaystyle\int_{0}^{t}\left\|e_{h}\left(\partial_{\tau}\boldsymbol{\xi}(t)\right)\right\|_{L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3})}d\tau.$ (2.30)

(2.24) now follows from (2.27), (2.29) and (2.30) by using Lemma 2.1, and
$\varepsilon\leq Mh$. ∎

The following proposition gives us necessary a priori estimates.

######  Proposition 2.3.

Let us suppose that

$\displaystyle\boldsymbol{F}^{h}\in H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{3})),$
$\displaystyle\|F^{h}_{3}\|_{L^{2}(\Omega;\mathbb{R}^{3})}+h\sum_{\alpha=1,2}\left(\left\|{F}^{h}_{\alpha}(0)\right\|_{L^{2}(\Omega;\mathbb{R}^{3})}+\left\|\partial_{t}F^{h}_{\alpha}\right\|_{L^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))}\right)\leq
C,$ (2.31)

where $C$ doesn’t depend on $h$ and 555this can always be achieved by
translation of coordinate system for every $t$

$\int_{\Omega}\boldsymbol{F}^{h}(t))\,dx=0,\quad\textrm{ for every
}t\in[0,T].$ (2.32)

In the case when $\eta(h)\to 0$ we additionally suppose that

$\left\|\pi_{h}\boldsymbol{F}^{h}\right\|_{H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))}\leq
C.$ (2.33)

Then we have:

$\displaystyle\displaystyle\sqrt{\eta}\left\|\partial_{t}\boldsymbol{u}^{h}\right\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))}+\displaystyle\frac{1}{h}\left\|e_{h}(\boldsymbol{u}^{h})\right\|_{L^{\infty}(0,T;L^{2}(\Omega^{h}_{s};\mathbb{R}^{3\times
3})}$
$\displaystyle+\frac{\varepsilon}{h^{2}}\left\|e_{h}\left(\partial_{t}\boldsymbol{u}^{h}\right)\right\|_{L^{2}(0,T;L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3}))}\leq C,$ (2.34)

where $C$ doesn’t depend on $h$. Here $(\boldsymbol{u}^{h},p^{h})$ is a
solution of (2.20) with the initial condition (2.21).

###### Proof.

Without loss of generality we can in any case assume that
$\int_{\Omega}\boldsymbol{F}^{h}(t)=0$, for all $t\in[0,T]$ and consequently
$\int_{\Omega}\kappa^{h}\boldsymbol{u}^{h}(t)=0$, for all $t\in[0,T]$ (this
can be achieved by a translation of a coordinate system). From (2.3) we
conclude

$\sum_{\alpha=1,2}\left\|\pi_{h}F^{h}_{\alpha}\right\|_{L^{\infty}(0,T,L^{2}(\Omega;\mathbb{R}^{3}))}\leq
C,$ (2.35)

and from (2.33) we can conclude

$\left\|\pi_{h}\boldsymbol{F}^{h}\right\|_{L^{\infty}(0,T,L^{2}(\Omega;\mathbb{R}^{3}))}\leq
C,$ (2.36)

with $C$ independent of $h$.

We take $\varphi=\partial_{t}u^{h}$ as test function in (2.20). This yields
for almost every $t\in(0,T)$

$\displaystyle\displaystyle\frac{1}{2}\frac{d}{dt}\left(\displaystyle\int_{\Omega}\eta\kappa^{h}|\partial_{t}\boldsymbol{u}^{h}(t)|^{2}\,dx+\frac{1}{h^{2}}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{u}^{h}(t))\,dx\right)$
$\displaystyle+\displaystyle\frac{\varepsilon^{2}}{h^{4}}\int_{\Omega^{h}_{f}}2|e_{h}(\partial_{t}\boldsymbol{u}^{h}(t))|^{2}\,dx=\int_{\Omega}\boldsymbol{F}^{h}\partial_{t}\boldsymbol{u}^{h}\,dx.$
(2.37)

By integration over $t\in[0,T]$, we obtain

$\displaystyle\displaystyle\frac{1}{2}\displaystyle\int_{\Omega}\eta\kappa^{h}|\partial_{t}\boldsymbol{u}^{h}(t)|^{2}\,dx+\frac{1}{h^{2}}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{u}^{h}(t))\,dx$
$\displaystyle+\displaystyle\frac{\varepsilon^{2}}{h^{4}}\int_{0}^{t}\int_{\Omega^{h}_{f}}2|e_{h}(\partial_{t}\boldsymbol{u}^{h}(\tau))|^{2}\,dx\,d\tau=\int_{0}^{t}\int_{\Omega}\boldsymbol{F}^{h}\partial_{t}\boldsymbol{u}^{h}(\tau)\,dx\,d\tau.$
(2.38)

On the other hand, we can obtain

$\displaystyle\left|\displaystyle\int_{0}^{t}\int_{\Omega}\boldsymbol{F}^{h}\partial_{t}\boldsymbol{u}^{h}(\tau)\,dx\,d\tau\right|$
$\displaystyle\leq\sum_{\alpha=1,2}\left(\left|\displaystyle\int_{\Omega}F_{\alpha}^{h}(t)\boldsymbol{u}^{h}(t)\,dx\right|+\left|\displaystyle\int_{0}^{t}\int_{\Omega}\partial_{t}F^{h}_{\alpha}\boldsymbol{u}^{h}(\tau)\,dx\,d\tau\right|\right)+C\|F^{h}_{3}\|_{L^{2}(\Omega)}\|\partial_{t}u_{3}(t)\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\Bigg{(}\sum_{\alpha=1,2}\left(\|hF^{h}_{\alpha}(t)\|_{L^{2}(\Omega)}\|\frac{u_{\alpha}^{h}(t)}{h}\|_{L^{2}(\Omega)}+\|h\partial_{t}F^{h}_{\alpha}\|_{L^{2}(0,t;L^{2}(\Omega))}\|\frac{u_{\alpha}^{h}}{h}\|_{L^{2}(0,t;L^{2}(\Omega;\mathbb{R}^{3}))}\right)$
(2.39) $\displaystyle\hskip
21.52771pt+\|F^{h}_{3}\|_{L^{2}(\Omega)}\|\partial_{t}u_{3}(t)\|_{L^{2}(\Omega)}\Bigg{)},$
(2.40)

i.e., in the case when $\eta(h)\to 0$

$\displaystyle\left|\displaystyle\int_{0}^{t}\int_{\Omega}\boldsymbol{F}^{h}\partial_{t}\boldsymbol{u}^{h}(\tau)\,dx\,d\tau\right|$
$\displaystyle\leq\left|\displaystyle\int_{\Omega}\boldsymbol{F}^{h}(t)\boldsymbol{u}^{h}(t)\,dx\right|+\left|\displaystyle\int_{0}^{t}\int_{\Omega}\partial_{t}\boldsymbol{F}^{h}\boldsymbol{u}^{h}(\tau)\,dx\,d\tau\right|$
$\displaystyle\leq
C\left(\|\pi_{h}\boldsymbol{F}^{h}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}\|\pi_{1/h}\boldsymbol{u}^{h}(t)\|_{L^{2}(\Omega;\mathbb{R}^{3})}+\|\pi_{h}\partial_{t}\boldsymbol{F}^{h}\|_{L^{2}(0,t;L^{2}(\Omega;\mathbb{R}^{3}))}\|\pi_{1/h}\boldsymbol{u}^{h}\|_{L^{2}(0,t;L^{2}(\Omega;\mathbb{R}^{3}))}\right).$

Then from (2.36), (2.37), (2.38), (2.40) and Gronwall’s inequality it follows

$\displaystyle\displaystyle\sqrt{\eta}\left\|\partial_{t}\boldsymbol{u}^{h}\right\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))}+\displaystyle\frac{1}{h}\left\|e_{h}(\boldsymbol{u}^{h})\right\|_{L^{\infty}(0,T;L^{2}(\Omega^{h}_{s};\mathbb{R}^{3\times
3}))}$
$\displaystyle+\frac{\varepsilon}{h^{2}}\left\|e_{h}\left(\partial_{t}\boldsymbol{u}^{h}\right)\right\|_{L^{2}(0,T;L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3}))}\leq C.$ (2.42)

∎

####  2.3.2 A priori estimate for pressure

With the existence of a strong solution guaranteed by proposition 2.3, it is
now possible for us to establish the apriori estimate for a pressure.

From (2.20), we obtain that $p^{h}\in L^{2}(0,T;L^{2}(\Omega^{h}_{f}))$
satisfies

$\displaystyle\displaystyle\frac{1}{h^{2}}\int_{\Omega^{h}_{f}}p^{h}\text{div}_{h}\,\boldsymbol{v}\,dx=$
$\displaystyle\int_{\Omega}\eta\kappa^{h}\displaystyle\partial_{tt}\boldsymbol{u}^{h}\boldsymbol{v}\,dx+\frac{\varepsilon^{2}}{h^{4}}\int_{\Omega^{h}_{f}}2e_{h}(\partial_{t}\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{v})\,dx$
$\displaystyle\quad+\frac{1}{h^{2}}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{v})\,dx-\int_{\Omega}\boldsymbol{F}^{h}(t)\boldsymbol{v}\,dx,$
$\displaystyle\quad\forall\boldsymbol{v}\in
H^{1}_{\text{\\#}}(\Omega;\mathbb{R}^{3}),\quad\text{a.e.}\,\,\text{in}\,\,(0,T).$
(2.43)

To obtain the a priori estimate for pressure we will need the following
proposition.

######  Proposition 2.4.

For every $g\in L^{2}(\Omega)$ and every $h$, there is $\boldsymbol{v}^{h}\in
H^{1}_{\\#}(\Omega;\mathbb{R}^{3})$, such that

$\operatorname{div}_{h}\boldsymbol{v}^{h}=g$

and

$\|\boldsymbol{v}^{h}\|_{L^{2}(\Omega;\mathbb{R}^{)}}+\|\nabla_{h}\boldsymbol{v}^{h}\|_{L^{2}(\Omega;\mathbb{R}^{)}}\leq
C\|g\|_{L^{2}(\Omega)},$

where $C$ is independent of $h$.

###### Proof.

We divide $\Omega$ in small plates
$\\{\Omega^{h}_{i}\\}_{i=1}^{\infty}\subset\Omega$ of size $h$, such that
$\Omega^{h}_{i}\cap\Omega^{h}_{j}=\emptyset$ for all $i\neq j$. Here

$\Omega^{h}_{k}=\omega\times\left(-\frac{h}{2}+kh,\frac{h}{2}+kh\right)\cap\Omega,\quad
k\in\mathbb{Z}.$

Notice that for every $h$ only finitely number of $\Omega_{i}^{h}$ are non-
empty, i.e. for every $h>0$ there exists $n(h)$ such that
$\Omega_{k}^{h}\neq\emptyset$ for $|k|\leq n(h)$ and
$\Omega_{k}^{h}=\emptyset$, for $|k|>n(h)$. We define $g^{h}$ on $\Omega$, by
defining it in the following way

$g^{h}(x_{1},x_{2},x_{3})=g(x_{1},x_{2},hx_{3}),\quad\textrm{if
}(x_{1},x_{2},x_{3})\in\Omega_{0}^{h},$

and we extend it by periodicity in $x_{3}$ to whole $\Omega$. We find
$\boldsymbol{k}^{h}\in H^{1}_{\\#}(\Omega;\mathbb{R}^{3})$, which satisfies

$\operatorname{div}\boldsymbol{k}^{h}=g^{h},\quad\|\boldsymbol{k}^{h}\|_{L^{2}(\Omega;\mathbb{R}^{3})}+\|\nabla\boldsymbol{k}^{h}\|_{L^{2}(\Omega;\mathbb{R}^{3\times
3})}\leq C\|g^{h}\|_{L^{2}(\Omega)}$ (2.44)

This can be obtained by solving $\Delta\boldsymbol{\phi}^{h}=g^{h}$ on
$\Omega$ with zero boundary condition on
$\omega\times\\{-\frac{1}{2},\frac{1}{2}\\}$ (and periodic on
$\partial\omega\times I$) and putting
$\boldsymbol{k}^{h}=\nabla\boldsymbol{\phi}^{h}$. From (2.44) it follows that
there exists $\Omega_{k}^{h}$, for $|k|<n(h)$ such that for
$\tilde{\boldsymbol{v}}^{h}=\boldsymbol{k}^{h}|_{\Omega_{k}^{h}}$ it is
satisfied

${\rm
div}\,\tilde{\boldsymbol{v}}^{h}=g^{h}|_{\Omega_{k}^{h}},\quad\|\tilde{\boldsymbol{v}}^{h}\|_{L^{2}(\Omega_{k}^{h};\mathbb{R}^{3})}+\|\nabla\tilde{\boldsymbol{v}}^{h}\|_{L^{2}(\Omega_{k}^{h};\mathbb{R}^{3\times
3})}\leq C\|g^{h}\|_{L^{2}(\Omega_{k}^{h})},$

where $C$ is independent of $h$. $\boldsymbol{v}^{h}$ can then be defined by
translating $\tilde{\boldsymbol{v}}^{h}$ on $\Omega_{0}^{h}$ and rescaling it
to whole $\Omega$.

∎

Next proposition establishes a priori estimate for the pressure.

######  Proposition 2.5.

Let conditions of Proposition 2.3 be satisfied. Let
$(\boldsymbol{u}^{h},p^{h})$ be a solution of (2.20) with the initial
condition (2.21). The following is satisfied:

$\|\tilde{p}^{h}\|_{H^{-1}(0,T;L^{2}(\Omega))}\leq Ch,$ (2.45)

where $C$ is independent of $h$ and $\tilde{p}^{h}$ is the extension by zero
of $p^{h}$ on whole $\Omega$.

###### Proof.

Using proposition 2.4 we take for $g\in L^{2}(\Omega)$ and $h>0$,
$\boldsymbol{v}^{h}$ that satisfies

${\rm
div}_{h}\,\boldsymbol{v}^{h}=g,\quad\|\boldsymbol{v}^{h}\|_{L^{2}(\Omega;\mathbb{R}^{3})}+\|\nabla_{h}\boldsymbol{v}^{h}\|_{L^{2}(\Omega;\mathbb{R}^{3})}\leq
C\|g\|_{L^{2}(\Omega)}.$

By using (2.34), after doing integration by parts in the first member, and
(2.43) we conclude that for $s\in C_{c}^{1}(0,T)$ we have

$\int_{0}^{T}\int_{\Omega}\frac{\tilde{p}^{h}}{h}\cdot sg\,dx\,dt\leq
C\|sg\|_{H^{1}(0,T;L^{2}(\Omega))},$

where $C>0$ is independent of $h$. By the density argument we have for every
$\varphi\in H_{0}^{1}(0,T;L^{2}(\Omega))$

$\int_{0}^{T}\int_{\Omega}\frac{\tilde{p}^{h}}{h}\cdot\varphi\,dx\,dt\leq
C\|\varphi\|_{H^{1}(0,T;L^{2}(\Omega))},$

from which the claim follows. ∎

##  3 Quasi-static case

In this section we analyze the case when $\lim_{h\to 0}\eta(h)=0$. In Section
3.1 we provide the compactness result for the sequence of solutions of (2.20),
in Section 3.2 we define the effective tensors appearing in the limit problem
and prove some of their properties, in Section 3.3 we obtain the limit model,
in Section 3.4 we provethe existence and uniqueness result and energy type
equality for the limit problem, in Section 3.5 we prove the strong convergence
of the solutions of (2.20) to the solution of the limit problem with
appropriate correctors, while in Section 3.6 we discuss the possibility of
having surface loads as well as the situation when fluid part touches the
longitudinal boundary of $\Omega^{h}$. We denote by
$L_{0}=\dot{H}^{1}_{\\#}(\omega;\mathbb{R}^{2})\times\dot{H}^{2}_{\\#}(\omega)$
and by $L_{0}^{\prime}$ its dual. Also we denote by $M=L^{2}(\omega;H^{1}(I))$
and by $M^{\prime}$ its dual. For $\boldsymbol{w}\in
H^{1}(\mathcal{Y};\mathbb{R}^{3})$, $\boldsymbol{g}\in\mathbb{R}^{3}$ we
define

$\mathfrak{C}_{\infty}(\boldsymbol{w},\boldsymbol{g})(y):=e_{y}(\boldsymbol{w})+\text{sym}\left(0|0|\boldsymbol{g}\right).$

The definition naturally extends to $H^{1}(\mathcal{Y};\mathbb{R}^{3})$ and
$\mathbb{R}^{3}$ valued functions. Appropriate definitions of two-scale
convergences are given in Section A.2.

###  3.1 Compactness result

For a given $\nu\in L^{1}(0,T)$ and a Hilbert space $X$ we define the operator
$\cdot^{\nu}:L^{\infty}(0,T;X)\to L^{\infty}(X)$ in the following way

$\boldsymbol{u}^{\nu}=\int_{0}^{T}\boldsymbol{u}(t)\nu(t)\,dt.$

The following theorem gives us the compactness result.

######  Theorem 3.1.

Let assumptions (2.32) and (2.33) be satisfied. We also assume that

$\pi_{h}\boldsymbol{F}^{h}\xrightharpoonup{t,dr-2,2}\boldsymbol{F},$

where $\boldsymbol{F}\in
H^{1}(0,T;L^{2}(\Omega\times\mathcal{Y};\mathbb{R}^{3}))$. The following
statements hold: Let $(\boldsymbol{u}^{h},p^{h})$ be the variational solution
of (2.20) with initial condition (2.21). Then there exist limits (on a
subsequence):

$\displaystyle(\mathfrak{a},\mathfrak{b})$ $\displaystyle\in
L^{\infty}(0,T;L_{0}),$ (3.1) $\displaystyle\boldsymbol{w}$ $\displaystyle\in
L^{\infty}(0,T;L^{2}(\Omega,\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3}))),$ (3.2)
$\displaystyle\boldsymbol{g}$ $\displaystyle\in
L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3})),$ (3.3)
$\displaystyle\boldsymbol{u}^{0}_{f}$ $\displaystyle\in
H^{1}(0,T;L^{2}(\Omega;H^{1}(\mathcal{Y}_{f}(x);\mathbb{R}^{3}))),\
\text{div}_{y}\boldsymbol{u}_{f}^{0}=0,\ \boldsymbol{u}_{f}^{0}(0)=0$ (3.4)
$\displaystyle p$ $\displaystyle\in H^{-1}(0,T;L^{2}(\Omega)),\ p=0\textrm{ on
}\Omega\times\mathcal{Y}_{s}^{x},$ (3.5)

such that for we have (on a subsequence)

$\displaystyle(h^{-1}u^{h}_{\alpha})^{\nu}\overset{L^{2}}{\rightarrow}(\mathfrak{a}_{\alpha}-x_{3}\partial_{\alpha}\mathfrak{b})^{\nu},\quad(u^{h}_{3})^{\nu}\overset{L^{2}}{\rightarrow}(\mathfrak{b})^{\nu},\quad\forall\nu\in
L^{1}(0,T)$ (3.6) $\displaystyle
h^{-2}\boldsymbol{u}^{h}_{f}\xrightharpoonup{t,dr-2,\infty}\boldsymbol{u}^{0}_{f},$
(3.7) $\displaystyle
h^{-1}e_{h}(\hat{\boldsymbol{u}}^{h})\xrightharpoonup{t,dr-2,\infty}\iota(e_{\hat{x}}(\mathfrak{a})-x_{3}\nabla^{2}_{\hat{x}}\mathfrak{b})+\mathfrak{C}_{\infty}(\boldsymbol{w},\boldsymbol{g}),$
(3.8) $\displaystyle\varepsilon
h^{-2}e_{h}(\boldsymbol{u}^{h}_{f})\xrightharpoonup{t,dr-2,\infty}e_{y}(\boldsymbol{u}^{0}_{f}),$
(3.9) $\displaystyle\varepsilon
h^{-2}e_{h}(\partial_{t}\boldsymbol{u}^{h})\xrightharpoonup{t,dr-2,2}e_{y}(\partial_{t}\boldsymbol{u}^{0}_{f})$
(3.10) $\displaystyle
h^{-1}(p^{h})^{\nu}\xrightharpoonup{t,dr-2,2}p^{\nu},\quad\forall\nu\in
H_{0}^{1}(0,T).$ (3.11)

In the sense of distributions the following identity holds

$\displaystyle\displaystyle-|\mathcal{Y}_{f}(x)|\text{div}_{\hat{x}}\,\mathfrak{a}(\hat{x},t)+|\mathcal{Y}_{f}(x)|x_{3}\text{div}_{\hat{x}}\,\nabla_{\hat{x}}\mathfrak{b}(\hat{x},t)-\displaystyle\displaystyle\partial_{3}\int_{\mathcal{Y}_{f}(x)}u^{0}_{f,3}(x,y,t)dy$
$\displaystyle\quad-\displaystyle\int_{\mathcal{Y}_{f}(x)}\left[\text{tr}\,\mathfrak{C}_{\infty}\left(\boldsymbol{w},\boldsymbol{g}\right)(x,y,t)\right]\,dy=0,$
(3.12)

where

$\boldsymbol{u}^{h}=\hat{\boldsymbol{u}}^{h}+\boldsymbol{u}_{f}^{h},$

Here $\hat{\boldsymbol{u}}^{h}$ is the extension defined in Lemma 2.1.

###### Proof.

As in the proof of Proposition 2.3 we can assume that
$\int_{\Omega}\kappa^{h}\boldsymbol{u}^{h}(t)=0$, for every $t\in[0,T]$. For
every $t\in[0,T]$, we have from Proposition 2.3

$\displaystyle\frac{1}{h}\left\|e_{h}\left(\boldsymbol{u}^{h}\right)\right\|_{L^{\infty}(0,T;L^{2}(\Omega^{h}_{s};\mathbb{R}^{3\times
3}))}\leq C,$ (3.13)

using the extension operator properties, we have for almost every $t\in[0,T]$

$\displaystyle\frac{1}{h}\left\|e_{h}\left(\hat{\boldsymbol{u}}^{h}\right)\right\|_{L^{2}(\Omega;\mathbb{R}^{3\times
3})}\leq
C\displaystyle\frac{1}{h}\left\|e_{h}\left(\boldsymbol{u}^{h}\right)\right\|_{L^{2}(\Omega^{h}_{s};\mathbb{R}^{3\times
3})}\leq C,$

therefore

$\displaystyle\limsup_{h\to
0}\left\|e_{h}\left(\frac{1}{h}\hat{\boldsymbol{u}}^{h}\right)\right\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3\times
3}))}\leq C.$ (3.14)

We obtain that on a subsequence

$e_{h}\left(\frac{1}{h}\hat{\boldsymbol{u}}^{h}\right)\xrightharpoonup{t,dr-2,\infty}\mathbb{L},$
(3.15)

where $\mathbb{L}\in
L^{\infty}(0,T;L^{2}(\Omega\times\mathcal{Y};\mathbb{R}^{3\times 3}_{\rm
sym}))$.

Next we take arbitrary $\nu\in L^{1}(0,T)$.

Lemma A.1 and Remark A.2 yields the following decomposition of the sequence
$(\hat{\boldsymbol{u}}^{h})^{\nu}$:

$\displaystyle\frac{1}{h}(\hat{\boldsymbol{u}}^{h})^{\nu}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-x_{3}\partial_{1}\mathfrak{b}(\nu)\\\
-x_{3}\partial_{2}\mathfrak{b}(\nu)\\\
h^{-1}\mathfrak{b}(\nu)\end{pmatrix}+\begin{pmatrix}\mathfrak{a}_{1}(\nu)\\\
\mathfrak{a}_{2}(\nu)\\\
0\end{pmatrix}+\boldsymbol{\psi}^{h}(\nu)+C^{h}(\nu),$ (3.16)
$\displaystyle\displaystyle\frac{1}{h}e_{h}\left((\hat{\boldsymbol{u}}^{h})^{\nu}\right)$
$\displaystyle=$
$\displaystyle\iota\left(-x_{3}\nabla^{2}_{\hat{x}}\mathfrak{b}(\nu)+e_{\hat{x}}(\mathfrak{a}(\nu))\right)+e_{h}\left(\boldsymbol{\psi}^{h}(\nu)\right),$

where $\mathfrak{b}(\nu)\in H^{2}_{\\#}(\omega),\mathfrak{a}(\nu)\in
H^{1}_{\\#}(\omega;\mathbb{R}^{2}),(\boldsymbol{\psi}^{h}(\nu))_{h>0}\subset
H^{1}_{\\#}(\Omega;\mathbb{R}^{3})$ and
$h\pi_{1/h}\boldsymbol{\psi}^{h}(\nu)\overset{L^{2}}{\rightarrow}0$ and
$C^{h}(\nu)\in\mathbb{R}^{3}$ is chosen such that

$\int_{\Omega}\kappa^{h}\left((\hat{\boldsymbol{u}}^{h})^{\nu}-C^{h}(\nu)\right)=0.$
(3.17)

From (3.17) it follows that

$\int_{\omega}\mathfrak{a}(\nu)=0,\quad\int_{\omega}\mathfrak{b}(\nu)=0.$
(3.18)

Furthermore, by Lemma A.2 and Remark A.3 applied to
$\boldsymbol{\psi}^{h}(\nu)$, there exists another subsequences
$(\varphi^{h}(\nu))_{h>0}\subset
H^{2}_{\\#}(\omega)),(\tilde{\boldsymbol{\psi}}^{h}(\nu))_{h>0}\subset
H^{1}_{\\#}(\Omega;\mathbb{R}^{3})),(o^{h}(\nu))_{h>0}\subset$
$L^{2}(\Omega;\mathbb{R}^{3\times 3})$ such that

$e_{h}\left(\boldsymbol{\psi}^{h}(\nu)\right)=\iota\left(-x_{3}\nabla^{2}_{\hat{x}}\varphi^{h}(\nu)\right)+e_{h}\left(\tilde{\boldsymbol{\psi}}^{h}(\nu)\right)+o^{h}(\nu),$

where

$\displaystyle\varphi^{h}(\nu)\overset{H^{1}}{\rightarrow}0,\quad\left\|\nabla^{2}_{\hat{x}}\varphi^{h}(\nu)\right\|_{L^{2}}\leq
C,$ (3.19)
$\displaystyle\tilde{\boldsymbol{\psi}}^{h}(\nu)\overset{L^{2}}{\rightarrow}0,\quad\left\|\nabla_{h}\tilde{\boldsymbol{\psi}}^{h}(\nu)\right\|_{L^{2}}\leq
C,\quad o^{h}(\nu)\overset{L^{2}}{\rightarrow}0.$ (3.20)

By using Lemma A.3 and Lemma A.5 we have that there exist $\varphi(\nu)\in
L^{2}(\omega;\dot{H}^{2}(\mathcal{Y}))$, $\boldsymbol{g}(\nu)\in
L^{2}(\Omega,\mathbb{R}^{3})$ and $\boldsymbol{w}^{1}(\nu)\in
L^{2}(\Omega,\dot{H}^{1}(\mathcal{Y},\mathbb{R}^{3}))$ such that

$\displaystyle\nabla^{2}_{\hat{x}}\varphi^{h}(\nu)(\hat{x},t)$
$\displaystyle\xrightharpoonup{dr-2}\nabla^{2}_{y}\varphi(\nu)(\hat{x},y,t),$
$\displaystyle e_{h}\left(\tilde{\boldsymbol{\psi}}^{h}(\nu)\right)$
$\displaystyle\xrightharpoonup{dr-2}\mathfrak{C}_{\infty}\left(\boldsymbol{w}^{1}(\nu),\boldsymbol{g}(\nu)\right),$

where
$\mathfrak{C}_{\infty}\left(\boldsymbol{w}^{1}(\nu),\boldsymbol{g}(\nu)\right)=e_{y}\left(\boldsymbol{w}^{1}(\nu)\right)+\text{sym}\left(0|0|\boldsymbol{g}(\nu)\right)$.

By introducing the function

$\boldsymbol{w}(\nu)(x,y,t):=\begin{pmatrix}-x_{3}\partial_{y_{1}}\varphi(\nu)(\hat{x},y,t)\\\
-x_{3}\partial_{y_{2}}\varphi(\nu)(\hat{x},y,t)\\\
\varphi(\nu)(\hat{x},y,t)\end{pmatrix}+\boldsymbol{w}^{1}(\nu)(x,y,t),$

we have

$\mathfrak{C}_{\infty}(\boldsymbol{w}(\nu)(x,y,t),\boldsymbol{g}(\nu)(x,t))=-x_{3}\iota\left(\nabla^{2}_{y}\varphi(\nu)(\hat{x},y,t)\right)+\mathfrak{C}_{\infty}\left(\boldsymbol{w}^{1}(\nu)(x,y,t),\boldsymbol{g}(\nu)(x,t)\right).$

From (3.15) we conclude that

$\displaystyle e_{\hat{x}}(\mathfrak{a}(\nu))_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\int_{[0,T]\times I\times\cal
Y}\mathbb{L}_{\alpha\beta}\nu\,dy\,dx_{3}\,dt,$ (3.21)
$\displaystyle(\nabla^{2}_{\hat{x}}\mathfrak{b}(\nu))_{\alpha\beta}$
$\displaystyle=$ $\displaystyle-12\int_{[0,T]\times I\times\cal
Y}x_{3}\mathbb{L}_{\alpha\beta}\nu\,dy\,dx_{3}\,dt,\,\alpha,\beta\in\\{1,2\\},$
(3.22) $\displaystyle\boldsymbol{g}_{i3}(\nu)$ $\displaystyle=$
$\displaystyle\int_{[0,T]\times I\times\cal
Y}\mathbb{L}_{i3}\nu\,dy\,dx_{3}\,dt,\quad i\in\\{1,2,3\\},$ (3.23)
$\displaystyle e_{y}\left(\boldsymbol{w}(\nu)\right)$ $\displaystyle=$
$\displaystyle\int_{[0,T]}\mathbb{L}\nu\,dt-\iota\left(-x_{3}\nabla^{2}_{\hat{x}}\mathfrak{b}(\nu)+e_{\hat{x}}(\mathfrak{a}(\nu))\right)-\text{sym}\left(0|0|\boldsymbol{g}(\nu)\right).$
(3.24)

Since $\nu\in L^{1}(0,T)$ is arbitrary the existence of $\mathfrak{a}$,
$\mathfrak{b}$, $\boldsymbol{w}$, $\boldsymbol{g}$ in appropriate spaces (see
(3.1)-(3.3)) that satisfy (3.8) follows from (3.16), (3.18) and (3.21)-(3.24).

In order to obtain (3.7) and (3.9), we have from Proposition 2.3

$\displaystyle\frac{\varepsilon}{h^{2}}\left\|e_{h}\left(\boldsymbol{u}^{h}\right)\right\|_{H^{1}(0,T;L^{2}(\Omega^{h}_{f};\mathbb{R}^{3\times
3}))}\leq C,$ (3.25)

therefore from (3.14)

$\displaystyle\frac{\varepsilon}{h^{2}}\left\|e_{h}\left(\boldsymbol{u}^{h}_{f}\right)\right\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3\times
3}))}\leq C.$ (3.26)

From Lemma 2.2 we have

$\displaystyle\frac{1}{h^{2}}\|\boldsymbol{u}^{h}_{f}(t)\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))}\leq
C\frac{\varepsilon}{h^{2}}\|\nabla_{h}\boldsymbol{u}^{h}_{f}(t)\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3\times
3}))}\leq
C\frac{\varepsilon}{h^{2}}\|e_{h}(\boldsymbol{u}^{h}_{f}(t))\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3\times
3}))}.$ (3.27)

From, (3.26)-(3.27) we obtain

$\displaystyle\displaystyle\frac{1}{h^{2}}\|\boldsymbol{u}^{h}_{f}(t)\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))}$
$\displaystyle\leq C,$ (3.28)
$\displaystyle\displaystyle\frac{\varepsilon}{h^{2}}\|\nabla_{h}\boldsymbol{u}^{h}_{f}(t)\|_{L^{\infty}(0,T;L^{2}(\Omega;\mathbb{R}^{3\times
3}))}$ $\displaystyle\leq C.$ (3.29)

By application of Lemma A.4 for
$\boldsymbol{V}^{h}=\displaystyle\frac{1}{h^{2}}\boldsymbol{u}^{h}_{f}$ there
exists a function $\boldsymbol{u}^{0}_{f}\in
L^{\infty}(0,T;L^{2}(\Omega;H^{1}(\mathcal{Y};\mathbb{R}^{3})))$ and a
subsequence (not relabelled) such that (3.7) and (3.9) holds. (3.10) folows
from (3.9), (3.14) and (3.25). The fact that $\boldsymbol{u}_{f}^{0}$ is
supported in $\mathcal{Y}_{f}$ follows from the fact that
$\boldsymbol{u}_{f}^{h}$ is zero outside $\Omega_{f}^{h}$. In order to
conclude that $\boldsymbol{u}^{0}_{f}\in
H^{1}(0,T;L^{2}(\Omega;H^{1}(\mathcal{Y}_{f}(x);\mathbb{R}^{3})))$ we will use
(3.25). From (3.14) and (3.25), using the fact that $\varepsilon\ll h$, we
conclude that $e_{y}(\boldsymbol{u}_{f}^{0})\in H^{1}(0,T;L^{2}(\Omega\times
Y;\mathbb{R}^{3\times 3}))$. From this, by an approximation with convolution
and using Korn’s inequality we obtain that $\boldsymbol{u}^{0}_{f}\in
H^{1}(0,T;L^{2}(\Omega;H^{1}(\mathcal{Y}_{f}(x);\mathbb{R}^{3})))$. The
property ${\rm div}_{y}\boldsymbol{u}_{f}^{0}={\rm
tr}\,e_{y}(\boldsymbol{u}_{f}^{0})=0$ follows from convergence (3.9), (3.14)
and the property $\frac{\varepsilon}{h^{2}}{\rm div}\boldsymbol{u}^{h}=0$ on
$\Omega^{h}_{f}$. $\boldsymbol{u}_{f}^{0}(0)=0$ follows from (2.21), (3.9) and
(3.10). This establishes (3.4).

Since for every $\nu\in L^{1}(0,T)$ we have from (3.16),(3.28) and (3.17)

$0=\pi_{1/h}\int_{\Omega}\kappa^{h}(\boldsymbol{u}^{h})^{\nu}=\pi_{1/h}\int_{\Omega}\kappa^{h}(\hat{\boldsymbol{u}}^{h}+\boldsymbol{u}_{f}^{h})^{\nu}\to\lim_{h\to
0}\left(\pi_{1/h}C^{h}(\nu)\int_{\Omega}\kappa^{h}\right),$

we conclude that $\lim_{h\to 0}\left(\pi_{1/h}C^{h}(\nu)\right)=0$. From
(3.16) we conclude (3.6).

In order to show (3.12) we take $\varphi\in C^{1}_{\\#}(\Omega)$, which is
periodic in $x_{1},x_{2}$ and compute

$\displaystyle
0=-\displaystyle\int_{\Omega^{h}_{f}}\left(\frac{\partial_{1}u^{h}_{1}}{h}+\frac{\partial_{2}u^{h}_{2}}{h}+\frac{\partial_{3}u^{h}_{3}}{h^{2}}\right)\varphi(x)dx=-\displaystyle\int_{\Omega^{h}_{f}}\text{div}_{h}\frac{\boldsymbol{u}^{h}}{h}\varphi(x)dx$
$\displaystyle\quad=-\displaystyle\int_{\Omega^{h}_{f}}\left(\frac{\partial_{1}\hat{u}^{h}_{1}}{h}+\frac{\partial_{2}\hat{u}^{h}_{2}}{h}+\frac{\partial_{3}\hat{u}^{h}_{3}}{h^{2}}\right)\varphi(x)dx-\displaystyle\int_{\Omega^{h}_{f}}\left(\frac{\partial_{1}u^{h}_{f,1}}{h}+\frac{\partial_{2}u^{h}_{f,2}}{h}+\frac{\partial_{3}u^{h}_{f,3}}{h^{2}}\right)\varphi(x)dx$
$\displaystyle\quad=-\displaystyle\int_{\Omega^{h}_{f}}\left(\frac{\partial_{1}\hat{u}^{h}_{1}}{h}+\frac{\partial_{2}\hat{u}^{h}_{2}}{h}+\frac{\partial_{3}\hat{u}^{h}_{3}}{h^{2}}\right)\varphi(x)dx+\displaystyle\int_{\Omega^{h}_{f}}\left(\frac{u^{h}_{f,1}}{h}\partial_{1}\varphi(x)+\frac{u^{h}_{f,2}}{h}\partial_{2}\varphi(x)+\frac{u^{h}_{f,3}}{h^{2}}\partial_{3}\varphi(x)\right)dx$
$\displaystyle\rightarrow-\displaystyle\int_{\Omega}|\mathcal{Y}_{f}(x)|\text{div}_{\hat{x}}\mathfrak{a}(\hat{x},t)\,\varphi(x)dx+\displaystyle\int_{\Omega}|\mathcal{Y}_{f}(x)|x_{3}\text{div}_{\hat{x}}\nabla_{\hat{x}}\mathfrak{b}(\hat{x},t)\,\varphi(x)dx$
$\displaystyle\quad-\displaystyle\int_{\Omega}\int_{\mathcal{Y}_{f}(x)}\text{div}_{y}\boldsymbol{w}(t,x,y)\varphi(x)\,dydx$
$\displaystyle\quad-\displaystyle\int_{\Omega}|\mathcal{Y}_{f}(x)|g_{3}(x,t)\varphi(x)\,dx+\displaystyle\int_{\Omega}\int_{\mathcal{Y}_{f}(x)}u^{f}_{0,3}(t,x,y)\partial_{3}\varphi(x)dydx.$
(3.30)

(3.12) follows directly from (3.30) by taking the time derivative.

The convergence (3.11) (on a subsequence) for some $p\in
H^{-1}(0,T;L^{2}(\Omega\times\mathcal{Y}_{f}^{x}))$ is a direct consequence of
Proposition 2.5. To prove its independence of $y$

$p=p(x,t)\,\,\text{in}\,\,\Omega\times\mathcal{Y}_{f}^{x}\times(0,T),$ (3.31)

we can choose as test function $\boldsymbol{v}^{h}(x,t)=\varepsilon
h\sum_{i=1}^{m}\boldsymbol{\tau}^{i}\left(\displaystyle\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)\xi^{i}(x)\nu(t)$,
such that $\nu\in C^{1}_{c}[0,T]$, $\xi^{i}\in C_{c}^{1}(\Omega)$,
$\textrm{supp }\xi^{i}\subset U_{i}$, $\boldsymbol{\tau}^{i}\in
C_{c}^{1}(\mathcal{Y};\mathbb{R}^{3})$, $\text{supp
}\boldsymbol{\tau}^{i}\subset\mathcal{Y}_{f}^{i}$ for $i=1,\dots,m$, plug it
in (2.43) and integrate over $(0,T)$. We obtain

$\displaystyle
0\leftarrow-\displaystyle\int_{0}^{T}\int_{\Omega^{h}_{f}}\frac{p^{h}}{h}\cdot\frac{\text{div}_{h}\boldsymbol{v}^{h}}{h}dx\,dt=-\displaystyle\int_{0}^{T}\int_{\Omega}\frac{p^{h}}{h}\chi_{\Omega^{h}_{f}}\cdot\frac{\text{div}_{h}\boldsymbol{v}^{h}}{h}dx\,dt.$

We have as a consequence of Proposition 2.3

$\displaystyle 0=-\lim_{h\to
0}\displaystyle\int_{0}^{T}\int_{\Omega^{h}_{f}}\frac{p^{h}}{h}\cdot\frac{\text{div}_{h}\boldsymbol{v}^{h}}{h}dx\,dt=\displaystyle\sum_{i=1}^{n}\int_{0}^{T}\int_{\Omega}\int_{\mathcal{Y}}\chi_{\mathcal{Y}_{f}(x)}(y)\cdot
p(t,x,y)\cdot\text{div}_{y}\boldsymbol{\tau}^{i}(y)\xi^{i}(x)\nu(t)\,dy\,dx\,dt$
$\displaystyle=\displaystyle\sum_{i=1}^{n}\int_{0}^{T}\int_{\Omega}\int_{\mathcal{Y}_{f}(x)}p(t,x,y)\cdot\text{div}_{y}\boldsymbol{\tau}^{i}(y)\xi^{i}(x)\nu(t)\,dy\,dx\,dt$
$\displaystyle=-\displaystyle\sum_{i=1}^{n}\int_{0}^{T}\int_{\Omega}\int_{\mathcal{Y}_{f}(x)}\nabla_{y}p(t,x,y)\cdot\boldsymbol{\tau}^{i}(y)\xi^{i}(x)\nu(t)\,dy\,dx\,dt$
$\displaystyle=-\sum_{i=1}^{n}\displaystyle\int_{0}^{T}\int_{\Omega}\langle\nabla_{y}p(t,x,y),\boldsymbol{\tau}^{i}(y)\rangle_{\mathcal{Y}_{f}(x)}\xi^{i}(x)\nu(t)\,d\hat{x}\,dt.$
(3.32)

Here $\nabla_{y}p$ is the distributional gradient of $p$. Finally, (3.31)
follows directly from (3.32). ∎

######  Remark 3.1.

In (3.30) we used the fact that $u_{f,3}^{h}$ is zero on
$\omega\times\\{-\frac{1}{2},\frac{1}{2}\\}$ which is a consequence of our
geometrical assumption (2.1). We will discuss in Section 3.6 what if this is
not satisfied.

###  3.2 Effective tensors

Before stating the result on limit equations we need to define the effective
tensors.

  1. 1.

We define the effective forth order tensor: For $\hat{x}\in\omega$ and
$\boldsymbol{A},\boldsymbol{B},\boldsymbol{C},\boldsymbol{D}\in\mathbb{R}^{2\times
2}_{\text{sym}}$ we define

$\displaystyle\mathbb{A}^{hom}(\hat{x})(\boldsymbol{A},\boldsymbol{B}):(\boldsymbol{C},\boldsymbol{D})=$
$\displaystyle\displaystyle\quad\int_{I}\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\left[\iota(\boldsymbol{A}-x_{3}\boldsymbol{B})+\mathfrak{C}_{\infty}(\boldsymbol{w}_{1,x},\boldsymbol{g}_{1,x})\right]:\left[\iota(\boldsymbol{C}-x_{3}\boldsymbol{D})\right]dydx_{3},$
(3.33)

where $(\boldsymbol{w}_{1,x},\boldsymbol{g}_{1,x})\in
L^{2}(I;\dot{H}^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}))\times
L^{2}(I;\mathbb{R}^{3})$ is a unique solution of

$\displaystyle\displaystyle\int_{I}\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\left[\iota(\boldsymbol{A}-x_{3}\boldsymbol{B})+\mathfrak{C}_{\infty}(\boldsymbol{w}_{1,x},\boldsymbol{g}_{1,x})\right]:\mathfrak{C}_{\infty}(\boldsymbol{\zeta},\boldsymbol{r})\,dy\,dx_{3}=0;$
(3.34) $\displaystyle\forall\boldsymbol{\zeta}\in
L^{2}(I;H^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3})),\ \boldsymbol{r}\in
L^{2}(I;\mathbb{R}^{3}).$

  2. 2.

For $x\in\Omega$ we define the second order tensor

$\mathbb{B}^{H}(x)=\displaystyle\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\,\mathfrak{C}_{\infty}\left(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x}\right)\,dy,$
(3.35)

where
$(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x})\in\dot{H}^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3})\times\mathbb{R}^{3}$
are unique solutions of

$\displaystyle\displaystyle\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\,\mathfrak{C}_{\infty}(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x}):\mathfrak{C}_{\infty}(\boldsymbol{\zeta},\boldsymbol{r})\,dy$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\mathcal{Y}_{f}(x)}\,\left(\text{div}_{y}\,\boldsymbol{\zeta}+r_{3}\right)\,dy,$
(3.36)
$\displaystyle\forall\boldsymbol{\zeta}\in{H}^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}),\
\boldsymbol{r}\in\mathbb{R}^{3}.$

  3. 3.

For $x\in\Omega$ we define the second order tensor

$\mathbb{C}^{H}_{ij}(x)=-\displaystyle\int_{\mathcal{Y}_{f}(x)}\text{tr}\,\mathfrak{C}_{\infty}(\boldsymbol{w}^{ij}_{x},\boldsymbol{g}^{ij}_{x})\,dy,$
(3.37)

where for $i,j=1,2,3$
$\boldsymbol{w}^{ij}_{x}\in\dot{H}^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3})$ and
$\boldsymbol{g}^{ij}_{x}\in\mathbb{R}^{3}$ are unique solutions of the
following cell problems:

$\displaystyle\displaystyle\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\left[\boldsymbol{e}^{i}\odot\boldsymbol{e}^{j}+\mathfrak{C}_{\infty}(\boldsymbol{w}_{x}^{ij},\boldsymbol{g}_{x}^{ij})\right]:\mathfrak{C}_{\infty}(\boldsymbol{\zeta},\boldsymbol{r})\,dy=0,\quad\forall\boldsymbol{\zeta}\in{H}^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}),\
\boldsymbol{r}\in\mathbb{R}^{3}.$ (3.38)

  4. 4.

For $x\in\Omega$ we define the second order tensor

$\mathbb{K}_{ij}(x)=\displaystyle\int_{\mathcal{Y}_{f}(x)}q^{i}_{j,x}\,dy.\quad\text{(the
permeability tensor)},\quad i,j=1,2,3,$ (3.39)

where for $i=1,2,3$, $\mathbf{q}^{i}_{x}\in
H^{1}(\mathcal{Y}_{f}(x);\mathbb{R}^{3})$, $\pi^{i}_{x}\in
H^{-1}(\mathcal{Y}_{f}(x))$ are the solutions of

$\begin{cases}-\Delta_{y}\boldsymbol{q}^{i}_{x}+\nabla_{y}\pi^{i}_{x}=\boldsymbol{e}^{i},\quad\text{in}\,\mathcal{Y}_{f}(x)&\\\
\text{div}_{y}\boldsymbol{q}^{i}_{x}=0,\quad\text{in }\,\mathcal{Y}_{f}(x)&\\\
\left.\boldsymbol{q}^{i}_{x}\right|_{\partial\mathcal{Y}_{f}(x)\setminus\partial\mathcal{Y}}=0,\quad\\{\boldsymbol{q}^{i}_{x},\pi^{i}_{x}\\}\,\,\mathcal{Y}-\text{periodic}\end{cases}$
(3.40)

  5. 5.

For $x\in\Omega$, we define scalar

$M_{0}(x):=\displaystyle\int_{\mathcal{Y}_{f}(x)}\text{tr}\,\mathfrak{C}_{\infty}(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x})\,dy,$

where again
$(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x})\in\dot{H}^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3})\times\mathbb{R}^{3}$
are unique solutions of (3.37).

The following proposition gives us the important properties of the tensors
defined above

######  Proposition 3.1.

Let the geometry assumption be satisfied. The solutions of (3.34), (3.36),
(3.38), (3.57) exist and are unique. The tensors $\mathbb{A}^{hom}$,
$\mathbb{B}^{H}$, $\mathbb{C}^{H}$, $\mathbb{K}$, $M_{0}$ are piecewise
constant and following properties are satisfied:

  1. 1.

For all $\hat{x}\in\omega$, the tensor $\mathbb{A}^{hom}$ is symmetric and
there exists $\nu_{\mathbb{A}^{hom}}$ such that

$\nu_{\mathbb{A}^{hom}}^{-1}(|\mathbf{A}|^{2}+|\mathbf{B}|^{2})\leq\mathbb{A}^{hom}(\hat{x})(\boldsymbol{A},\boldsymbol{B}):(\boldsymbol{A},\boldsymbol{B})\leq\nu_{\mathbb{A}^{hom}}(|\mathbf{A}|^{2}+|\mathbf{B}|^{2}),\quad\forall\mathbf{A},\mathbf{B}\in\mathbb{R}_{\rm
sym}^{2\times 2},\forall\hat{x}\in\omega.$

  2. 2.

For all $x\in\Omega$, the tensors $\mathbb{B}^{H}(x)$, $\mathbb{C}^{H}(x)$ are
symmetric and $\mathbb{B}^{H}(x)=\mathbb{C}^{H}(x)$.

  3. 3.

For all $x\in\Omega$, the tensor $\mathbb{K}(x)$ is symmetric and positive
semidefinite and there exists $\nu_{\mathbb{K}}>0$ such that

$\nu_{\mathbb{K}}\leq\mathbb{K}_{ii}(x)\leq\nu_{\mathbb{K}}^{-1},\quad
i=1,2,3,\forall x\in\Omega.$ (3.41)

  4. 4.

There exists $\nu_{M_{0}}>0$ such that

$M_{0}(x)\geq\nu_{M_{0}}>0,\quad\forall x\in\Omega.$ (3.42)

###### Proof.

The existence and uniqueness of the solutions of (3.34), (3.36), (3.38),
(3.57) go by standard arguments, using Lax-Milgram and (2.5). To prove 1.
notice that

$\left(\boldsymbol{w}_{1,x},\boldsymbol{g}_{1,x}\right)=\displaystyle\sum_{\alpha,\beta=1,2}[\boldsymbol{A}_{\alpha\beta}\left(\boldsymbol{w}^{\alpha\beta}_{x},\boldsymbol{g}^{\alpha\beta}_{x}\right)-x_{3}\boldsymbol{B}_{\alpha\beta}\left(\boldsymbol{w}^{\alpha\beta}_{x},\boldsymbol{g}^{\alpha\beta}_{x}\right)],$
(3.43)

where
$\left(\boldsymbol{w}^{\alpha\beta}_{x},\boldsymbol{g}^{\alpha\beta}_{x}\right)$
are defined in (3.38). By using the extension on $\mathcal{Y}_{s}(x)$ (see
[20, Chapter 4]), for each $\boldsymbol{w}^{\alpha\beta}_{x}$, for
$\alpha,\beta=1,2$, 1. follows the proof of [7, Proposition 3.4]. To prove 2.,
notice that the symmetricity of $\mathbb{C}^{H}$ is obvious, since
$\boldsymbol{w}^{ij}_{x}=\boldsymbol{w}^{ji}_{x}$,
$\boldsymbol{g}^{ij}_{x}=\boldsymbol{g}^{ji}_{x}$, for $i,j=1,2,3$. We prove
$\mathbb{B}^{h}=\mathbb{C}^{H}$.

Using
$\left(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x}\right)$
as test functions in (3.38) and by virtue of (3.36), we obtain that

$\displaystyle\mathbb{B}^{H}_{ij}(x)$
$\displaystyle=\displaystyle\int_{\mathcal{Y}_{s}(x)}\left(\mathbb{A}(x)\mathfrak{C}_{\infty}\left(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x}\right)\right)_{ij}\,dy=\displaystyle\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\mathfrak{C}_{\infty}\left(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x}\right):\left(\boldsymbol{e}^{i}\odot\boldsymbol{e}^{j}\right)\,dy$
$\displaystyle=-\displaystyle\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\mathfrak{C}_{\infty}\left(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x}\right):\mathfrak{C}_{\infty}\left(\boldsymbol{w}^{ij}_{x},\boldsymbol{g}^{ij}_{x}\right)\,dy=-\displaystyle\int_{\mathcal{Y}_{f}(x)}\text{tr}\,\mathfrak{C}_{\infty}(\boldsymbol{w}^{ij}_{x},\boldsymbol{g}^{ij}_{x})\,dy=\mathbb{C}^{H}_{ij}(x).$

We prove 3. By testing (3.40) with $\mathbf{q}^{j}_{x}$ for $j=1,2,3$ we
obtain

$\mathbb{K}_{ij}(x)=\langle\nabla\mathbf{q}^{i}_{x},\nabla\mathbf{q}^{j}_{x}\rangle_{L^{2}(\mathcal{Y}_{f}(x);\mathbb{R}^{3})},\quad
i,j=1,2,3.$

(3.41) follows from the fact that $\mathbf{q}^{i}_{x}$ cannot be constant
(uniformity with respect to $x$ is the consequence of the geometric
assumptions, where only finite number of different domains $\mathcal{Y}_{f}$
appear).

It remains to prove 4. By virtue of (3.36) we have

$\displaystyle M_{0}(x)$
$\displaystyle=\displaystyle\int_{\mathcal{Y}_{f}(x)}\text{tr}\,\mathfrak{C}_{\infty}(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x})\,dy=\displaystyle\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\,\mathfrak{C}_{\infty}(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x}):\mathfrak{C}_{\infty}(\widetilde{\boldsymbol{w}}_{2,x},\widetilde{\boldsymbol{g}}_{2,x})\,dy>0.$
(3.44)

Again, the uniformity with respect to $x$, is the consequence of the geometric
assumptions.

∎

###  3.3 Limit equations

The following theorem characterizes the limit obtained by the compactness
result in Theorem 3.1.

######  Theorem 3.2.

Let assumptions (2.32) and (2.33) be satisfied. We also assume that

$\pi_{h}\boldsymbol{F}^{h}\xrightharpoonup{t,dr-2,2}\boldsymbol{F},$

where $\boldsymbol{F}\in
H^{1}(0,T;L^{2}(\Omega\times\mathcal{Y};\mathbb{R}^{3}))$. Then for the limit
$p$ obtained in obtained in (3.5) of Theorem 3.1, we have the additional
regularity $p\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;M)$ and the
limits (3.1)-(3.5) satisfy the following equations:

$\displaystyle\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b}):(e_{\hat{x}}(\boldsymbol{\theta}_{*}),\nabla^{2}_{\hat{x}}\theta_{3})d\hat{x}-\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)pIdx_{3}:\left[\iota(e_{\hat{x}}(\boldsymbol{\theta}_{*}))\right]\,d\hat{x}$
$\displaystyle+\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}pIdx_{3}:\left[\iota(\nabla^{2}_{\hat{x}}\theta_{3})\right]\,d\hat{x}$
$\displaystyle=\displaystyle\int_{\omega}\overline{\langle\boldsymbol{F}\rangle_{Y}}\cdot(\boldsymbol{\theta}_{*},\theta_{3})\,d\hat{x}-\displaystyle\int_{\omega}\overline{\langle
x_{3}\boldsymbol{F}_{*}\rangle_{Y}}\cdot\nabla_{\hat{x}}\theta_{3}\,d\hat{x},$
$\displaystyle\quad\forall\left(\boldsymbol{\theta}_{*},{\theta}_{3}\right)\in
L_{0},\quad\textrm{for a.e. }t\in(0,T),$ (3.45)
$\displaystyle\displaystyle-\int_{0}^{T}\int_{\Omega}M_{0}(x)p\,\partial_{t}\varphi
dx\,dt+\displaystyle\int_{0}^{T}\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}p\,\partial_{3}\varphi
dx\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))\partial_{t}\varphi
Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}\right)\right)d\hat{x}\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\partial_{t}\varphi
Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}\right)\,d\hat{x}\,dt=0,$
$\displaystyle\forall\varphi\in L^{2}(0,T;M)\cap
H^{1}(0,T;L^{2}(\Omega))\textrm{ such that }\varphi(T)=0.$ (3.46)

###### Proof.

To obtain the limit equations, we choose the test function
$\boldsymbol{v}^{h}\varphi$ in $(\ref{varformporous2gamma0may10})$, where
$\varphi\in C^{1}([0,T])$ such that $\varphi(T)=0$ and $\boldsymbol{v}^{h}$ is
defined with

$\boldsymbol{v}^{h}(x)=\displaystyle
h\boldsymbol{\theta}^{h}(x)+h\varepsilon\boldsymbol{\zeta}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\boldsymbol{\xi}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\displaystyle\int_{0}^{x_{3}}\boldsymbol{r}(x)\,dx_{3},$

and integrate (2.20) over the interval $[0,T]$. Here

$\boldsymbol{\theta}^{h}(x)=\begin{pmatrix}\theta_{1}(\hat{x})-x_{3}\partial_{1}\theta_{3}(\hat{x})\\\
\theta_{2}(\hat{x})-x_{3}\partial_{2}\theta_{3}(\hat{x})\\\
h^{-1}\theta_{3}(\hat{x})\end{pmatrix},$

$\boldsymbol{\theta}_{*}\in\dot{C}^{1}_{c}(\omega;\mathbb{R}^{2})$,
${\theta}_{3}\in\dot{C}^{2}_{c}(\omega)$, $\boldsymbol{\zeta}\in
C^{1}_{c}(\Omega;C^{1}_{\\#}(\mathcal{Y};\mathbb{R}^{3}))$,
$\boldsymbol{\xi}\in
C^{1}_{c}(\Omega;C^{1}_{\\#}(\mathcal{Y};\mathbb{R}^{3}))$,
$\boldsymbol{\xi}=\sum_{i=1}^{m}\boldsymbol{\tau}^{i}\left(\displaystyle\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)\xi^{i}(x)$,
such that $\xi^{i}\in C_{c}^{1}(\Omega)$, $\textrm{supp }\xi^{i}\subset
U_{i}$, $\boldsymbol{\tau}^{i}\in C_{c}^{1}(\mathcal{Y};\mathbb{R}^{3})$,
$\text{supp }\boldsymbol{\tau}^{i}\subset\mathcal{Y}_{f}^{i}$,
$\textrm{div}_{y}\boldsymbol{\tau}^{i}=0$, for $i=1,\dots,n$ and
$\boldsymbol{r}\in C^{1}_{c}(\Omega)^{3}$. Then as a consequence of (2.6),
(2.42) we have

$\displaystyle\displaystyle\eta\int_{0}^{T}\frac{d^{2}}{dt^{2}}\int_{\Omega}\kappa^{h}\boldsymbol{u}^{h}(t)\boldsymbol{v}^{h}(x)\varphi(t)\,dx\,dt=\eta\int_{0}^{T}\int_{\Omega}\kappa^{h}\partial_{t}\boldsymbol{u}^{h}(t)\boldsymbol{v}^{h}(x)\varphi^{\prime}(t)\,dx\,dt\rightarrow
0,$ (3.47) $\displaystyle
2\displaystyle\frac{\varepsilon}{h^{2}}\int_{0}^{T}\int_{\Omega_{1}}e_{h}\left(\partial_{t}\boldsymbol{u}^{h}_{f}\right):\frac{1}{h^{2}}e_{h}\left(\varepsilon\boldsymbol{v}^{h}\right)\varphi\,dx\,dt$
$\displaystyle\quad\rightarrow\displaystyle
2\int_{0}^{T}\int_{\Omega}\int_{\mathcal{Y}_{f}}e_{y}\left(\partial_{t}\boldsymbol{u}^{0}_{f}\right):e_{y}(\boldsymbol{\xi})\varphi
dydxdt,$ (3.48)
$\displaystyle\displaystyle\int_{0}^{T}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}(x)\frac{1}{h}e_{h}(\hat{\boldsymbol{u}}^{h}(t)):$
$\displaystyle\hskip 86.11084pt\frac{1}{h}e_{h}\left(\displaystyle
h\boldsymbol{\theta}^{h}(x)+h\varepsilon\boldsymbol{\zeta}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\boldsymbol{\xi}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\displaystyle\int_{0}^{x_{3}}\boldsymbol{r}(x)\,dx_{3}\right)\varphi\,dx\,dt$
$\displaystyle\quad\rightarrow\displaystyle\int_{0}^{T}\int_{\Omega}\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\left[\iota(e_{\hat{x}}(\mathfrak{a})-x_{3}\nabla^{2}_{\hat{x}}\mathfrak{b})+\mathfrak{C}_{\infty}(\boldsymbol{w},\boldsymbol{g})\right]:$
$\displaystyle\hskip
129.16626pt\left[\iota(e_{\hat{x}}(\boldsymbol{\theta}_{*})-x_{3}\nabla^{2}_{\hat{x}}\theta_{3})+\mathfrak{C}_{\infty}(\boldsymbol{\zeta},\boldsymbol{r})\right]\varphi\,dy\,dx\,dt,$
(3.49)
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\Omega_{f}^{h}}\frac{p^{h}}{h}\cdot\frac{1}{h}\text{div}_{h}\,\left(\displaystyle
h\boldsymbol{\theta}^{h}(x)+h\varepsilon\boldsymbol{\zeta}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\boldsymbol{\xi}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\displaystyle\int_{0}^{x_{3}}\boldsymbol{r}(x)\,dx_{3}\right)\varphi\,dx\,dt$
$\displaystyle\quad\rightarrow\displaystyle\int_{0}^{T}\int_{\Omega}\int_{\mathcal{Y}_{f}}-p\left(\text{div}_{\hat{x}}\,\left(\boldsymbol{\theta}_{*}-x_{3}\nabla_{\hat{x}}\theta_{3}\right)+\text{div}_{y}\,\boldsymbol{\zeta}+r_{3}(x)+\partial_{3}\xi_{3}\right)\varphi\,dy\,dx\,dt,$
(3.50)
$\displaystyle\displaystyle\int_{0}^{T}\int_{\Omega}\left(\frac{F_{1}}{h},\frac{F_{2}}{h},F_{3}\right)\left(\displaystyle
h\boldsymbol{\theta}^{h}(x)+h\varepsilon\boldsymbol{\zeta}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\boldsymbol{\xi}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)+h^{2}\displaystyle\int_{0}^{x_{3}}\boldsymbol{r}(x)\,dx_{3}\right)\varphi\,dx\,dt$
$\displaystyle\rightarrow\displaystyle\int_{0}^{T}\int_{\omega}\overline{\langle\boldsymbol{F}\rangle_{Y}}\cdot(\boldsymbol{\theta}_{*},\theta_{3})\varphi\,d\hat{x}\,dt-\displaystyle\int_{0}^{T}\int_{\omega}\overline{\langle
x_{3}\boldsymbol{F}_{*}\rangle_{\mathcal{Y}}}\cdot\nabla_{\hat{x}}\theta_{3}\varphi\,d\hat{x}\,dt.$
(3.51)

From this we conclude for a.e. $t\in[0,T]$:

$\displaystyle\displaystyle\int_{\Omega}\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\left[\iota(e_{\hat{x}}(\mathfrak{a})-x_{3}\nabla^{2}_{\hat{x}}\mathfrak{b})+\mathfrak{C}_{\infty}(\boldsymbol{w},\boldsymbol{g})\right]:\left[\iota(e_{\hat{x}}(\boldsymbol{\theta}_{*})-x_{3}\nabla^{2}_{\hat{x}}\theta_{3})\right]\,dy\,dx$
$\displaystyle\quad+\int_{\Omega}|\mathcal{Y}_{f}(x)|x_{3}p\,(\text{div}_{\hat{x}}\nabla_{\hat{x}}\theta_{3})\,d\hat{x}-\displaystyle\int_{\Omega}|\mathcal{Y}_{f}(x)|p\,\text{div}_{\hat{x}}\left(\boldsymbol{\theta}_{*}\right)\,dx$
$\displaystyle=\displaystyle\int_{\omega}\overline{\langle\boldsymbol{F}\rangle_{Y}}\cdot(\boldsymbol{\theta}_{*},\theta_{3})\,d\hat{x}-\displaystyle\int_{\omega}\overline{\langle
x_{3}\boldsymbol{F}_{*}\rangle_{Y}}\cdot\nabla_{\hat{x}}\theta_{3}\,d\hat{x},\quad\forall\left(\boldsymbol{\theta}_{*},{\theta}_{3}\right)\in
L_{0},$ (3.52)
$\displaystyle\displaystyle\int_{I}\int_{\mathcal{Y}_{s}(x)}\mathbb{A}(x)\left[\iota(e_{\hat{x}}(\mathfrak{a})-x_{3}\nabla^{2}_{\hat{x}}\mathfrak{b})+\mathfrak{C}_{\infty}(\boldsymbol{w},\boldsymbol{g})\right]:\mathfrak{C}_{\infty}(\boldsymbol{\zeta},\boldsymbol{r})\,dy\,dx_{3}$
$\displaystyle\hskip
8.61108pt-\displaystyle\int_{I}\int_{\mathcal{Y}_{f}(x)}p\,\left(\text{div}_{y}\,\boldsymbol{\zeta}+r_{3}\right)\,dy\,dx_{3}=0,\
\quad\forall(\boldsymbol{\zeta},\boldsymbol{r})\in
L^{2}(\Omega,\dot{H}^{1}(\mathcal{Y};\mathbb{R}^{3}))\times
L^{2}(\Omega\times\mathcal{Y};\mathbb{R}^{3}),$ (3.53)
$\displaystyle\displaystyle
2\int_{\Omega}\int_{\mathcal{Y}_{f}(x)}e_{y}\left(\partial_{t}\boldsymbol{u}^{0}_{f}\right):e_{y}(\boldsymbol{\xi})dydx\displaystyle-\displaystyle\int_{\Omega}\int_{\mathcal{Y}_{f}(x)}p\,\partial_{3}\xi_{3}\,dx\,dy=0,\quad\forall\boldsymbol{\xi}\in
L^{2}(\omega;H^{1}(I;H^{1}(\mathcal{Y};\mathbb{R}^{3})))$
$\displaystyle\text{such
that}\,\,\boldsymbol{\xi}=0\quad\text{on}\,\,\Omega\times\mathcal{Y}^{x}_{s}\quad\text{and}\quad\text{div}_{y}\,\boldsymbol{\xi}=0\quad\text{on}\quad\Omega\times\mathcal{Y}^{x}_{f}.$
(3.54)

From (3.53) we conclude

$\displaystyle\left(\boldsymbol{w},\boldsymbol{g}\right)(t,x,y)$
$\displaystyle=p(x,t)\left(\widetilde{\boldsymbol{w}}_{2,x}(y),\widetilde{\boldsymbol{g}}_{2,x}\right)$
$\displaystyle\quad+\displaystyle\sum_{\alpha,\beta=1,2}\left[\left(e_{\hat{x}}(\boldsymbol{a}(t,\hat{x}))\right)_{\alpha\beta}\left(\boldsymbol{w}^{\alpha\beta}_{x}(y),\boldsymbol{g}^{\alpha\beta}_{x}\right)-x_{3}\partial_{\alpha\beta}\mathfrak{b}(t,\hat{x})\left(\boldsymbol{w}^{\alpha\beta}_{x}(y),\boldsymbol{g}^{\alpha\beta}_{x}\right)\right],$
(3.55)

where $\widetilde{\boldsymbol{w}}_{2},\widetilde{g}_{2}$ and
$\boldsymbol{w}^{\alpha\beta},\boldsymbol{g}^{\alpha\beta}$ are defined in
(3.36) and (3.38) respectively. We conclude that $p\in
L^{\infty}(0,T;L^{2}(\Omega))$. By substituting (3.55) into (3.52) we obtain
(3.45).

On the other hand, the differential form of (3.54) is

$-\Delta_{y}\partial_{t}\boldsymbol{u}^{0}_{f}+(0,0,\partial_{3}p)+\nabla_{y}\pi=(0,0,0)$
(3.56)

which we rewrite as

$\begin{cases}-\Delta_{y}\displaystyle\partial_{t}\boldsymbol{u}^{0}_{f}+\nabla_{y}\pi=-(0,0,\partial_{3}p),&\\\
\text{div}_{y}\displaystyle\partial_{t}\boldsymbol{u}^{0}_{f}=0,&\\\
\left.\boldsymbol{u}^{0}_{f}\right|_{\partial\mathcal{Y}_{f}}=0,\quad\\{\boldsymbol{u}^{0}_{f},\pi\\}\,\,\mathcal{Y}-\text{periodic}.&\end{cases}$
(3.57)

We conclude that $\partial_{t}\boldsymbol{u}^{0}_{f}$ and $\pi$ have a
representation:

$\begin{cases}\displaystyle\partial_{t}\boldsymbol{u}^{0}_{f}=-\boldsymbol{q}_{x}^{3}(y)\partial_{3}p(x,t),&\\\
\displaystyle\pi=-\pi^{3}_{\gamma}(y)\partial_{3}p(x,t),&\end{cases}$ (3.58)

where $\boldsymbol{q}^{3}_{x}$ is defined in (3.40). From (3.58) it follows
that $\partial_{3}p\in L^{2}(0,T;L^{2}(\Omega))$.

To prove (3.46) we use (3.12), which we multiply with $\varphi^{\prime}$, for
$\varphi\in H^{1}(0,T;M)$, $\varphi(T)=0$, and then integrate over
$[0,T]\times\Omega$. We then plug in (3.55) and (3.58) and use Proposition 3.1
2.

∎

######  Remark 3.2.

The first equation in (3.58) tells us that in the effective Darcy’s law only
the derivative of the pressure in the vertical direction is present (unlike in
the bulk model where the full gradient is present).

######  Remark 3.3.

To conclude (3.49) we needed (2.4), in order to neglect the term
$h^{2}\boldsymbol{\xi}$.

###  3.4 Analysis of the limit equations

We will slightly modify the system (3.45)-(3.46). We will analyze for
$\boldsymbol{F}\in H^{1}(0,T;L_{0}^{\prime})$, $\boldsymbol{G}\in
L^{2}(0,T;M^{\prime})$ and $t_{0}\in L^{2}(\Omega)$ the following problem:
find $(\mathfrak{a},\mathfrak{b},p)\in L^{2}(0,T;L_{0})\times L^{2}(0,T;M)$
that satisfy

$\displaystyle\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b}):(e_{\hat{x}}(\boldsymbol{\theta}_{*}),\nabla^{2}_{\hat{x}}\theta_{3})d\hat{x}-\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)pIdx_{3}:\left[\iota(e_{\hat{x}}(\boldsymbol{\theta}_{*}))\right]\,d\hat{x}$
$\displaystyle+\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}pIdx_{3}:\left[\iota(\nabla^{2}_{\hat{x}}\theta_{3})\right]\,d\hat{x}$
$\displaystyle={}_{L_{0}^{\prime}}\langle\boldsymbol{F}(t),(\boldsymbol{\theta}_{*},{\theta}_{3})\rangle_{L_{0}},$
$\displaystyle\quad\forall\left(\boldsymbol{\theta}_{*},{\theta}_{3}\right)\in
L_{0},\quad\textrm{for a.e. }t\in(0,T),$ (3.59)
$\displaystyle\displaystyle-\int_{0}^{T}\int_{\Omega}M_{0}(x)p\,\partial_{t}\varphi
dx\,dt+\displaystyle\int_{0}^{T}\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}p\,\partial_{3}\varphi
dx\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))\partial_{t}\varphi
Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}\right)\right)d\hat{x}\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\partial_{t}\varphi
Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}\right)\,d\hat{x}\,dt=\int_{0}^{T}{{}_{M^{\prime}}}\langle\boldsymbol{G},\varphi\rangle_{M}\,dt+\int_{\Omega}t_{0}\cdot\varphi(0)\,dx,$
$\displaystyle\forall\varphi\in L^{2}(0,T;M)\cap
H^{1}(0,T;L^{2}(\Omega))\textrm{ such that }\varphi(T)=0.$ (3.60)

Thus the loads $\overline{\langle\boldsymbol{F}\rangle_{Y}}$,
$\overline{\langle x_{3}\boldsymbol{F}_{*}\rangle_{\mathcal{Y}}}$ for
$\boldsymbol{F}\in H^{1}(0,T;L^{2}(\Omega\times\mathcal{Y};\mathbb{R}^{3}))$
of the system (3.45)-(3.46) are naturally replaced simply with
$\boldsymbol{F}\in H^{1}(0,T;L_{0}^{\prime})$, and we have additional terms on
the right hand side of (3.60) when compared with (3.46)

We will give the abstract framework for this problem. As we will see adding
the additional term on the right hand side of (3.46) will change the initial
condition for the pressure.

First we give the abstract framework for the system. We will adapt the
approach from [17]. Let $V$ be a separable Hilbert space with dual
$V^{\prime}$ (which is not identified with $V$ here). Assume that $V$ is
densely and continuously embedded into another Hilbert space $H$, which is
identified with its dual:

$V\hookrightarrow H\equiv H^{\prime}\hookrightarrow V^{\prime}.$

Consequently $H$ is densely and continuously embedded into $V^{\prime}$. We
denote the inner product on $H$ by $\langle\cdot,\cdot\rangle_{H}$ and the
norm induced by that scalar product by $\|\cdot\|_{H}$. We also denote by
${}_{V^{\prime}}\langle\cdot,\cdot\rangle_{V}$ the duality pairing between $V$
and $V^{\prime}$ and by $\|\cdot\|_{V}$ the norm on $V$. Suppose that
$\mathcal{B}_{0}\in L(H)$ and $\mathcal{A}_{0}\in L(V,V^{\prime})$. Let
$u_{0}\in H$ and $S\in L^{2}(0,T;V^{\prime})$ be given. We look the following
Cauchy problem: Find $u\in L^{2}(0,T;V)$ such that

$\begin{cases}\frac{d}{dt}[\mathcal{B}_{0}u]+\mathcal{A}_{0}u=S\in
L^{2}(0,T;V^{\prime})\\\ [\mathcal{B}_{0}u](0)=\mathcal{B}u_{0}\in
H.\end{cases}$ (3.61)

The time derivative in (3.61) is taken in the sense of distributions. We give
the following definition of the weak solution of (3.61) on the time interval
$(0,T)$.

######  Definition 3.1.

The function $u\in L^{2}(0,T;V)$ with

$-\int_{0}^{T}\langle\mathcal{B}_{0}u(t),v^{\prime}(t)\rangle_{H}\,dt+\int_{0}^{T}{{}_{V^{\prime}}}\langle\mathcal{A}_{0}u(t),v(t)\rangle_{V}\,dt=\int_{0}^{T}{{}_{V^{\prime}}}\langle
S(t),v(t)\rangle_{V}\,dt+\langle\mathcal{B}_{0}u_{0},v(0)\rangle_{H},$ (3.62)

holding for all

$v\in\\{w\in L^{2}(0,T;V)\cap H^{1}(0,T;H)\ :\ w(T)=0\\}$

is called a weak solution to (3.61).

######  Remark 3.4.

Notice that the solution of (3.61) satisfies $\mathcal{B}_{0}u\in
H^{1}(0,T;V^{\prime})$ and thus we can give a meaning to the initial value in
(3.61).

The following assumption will be needed for the existence result.

######  Assumption 3.1.

We assume that

  1. 1.

$\mathcal{A}_{0}\textrm{ is monotone on }V$, i.e.

${}_{V^{\prime}}\langle\mathcal{A}_{0}v,v\rangle_{V}\geq 0,\quad\forall v\in
V;$

  2. 2.

$\mathcal{B}_{0}$ is self-adjoint positive semidefinite on $H$;

  3. 3.

there exist constants $\lambda,c>0$ such that

$_{V^{\prime}}\langle\mathcal{A}_{0}v,v\rangle_{V}+\lambda\langle\mathcal{B}_{0}v,v\rangle_{H}\geq
c\|v\|_{V}^{2},\quad\forall v\in V.$ (3.63)

We have the following theorem as a consequence of [17, Theorem 2.1, Theorem
2.2].

######  Theorem 3.3.

Under the Assumption 3.1, there exist a unique solution of (3.61) in the sense
of Definition 3.1. The solution satisfies the following stability estimate:

$\|u\|^{2}_{L^{2}(0,T;V)}\leq
C(c,\lambda)\left[\|S\|^{2}_{L^{2}(0,T;V^{\prime})}+\langle\mathcal{B}_{0}u_{0},u_{0}\rangle_{H}\right],$
(3.64)

where $C(c,\lambda)>0$ depends only on $c$ and $\lambda$.

######  Remark 3.5.

If, in addition, we have that $\mathcal{B}_{0}$ is positive definite on $H$
then it follows from [11, Remark 1, Chapter XVIII.5] that the solution of
(3.62) is in $C([0,T];H)$ and there exists $C(c,\lambda)>0$ such that

$\|u\|^{2}_{C(0,T;H)}\leq
C(c,\lambda)\left[\|S\|^{2}_{L^{2}(0,T;V^{\prime})}+\langle\mathcal{B}_{0}u_{0},u_{0}\rangle_{H}\right].$
(3.65)

Next we state and prove the main result of this section.

######  Theorem 3.4.

For $\boldsymbol{F}\in H^{1}(0,T;L_{0}^{\prime})$, $\boldsymbol{G}\in
L^{2}(0,T;M^{\prime})$ there exist a unique solution
$(\mathfrak{a},\mathfrak{b},p)\in L^{2}(0,T;L_{0})\times L^{2}(0,T;M)$ of
(3.59)- (3.60).

###### Proof.

For $\hat{p}\in L^{2}(\Omega)$ and $\hat{\boldsymbol{F}}\in L_{0}^{\prime}$ we
introduce $(\mathfrak{a}^{\hat{p}}_{0,1},\mathfrak{b}^{\hat{p}}_{0,1})\in
L_{0}$ and
$(\mathfrak{a}_{0,2}^{\hat{\boldsymbol{F}}},\mathfrak{b}_{0,2}^{\hat{\boldsymbol{F}}})\in
L_{0}$ respectively that satisfy:

$\displaystyle\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}_{0,1}^{\hat{p}}),\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{\hat{p}}):(e_{\hat{x}}(\boldsymbol{\theta}_{*}),\nabla^{2}_{\hat{x}}\theta_{3})d\hat{x}=\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)\hat{p}Idx_{3}:\left[\iota(e_{\hat{x}}(\boldsymbol{\theta}_{*}))\right]\,d\hat{x}$
$\displaystyle-\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\hat{p}Idx_{3}:\left[\iota(\nabla^{2}_{\hat{x}}\theta_{3})\right]\,d\hat{x},$
$\displaystyle\quad\forall\left(\boldsymbol{\theta}_{*},{\theta}_{3}\right)\in
L_{0};$ (3.66)
$\displaystyle\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}_{0,2}^{\hat{\boldsymbol{F}}}),\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,2}^{\hat{\boldsymbol{F}}}):(e_{\hat{x}}(\boldsymbol{\theta}_{*}),\nabla^{2}_{\hat{x}}\theta_{3})d\hat{x}=_{L_{0}^{\prime}}\langle\boldsymbol{F}(t),(\boldsymbol{\theta}_{*},{\theta}_{3})\rangle_{L_{0}}$
$\displaystyle\quad\forall\left(\boldsymbol{\theta}_{*},{\theta}_{3}\right)\in
L_{0}.$ (3.67)

Assuming $\hat{p}$ and $\hat{\boldsymbol{F}}$ are known, the solutions of
(3.66)-(3.67) are unique as a consequence of Proposition 2.5 and Lax-Milgram.
The solution of (3.60) we write in the following way:

$\displaystyle-\displaystyle\int_{0}^{T}\int_{\Omega}M_{0}(x)p\,\partial_{t}\varphi
dx\,dt-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))\partial_{t}\varphi
Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}_{0,1}^{p}\right)\right)d\hat{x}\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\partial_{t}\varphi
Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{p}\right)\,d\hat{x}\,dt+\displaystyle\int_{0}^{T}\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}p\,\partial_{3}\varphi
dx\,dt$
$\displaystyle=\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))\partial_{t}\varphi
Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}_{0,2}^{\boldsymbol{F}}\right)\right)d\hat{x}\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\partial_{t}\varphi
Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,2}^{\boldsymbol{F}}\right)\,d\hat{x}\,dt+\int_{0}^{T}{{}_{M^{\prime}}}\langle
G,\varphi\rangle_{M}\,dt$
$\displaystyle+\int_{\Omega}t_{0}\cdot\varphi(0)\,dx,$
$\displaystyle\forall\varphi\in L^{2}(0,T;M)\cap
H^{1}(0,T;L^{2}(\Omega))\textrm{ such that }\varphi(T)=0.$ (3.68)

We put (3.68) in the framework of weak solution of (3.61), after integration
by parts on the right hand side. We now put $V:=M$, $H:=L^{2}(\Omega)$. The
operator $\mathcal{A}_{0}$ is defined through bilinear form

$_{V^{\prime}}\langle\mathcal{A}_{0}v_{1},v_{2}\rangle_{V}:=\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}v_{1}(x)\cdot\partial_{3}v_{2}(x)\,dx,\quad\forall
v_{1},v_{2}\in V.$ (3.69)

Obviously 1 of (3.1) is satisfied. The operator $\mathcal{B}_{0}$ is also
defined through bilinear form

$\displaystyle\langle\mathcal{B}_{0}h_{1},h_{2}\rangle_{H}$ $\displaystyle:=$
$\displaystyle\int_{\Omega}M_{0}(x)h_{1}(x)\cdot h_{2}(x)\,dx$
$\displaystyle+\displaystyle\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))h_{2}Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}_{0,1}^{h_{1}}\right)\right)d\hat{x}$
$\displaystyle-\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}h_{2}Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{h_{1}}\right)\,d\hat{x},$

where $(\mathfrak{a}_{0,1}^{h_{1}},\mathfrak{b}_{0,1}^{h_{1}})\in L_{0}$ are
defined as the solutions of (3.66) once $\hat{p}$ is replaced by $h_{1}$. From
(3.66) it follows:

$\displaystyle\displaystyle\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))h_{2}Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}_{0,1}^{h_{1}}\right)\right)d\hat{x}$
$\displaystyle-\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}h_{2}Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{h_{1}}\right)\,d\hat{x}$
$\displaystyle=\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}_{0,1}^{h_{1}}),\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{h_{1}}):(e_{\hat{x}}(\mathfrak{a}_{0,1}^{h_{2}}),\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{h_{2}})d\hat{x}$
$\displaystyle=\displaystyle\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))h_{1}Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}_{0,1}^{h_{2}}\right)\right)d\hat{x}$
$\displaystyle-\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}h_{1}Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{h_{2}}\right)\,d\hat{x}$

Using this and 4 of Proposition 2.5 we have the positive definitness of
$\mathcal{B}_{0}$ and thus Assumption 3.1 2 is satisfied. Also, it is easy to
see that Assumption 3.1 3 is satisfied. The operator $S(t)\in H$ is defined
for $t\in[0,T]$ in the following way

$\displaystyle\langle S(t),v\rangle_{H}$ $\displaystyle:=$
$\displaystyle-\displaystyle\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))vIdx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}_{0,2}^{\partial_{t}\boldsymbol{F}(t)}\right)\right)d\hat{x}$
$\displaystyle+\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}vIdx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,2}^{\partial_{t}\boldsymbol{F}(t)}\right)\,d\hat{x}$
$\displaystyle+_{V^{\prime}}\langle\boldsymbol{G}(t),v\rangle_{V},\ \forall
v\in V,$

where for $t\in[0,T]$,
$(\mathfrak{a}_{0,2}^{\partial_{t}\boldsymbol{F}(t)},\mathfrak{b}_{0,2}^{\partial_{t}\boldsymbol{F}(t)})\in
L_{0}$ are defined as solutions of (3.67) once $\hat{\boldsymbol{F}}$ is
replaced by $\partial_{t}\boldsymbol{F}(t)$. Since $H\hookrightarrow
V^{\prime}$, we have $S(t)\in V^{\prime}$. $p_{0}$ is defined through the
following relation:

$\displaystyle\langle\mathcal{B}_{0}p_{0},h\rangle_{H}$ $\displaystyle=$
$\displaystyle\displaystyle\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))hIdx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}_{0,2}^{\boldsymbol{F}(0)}\right)\right)d\hat{x}$
$\displaystyle-\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}hIdx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,2}^{\boldsymbol{F}(0)}\right)\,d\hat{x}+\int_{\Omega}t_{0}\cdot
h\,dx$ $\displaystyle=$
$\displaystyle\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}_{0,1}^{h}),\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,1}^{h}):(e_{\hat{x}}(\mathfrak{a}_{0,2}^{\boldsymbol{F}(0)}),\nabla^{2}_{\hat{x}}\mathfrak{b}_{0,2}^{\boldsymbol{F}(0)})d\hat{x}+\int_{\Omega}t_{0}\cdot
h\,dx,\quad\forall h\in H,$

where $(\mathfrak{a}^{\boldsymbol{F}(0)},\mathfrak{b}^{\boldsymbol{F}(0)})\in
L_{0}$ is defined by (3.67). Since $\mathcal{B}_{0}$ is a positive definite
bounded operator this defines unique $p_{0}\in H$ by Riesz representation
theorem. We apply Theorem 3.3 to obtain the unique $p\in L^{2}(0,T;V)$. The
uniqueness of $(\mathfrak{a},\mathfrak{b})\in L^{2}(0,T;L_{0})$ follows from
(3.45) and Lax-Milgram. ∎

From (3.4) we conclude that the initial condition $p_{0}$ of the system
(3.45)-(3.46) is related with the initial condition $\tilde{p}_{0}$ of the
system (3.59)-(3.60), after simple substitution of the loads, in the following
way

$\tilde{p}_{0}=p_{0}+\mathcal{B}^{-1}_{0}t_{0},$ (3.72)

where $\mathcal{B}_{0}$ is defined by $\eqref{krnj2}$.

Next, we define the operator $\widetilde{\mathcal{A}}_{0}$ as a self-adjoint
operator on $L^{2}(\Omega)$ defined through the bilinear form:

$\tilde{a}(v_{1},v_{2})=\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}v_{1}(x)\cdot\partial_{3}v_{2}(x)\,dx,\quad\forall
v_{1},v_{2}\in V.$ (3.73)

Note that $\mathcal{D}(\widetilde{\mathcal{A}}_{0})\subset V$ and the operator
$\mathcal{A}_{0}$ defined by (3.69) coincides with the operator
$\widetilde{\mathcal{A}}_{0}$ on $\mathcal{D}(\widetilde{\mathcal{A}}_{0})$.

######  Proposition 3.2.

If $\boldsymbol{F}\in H^{1}(0,T;L_{0}^{\prime})$ , $\boldsymbol{G}\in
L^{2}(0,T;V^{\prime})$, $t_{0}\in L^{2}(\Omega)$, then the solution of (3.59)-
(3.60) satisfies $(\mathfrak{a},\mathfrak{b},p)\in C([0,T];L_{0})\times
C([0,T];L^{2}(\Omega))$.

###### Proof.

The fact that $p\in C([0,T];L^{2}(\Omega))$ follows from (3.68) and Remark
3.5. The regularity of $\mathfrak{a}$, $\mathfrak{b}$ follows then from
(3.59). ∎

######  Remark 3.6.

The previous proposition fills the gap between the result of Theorem 3.2 and
Theorem 3.4. Namely as a result of Theorem 3.2 we have that the obtained limit
is in $L^{\infty}$ in time, while Theorem 3.4 guarantees only the existence of
solution, which is in $L^{2}$ in time.

######  Proposition 3.3.

If $\boldsymbol{F}\in H^{2}(0,T;L_{0}^{\prime})$, $\boldsymbol{G}\in
H^{1}(0,T;V^{\prime})$ and
$\tilde{p}_{0}\in\mathcal{D}(\widetilde{\mathcal{A}}_{0})$, where
$\tilde{p}_{0}$ is defined by (3.72) and $\widetilde{\mathcal{A}}_{0}$ is
defined through bilinear form (3.73), then the solution of (3.59)- (3.60)
satisfies $(\mathfrak{a},\mathfrak{b},p)\in H^{1}(0,T;L_{0})\times
H^{1}(0,T;M)$.

###### Proof.

It is easy to see, by taking the derivative in time in (3.61), that if in the
equation (3.61), one has that $\frac{d}{dt}S\in L^{2}(0,T;V^{\prime})$ and
$u_{0}\in V$ such that $\mathcal{A}_{0}u_{0}\in H$ then we have that
$\frac{d}{dt}u\in L^{2}(0,T;V)$. This is satisfied when one writes the system
(3.59) and (3.60) in the form of (3.61) if the conditions of the proposition
are satisfied (cf. (3.68)). From this we conclude the regularity in time for
$p$. To conclude the regularity in time for $\mathfrak{a}$ and $\mathfrak{b}$
we use (3.45). ∎

The following proposition is crucial for proving the convergence results in
Section 3.5.

######  Proposition 3.4.

If $\boldsymbol{F}\in H^{1}(0,T;L_{0}^{\prime})$ $\boldsymbol{G}\in
L^{2}(0,T;V^{\prime})$, $t_{0}\in L^{2}(\Omega)$, then the solution of
(3.59)-(3.60) satisfies the following identity for every $t\in(0,T)$ :

$\displaystyle\frac{1}{2}\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}(t)),\nabla^{2}_{\hat{x}}\mathfrak{b}(t))):(e_{\hat{x}}(\mathfrak{a}(t)),\nabla^{2}_{\hat{x}}\mathfrak{b}(t))d\hat{x}+\displaystyle\frac{1}{2}\int_{\Omega}M_{0}(x)p^{2}(t)dx$
$\displaystyle+\int_{0}^{t}\int_{\Omega}\mathbb{K}_{33}|\partial_{3}p|^{2}dx\,dt$
$\displaystyle={}_{M_{0}^{\prime}}\langle\boldsymbol{F}(t),(\mathfrak{a}(t),\mathfrak{b}(t))\rangle_{M_{0}}-_{M_{0}^{\prime}}\langle\boldsymbol{F}(0),(\mathfrak{a}(0),\mathfrak{b}(0))\rangle_{M_{0}}$
$\displaystyle-\int_{0}^{t}{{}_{M_{0}^{\prime}}}\langle\partial_{t}\boldsymbol{F}(\tau),(\mathfrak{a}(\tau),\mathfrak{b}(\tau))\rangle_{M_{0}}\,d\tau$
$\displaystyle+\int_{0}^{t}{{}_{V^{\prime}}}\langle\boldsymbol{G},p\rangle_{V}\,dt+\frac{1}{2}\int_{\Omega}M_{0}(x)(\tilde{p}_{0})^{2}\,dx,$
(3.74)

where $\tilde{p}_{0}$ is defined with (3.72) (and $p_{0}$ with (3.4)).

###### Proof.

First we look the system (3.59)-(3.60) assume that $\boldsymbol{F}\in
H^{2}(0,T;M_{0}^{\prime})$,
$\tilde{p}_{0}\in\mathcal{D}(\widetilde{\mathcal{A}})$. Using Proposition 3.3
we conclude that the solution of (3.45) and (3.60) satisfies
$(\mathfrak{a},\mathfrak{b},p)\in H^{1}(0,T;L_{0})\times H^{1}(0,T;V)$. By
doing integration by parts and density argument we conclude from (3.60) that
$p(0)=\tilde{p}_{0}$ and for a.e. $t\in(0,T)$ we have

$\displaystyle\displaystyle\int_{\Omega}M_{0}(x)\partial_{t}p\,\varphi
dx+\displaystyle\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}p\,\partial_{3}\varphi
dx$
$\displaystyle+\displaystyle\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))\varphi
Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{\partial_{t}a}\right)\right)d\hat{x}$
$\displaystyle+\displaystyle\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\varphi
Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\partial_{t}\mathfrak{b}\right)\,d\hat{x}={{}_{V^{\prime}}}\langle\boldsymbol{G},p\rangle_{V}\,dt$
$\displaystyle\forall\varphi\in V.$ (3.75)

We take $t\in[0,T]$. By testing (3.59) with
$(\partial_{\tau}\mathfrak{a},\partial_{\tau}\mathfrak{b})$, integrating over
$[0,t]$ and adding (3.75) tested with $\varphi=p$ and then integrated over
$(0,t)$ we obtain

$\displaystyle\frac{1}{2}\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}(t)),\nabla^{2}_{\hat{x}}\mathfrak{b}(t)):(e_{\hat{x}}(\mathfrak{a}(t)),\nabla^{2}_{\hat{x}}\mathfrak{b}(t))d\hat{x}dt+\displaystyle\frac{1}{2}\int_{\Omega}M_{0}(x)p^{2}(t)dxdt$
$\displaystyle+\int_{0}^{t}\int_{\Omega}\mathbb{K}_{33}|\partial_{3}p|^{2}dx\,d\tau$
$\displaystyle=\displaystyle\int_{0}^{t}{{}_{M_{0}^{\prime}}}\langle\boldsymbol{F},(\partial_{t}\mathfrak{a},\partial_{t}\mathfrak{b})\rangle_{M_{0}}\,d\tau+\int_{0}^{T}{{}_{V^{\prime}}}\langle\boldsymbol{G},p\rangle_{V}\,dt+\frac{1}{2}\int_{\Omega}M_{0}(x)(\tilde{p}_{0})^{2}\,dx.$
(3.76)

We obtain (3.74) by doing integration by parts on the right hand side of
(3.76) and by approximation from the stability results (3.64) and (3.65) and
the stability estimate for the equation (3.59), using the fact that
$H^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))$ is dense in
$H^{1}(0,T;L^{2}(\Omega;\mathbb{R}^{3}))$ and that
$\mathcal{D}(\widetilde{\mathcal{A}}_{0})$ is dense in $L^{2}(\Omega)$.

∎

######  Remark 3.7.

By using the operator $\widetilde{\mathcal{A}}_{0}$ and under the assumption
that $\mathcal{B}_{0}$ is coercive, one can use the semigroup theory to obtain
the existence for the problem (3.61), cf. Section 4.2 below. This, however,
gives only the existence in $C([0,T];H)$ and in order to conclude the
existence in $L^{2}(0,T;V)$ as well as the stability estimate (3.64) one would
have to do approximation of the solution by the Galerkin method as is done in
Section 4.2. Note, however, that putting (3.61) in the framework (4.8) would
be simpler, since the corresponding operator $\mathcal{A}$ would then be
positive definite (after doing the transformation of the solution
$\boldsymbol{u}(t)\mapsto e^{-\lambda t}u(t)$ which converts the condition
(3.63) in the positive definitness of the corresponding operator). In this way
the claim of Proposition 3.3 becomes clear: if the initial condition is
regular and the forces are regular the solution of the problem (4.8) is, in
fact, strong. In Section 4.2 we used both approaches: semigroup approach and
Galerkin approximation. On the one hand, semigroup approach doesn’t give us
enough regularity to prove the existence of the solution, but it gives us a
good way to approximate the initial condition and loads to obtain regular
enough solution. This is used to prove energy type equality. The existence of
the solution is proved via Galerkin approximation.

###  3.5 Strong convergence

We will firstly prove the convergence of energies under the assumption
$\boldsymbol{F}(0)=0$. This seems to be reasonable condition, since under that
condition we have that $\mathfrak{a}(0)=0$, $\mathfrak{b}(0)=0$, i.e. the
initial conditions are preserved. This then implies the strong convergence
result.

######  Theorem 3.5.

Let assumptions (2.32) and (2.33) be satisfied. We also assume that

$\pi_{h}\boldsymbol{F}^{h}\xrightarrow{t,dr-2,2}\boldsymbol{F},\quad\pi_{h}\partial_{t}\boldsymbol{F}^{h}\xrightarrow{t,dr-2,2}\partial_{t}\boldsymbol{F}$

where $\boldsymbol{F}\in
H^{1}(0,T;L^{2}(\Omega\times\mathcal{Y};\mathbb{R}^{3}))$ and
$\boldsymbol{F}(0)=0$. We have the following convergences:

$\displaystyle\displaystyle\lim_{h\to
0}\frac{1}{h^{2}}\int_{0}^{T}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}(x)e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{u}^{h}(t))\,dx$
$\displaystyle=\displaystyle\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}(t)),\nabla^{2}_{\hat{x}}\mathfrak{b}(t)):(e_{\hat{x}}(\mathfrak{a}(t)),\nabla^{2}_{\hat{x}}\mathfrak{b}(t))d\hat{x}$
$\displaystyle+\displaystyle\int_{\Omega}M_{0}(x)p^{2}(t)dx,$ (3.77)
$\displaystyle\displaystyle\lim_{h\to
0}\frac{\varepsilon^{2}}{h^{4}}\int_{0}^{T}\int_{\Omega^{h}_{f}}|e_{h}\left(\partial_{t}\boldsymbol{u}^{h}\right)|^{2}dx\,dt$
$\displaystyle=\int_{0}^{T}\int_{\Omega\times\mathcal{Y}_{f}(x)}\left|e_{y}\left(\partial_{t}\boldsymbol{u}^{f}_{0}\right)\right|^{2}dx\,dy\,dt,$
(3.78) $\displaystyle\lim_{h\to
0}\eta\int_{0}^{T}\left\|\partial_{t}\boldsymbol{u}^{h}(t)\right\|^{2}_{L^{2}(\Omega;\mathbb{R}^{3})}\,dt$
$\displaystyle=0.$ (3.79)

Here $(\mathfrak{a},\mathfrak{b},p)\in L^{2}(0,T;L_{0})\times L^{2}(0,T;M)$
are solutions of (3.45)-(3.46),

$\boldsymbol{u}^{f}_{0}\in
H^{1}(0,T;L^{2}(\Omega;H^{1}(\mathcal{Y}_{f}(x);\mathbb{R}^{3})))$

is defined with (3.57) and $(\boldsymbol{u}^{h},p^{h})$ is the solution of
(2.20) with initial condition (2.21).

###### Proof.

We will prove the convergences (3.77) and (3.79) and instead of (3.78) we will
firstly prove

$\displaystyle\displaystyle\lim_{h\to
0}\frac{\varepsilon^{2}}{h^{4}}\int_{0}^{T}\int_{0}^{t}\int_{\Omega^{h}_{f}}|e_{h}\left(\partial_{t}\boldsymbol{u}^{h}\right)|^{2}dx\,d\tau\,dt$
$\displaystyle=\int_{0}^{T}\int_{0}^{t}\int_{\Omega\times\mathcal{Y}_{f}^{x}}\left|e_{y}\left(\partial_{t}\boldsymbol{u}^{f}_{0}\right)\right|^{2}dx\,dy\,d\tau\,dt.$
(3.80)

Take $\boldsymbol{v}=\partial_{t}\boldsymbol{u}^{h}$ as test function in
(2.20). This yields for every $t\in[0,T]$

$\displaystyle\displaystyle\frac{1}{2}\frac{d}{dt}\left(\displaystyle\int_{\Omega}\eta\kappa^{h}|\partial_{t}\boldsymbol{u}^{h}(t)|^{2}\,dx+\frac{1}{h^{2}}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{u}^{h}(t))\,dx\right)$
$\displaystyle\quad+\displaystyle\frac{\varepsilon^{2}}{h^{4}}\int_{\Omega^{h}_{f}}2|e_{h}(\partial_{t}\boldsymbol{u}^{h}(t))|^{2}\,dx=\int_{\Omega}\boldsymbol{F}^{h}\partial_{t}\boldsymbol{u}^{h}\,dx.$

Fixing $t\in[0,T]$ and integrating over interval $[0,t]$ and then over
interval $[0,T]$ we obtain

$\displaystyle\displaystyle\frac{1}{2}\displaystyle\int_{0}^{T}\int_{\Omega}\eta\kappa^{h}|\partial_{t}\boldsymbol{u}^{h}(t)|^{2}\,dx\,dt+\frac{1}{2h^{2}}\int_{0}^{T}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{u}^{h}(t))\,dx\,dt$
$\displaystyle+\displaystyle\frac{\varepsilon^{2}}{h^{4}}\int_{0}^{T}\int_{0}^{t}\int_{\Omega^{h}_{f}}|e_{h}(\partial_{t}\boldsymbol{u}^{h}(\tau))|^{2}\,dx\,d\tau\,dt=\int_{0}^{T}\int_{0}^{t}\int_{\Omega}\boldsymbol{F}^{h}(\tau)\partial_{t}\boldsymbol{u}^{h}(\tau)\,dx\,d\tau\,dt.$
(3.81)

Notice that

$\displaystyle\int_{0}^{T}\int_{0}^{t}\int_{\Omega}\boldsymbol{F}^{h}(\tau)\partial_{t}\boldsymbol{u}^{h}(\tau)\,dx\,d\tau\,dt$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\boldsymbol{F}^{h}(t)\boldsymbol{u}^{h}(t)\,dx\,dt-\int_{0}^{T}\int_{0}^{t}\int_{\Omega}\partial_{t}\boldsymbol{F}^{h}(\tau)\boldsymbol{u}^{h}(\tau)\,dx\,d\tau\,dt$
$\displaystyle\to\displaystyle\int_{0}^{T}\int_{\omega}\overline{\langle\boldsymbol{F}\rangle_{Y}}\cdot(\mathfrak{a},\mathfrak{b})\,d\hat{x}\,dt-\displaystyle\int_{0}^{T}\int_{\omega}\overline{\langle
x_{3}\boldsymbol{F}_{*}\rangle_{Y}}\cdot\nabla_{\hat{x}}\mathfrak{b}\,d\hat{x}\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{0}^{t}\int_{\omega}\overline{\langle\partial_{t}\boldsymbol{F}(\tau)\rangle_{Y}}\cdot(\mathfrak{a}(\tau),\mathfrak{b}(\tau))\,d\hat{x}\,d\tau\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{0}^{t}\int_{\omega}\overline{\langle
x_{3}\partial_{t}\boldsymbol{F}_{*}(\tau)\rangle_{Y}}\cdot\nabla_{\hat{x}}\mathfrak{b}(\tau)\,d\hat{x}\,d\tau\,dt.$
(3.82)

Notice also that

$\int_{\Omega\times\mathcal{Y}_{f}(x)}\left|e_{y}\left(\partial_{t}\boldsymbol{u}^{0}_{f}(x,y)\right)\right|^{2}dx\,dy=\int_{\Omega}K_{33}(x)|\partial_{3}p(x,\tau)^{2}dx,$

where we use (3.58) and the fact that

$\mathbb{K}_{33}(x)=\int_{\mathcal{Y}_{f}(x)}q^{3}_{3}(y)\,dy=\langle\boldsymbol{q}^{3},\boldsymbol{q}^{3}\rangle_{L^{2}(\mathcal{Y}_{f}(x);\mathbb{R}^{3}}=\int_{\mathcal{Y}_{f}(x)}e_{y}(\boldsymbol{q}^{3}):e_{y}(\boldsymbol{q}^{3})\,dy.$
(3.83)

Using (3.74) integrated over $[0,T]$ (with $\mathfrak{a}(0)=0$,
$\mathfrak{b}(0)=0$, $\tilde{p}_{0}=0$), (3.82) and (3.83) we conclude that

$\displaystyle\displaystyle\frac{1}{2}\displaystyle\int_{0}^{T}\int_{\Omega}\eta\kappa^{h}|\partial_{t}\boldsymbol{u}^{h}(t)|^{2}\,dx\,dt+\frac{1}{2h^{2}}\int_{0}^{T}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{u}^{h}(t))\,dx\,dt$
$\displaystyle+\displaystyle\frac{\varepsilon^{2}}{h^{4}}\int_{0}^{T}\int_{0}^{t}\int_{\Omega^{h}_{f}}|e_{h}(\partial_{t}\boldsymbol{u}^{h}(\tau))|^{2}\,dx\,d\tau\,dt\to\frac{1}{2}\displaystyle\int_{0}^{T}\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b}):(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b})d\hat{x}dt$
$\displaystyle+\displaystyle\frac{1}{2}\int_{0}^{T}\int_{\Omega}M_{0}(x)p^{2}dx\,dt+\int_{0}^{T}\int_{0}^{t}\int_{\Omega}\mathbb{K}_{33}|\partial_{3}p|^{2}dx\,d\tau\,dt$
(3.84)

By using lower semicontinuity of the convex energy we have the following
inequalities (using (2.6), (3.8)-(3.10), (3.55)):

$\displaystyle\liminf_{h\to
0}\frac{1}{h^{2}}\int_{0}^{T}\int_{\Omega_{s}^{h}}\mathbb{A}^{h}e_{h}(\boldsymbol{u}^{h}(t)):e_{h}(\boldsymbol{u}^{h}(t))\,dx\,dt\geq$
$\displaystyle\displaystyle\int_{0}^{T}\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b}):(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b})d\hat{x}dt+\int_{0}^{T}\int_{\Omega}M_{0}(x)p^{2}dx\,dt,$
(3.85) $\displaystyle\liminf_{h\to
0}\displaystyle\frac{\varepsilon^{2}}{h^{4}}\int_{0}^{T}\int_{0}^{t}\int_{\Omega^{h}_{f}}|e_{h}(\partial_{t}\boldsymbol{u}^{h}(\tau))|^{2}\,dx\,d\tau\,dt\geq\int_{0}^{T}\int_{0}^{t}\int_{\Omega}K_{33}(x)|\partial_{3}p(x,\tau)^{2}dx\,d\tau\,dt,$
(3.86) $\displaystyle\liminf_{h\to
0}\frac{1}{2}\displaystyle\int_{0}^{T}\int_{\Omega}\eta\kappa^{h}|\partial_{t}\boldsymbol{u}^{h}(t)|^{2}\,dx\,dt\geq\min\\{\kappa_{f}^{0},\kappa_{s}^{0}\\}\eta\liminf_{h\to
0}\frac{1}{2}\displaystyle\int_{0}^{T}\int_{\Omega}\eta|\partial_{t}\boldsymbol{u}^{h}(t)|^{2}\,dx\,dt\geq
0.$ (3.87)

By combinig (3.85)-(3.87) with (3.84) we obtain (3.77), (3.79) and (3.80). It
remains to prove (3.78). We take a sequence $(\boldsymbol{u}_{0}^{f})^{n}\in
H^{1}(0,T;C^{1}(\Omega;H^{1}(\mathcal{Y};\mathbb{R}^{3})))$, such that

$\varepsilon
1_{\Omega_{f}^{h}}(x)e_{x}\left(\partial_{t}\boldsymbol{u}_{0}^{f,n}(x,\frac{x}{\varepsilon})\right)\xrightarrow{2,dr-2,2}1_{\mathcal{Y}_{f}(x)}(y)e_{y}(\partial_{t}\boldsymbol{u}_{f}^{0}).$

Notice that (3.80) is equivalent with (taking into account (3.10)):

$\lim_{n\to\infty}\lim_{h\to
0}\int_{0}^{T}\int_{0}^{t}\int_{\Omega_{f}^{h}}\left|\frac{\varepsilon}{h^{2}}e_{h}(\partial_{t}\boldsymbol{u}^{h})-\varepsilon
e_{x}\left(\partial_{t}\boldsymbol{u}_{0}^{f,n}(x,\frac{x}{\varepsilon})\right)\right|^{2}dx\,d\tau\,dt=0.$
(3.88)

(3.88) implies that

$\lim_{h\to
0}\frac{\varepsilon^{2}}{h^{4}}\int_{0}^{T^{\prime}}\int_{\Omega^{h}_{f}}|e_{h}\left(\partial_{t}\boldsymbol{u}^{h}\right)|^{2}dx\,dt=\int_{0}^{T^{\prime}}\int_{\Omega\times\mathcal{Y}_{f}(x)}\left|e_{y}\left(\partial_{t}\boldsymbol{u}^{f}_{0}\right)\right|^{2}dx\,dy\,dt,\quad\forall
T^{\prime}<T.$

In order to conclude it also for $t=T$ one can extend the forces on the
interval $[0,T^{\prime\prime}]$, where $T^{\prime\prime}>T$ with constant
forces $\boldsymbol{F}^{h}(T)$ and then use the uniqueness of the solution and
the above result. ∎

We have the following strong convergence result which includes correctors.

######  Theorem 3.6.

Under the conditions of Theorem 3.5 the following convergences hold:

$\displaystyle\displaystyle\lim_{n\to\infty}\lim_{h\to
0}\int_{0}^{T}\int_{\Omega^{h}_{f}}\left|\frac{\varepsilon}{h^{2}}e_{h}(\partial_{\tau}\boldsymbol{u}^{h})(x,\tau)+e_{y}(\boldsymbol{q}^{3}_{n})\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)\frac{\partial
p(x,\tau)}{\partial x_{3}}\right|^{2}\,dx\,d\tau=0,$ (3.89)
$\displaystyle\displaystyle\lim_{n\to\infty}\lim_{h\to
0}\int_{0}^{T}\int_{\Omega^{h}_{s}}\left|\frac{1}{h}e_{h}\left(\boldsymbol{u}^{h}\right)-e_{h}\left(\begin{pmatrix}\mathfrak{a}_{1}(\hat{x},t)-x_{3}\partial_{1}\mathfrak{b}(\hat{x},t)\\\
\mathfrak{a}_{2}(\hat{x},t)-x_{3}\partial_{2}\mathfrak{b}(\hat{x},t)\\\
h^{-1}\mathfrak{b}(\hat{x},t)\end{pmatrix}\right)\right.$ $\displaystyle\quad-
e_{h}\left(\varepsilon\displaystyle\sum_{\alpha,\beta}\left(e_{\hat{x}}(\mathfrak{a}_{n}(\hat{x},t))\right)_{\alpha\beta}\boldsymbol{w}^{\alpha\beta}_{n}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)\right)$
$\displaystyle\quad+e_{h}\left(\varepsilon
x_{3}\displaystyle\sum_{\alpha,\beta}\partial_{\alpha\beta}\mathfrak{b}_{n}(\hat{x},t)\boldsymbol{w}^{\alpha\beta}_{n}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)\right)-e_{h}\left(\varepsilon
p_{n}(x,t)\widetilde{\boldsymbol{w}}_{2,n}\left(x,\frac{\hat{x}}{\varepsilon},\frac{x_{3}}{\frac{\varepsilon}{h}}\right)\right)$
$\displaystyle\quad-
e_{h}\left(h\displaystyle\sum_{\alpha,\beta}\left(e_{\hat{x}}(\mathfrak{a}_{n}(\hat{x},t))\right)_{\alpha\beta}\int_{0}^{x_{3}}\boldsymbol{g}^{\alpha\beta}_{n}(x)\,dx_{3}\right)$
$\displaystyle\left.\quad+e_{h}\left(hx_{3}\displaystyle\sum_{\alpha,\beta}\partial_{\alpha\beta}\mathfrak{b}_{n}(\hat{x},t)\int_{0}^{x_{3}}\boldsymbol{g}^{\alpha\beta}_{n}(x)\,dx_{3}\right)-e_{h}\left(hp_{n}(x,t)\int_{0}^{x_{3}}\widetilde{\boldsymbol{g}}_{2,n}(x)\,dx_{3}\right)\right|^{2}dx\,dt=0.$

Here $\boldsymbol{q}^{3}_{n}\in
L^{2}(\Omega;C^{1}(\mathcal{Y}_{f}(x),\mathbb{R}^{3}))$,
$\mathfrak{a}_{n}(\cdot,t)\in C^{2}_{\\#}(\omega;\mathbb{R}^{2})$,
$\mathfrak{b}_{n}\in C_{\\#}^{3}(\omega)$, for $\alpha,\beta=1,2$,
$\boldsymbol{w}^{\alpha\beta}_{n}\in
C^{1}(\Omega;C^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}))$,
$\widetilde{\boldsymbol{w}}_{2,n}\in
C^{1}(\Omega;C^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}))$,
$\boldsymbol{g}^{\alpha\beta}_{n}\in C^{1}(\Omega;\mathbb{R}^{3})$,
$\widetilde{\boldsymbol{g}}_{2,n}\in C^{1}(\Omega;\mathbb{R}^{3})$ are chosen
such that (recall (3.36), (3.38), (3.40) ):

$\displaystyle\|\boldsymbol{q}_{n}^{3}-\boldsymbol{q}_{x}^{3}\|_{L^{2}(\Omega;H^{1}(\mathcal{Y}_{f}(x);\mathbb{R}^{3}))}\to
0,\quad\|\boldsymbol{q}_{n}^{3}\|_{L^{\infty}(\Omega;H^{1}(\mathcal{Y}_{f}(x);\mathbb{R}^{3}))}\textrm{
bounded};$
$\displaystyle\|\mathfrak{a}_{n}-\mathfrak{a}\|_{L^{2}(0,T;H_{\\#}^{1}(\omega;\mathbb{R}^{2}))}\to
0,\quad
h\|\mathfrak{a}_{n}\|_{L^{\infty}(0,T;C^{2}_{\\#}(\omega;\mathbb{R}^{2}))}\to
0;$
$\displaystyle\|\mathfrak{b}_{n}-\mathfrak{b}\|_{L^{2}(0,T;H_{\\#}^{2}(\omega))}\to
0,\quad h\|\mathfrak{b}_{n}\|_{L^{\infty}(0,T;C^{3}_{\\#}(\omega))}\to 0;$
$\displaystyle\|\boldsymbol{w}^{\alpha\beta}_{n}-\boldsymbol{w}_{x}^{\alpha\beta}\|_{L^{2}(\Omega;H^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}))}\to
0,\quad\|\boldsymbol{w}^{\alpha\beta}_{n}\|_{L^{\infty}(\Omega;H^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}))}\textrm{
bounded; }$ $\displaystyle\|p_{n}-p\|_{L^{2}(0,T;L^{2}(\omega;H^{1}(I)))}\to
0,\quad\min\\{\frac{\varepsilon}{h},h\\}\|p_{n}\|_{L^{\infty}(0,T;C^{1}(\omega))}\to
0;$
$\displaystyle\|\widetilde{\boldsymbol{w}}_{2,n}-\widetilde{\boldsymbol{w}}_{2,x}\|_{L^{2}(\Omega;H^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}))}\to
0,\quad\|\widetilde{\boldsymbol{w}}_{2,n}\|_{L^{\infty}(\Omega;H^{1}(\mathcal{Y}_{s}(x);\mathbb{R}^{3}))}\textrm{
bounded };$
$\displaystyle\|\boldsymbol{g}_{n}^{\alpha\beta}-\boldsymbol{g}^{\alpha\beta}_{x}\|_{L^{2}(\Omega;\mathbb{R}^{3})}\to
0,\quad
h\|\boldsymbol{g}_{n}^{\alpha\beta}\|_{C^{1}(\Omega;\mathbb{R}^{3})}\to
0,\quad\|\boldsymbol{g}_{n}^{\alpha\beta}\|_{L^{\infty}(\Omega;\mathbb{R}^{3})}\textrm{
bounded};$
$\displaystyle\|\widetilde{\boldsymbol{g}}_{2,n}-\widetilde{\boldsymbol{g}}_{2,x}\|_{L^{2}(\Omega;\mathbb{R}^{3})}\to
0,\quad
h\|\widetilde{\boldsymbol{g}}_{2,n}\|_{C^{1}(\Omega;\mathbb{R}^{3})}\to
0,\quad\|\widetilde{\boldsymbol{g}}_{2,n}\|_{L^{\infty}(\Omega;\mathbb{R}^{3})}\textrm{
bounded}.$

###### Proof.

Using the convexity of the quadratic energy, (3.89) is equivalent with (3.78)
as a consequence of (3.58). In the similar way (LABEL:strongtwoscale4may16) is
equivalent with (3.77), using (3.33), (3.34) and (3.55).

∎

###  3.6 Further discussions on the limit model

In this section we discuss the possibility of surface loads as well as the
situation when condition (2.1) is not satisfied. Everything discussed here can
also be concluded for the situation with inertia term analyzed in Section 4.

####  3.6.1 Surface loads

From the application point of view it is more important to have surface loads
on the longitudinal part of the boundary
$\omega\times\\{-\frac{1}{2},\frac{1}{2}\\}$. The discussion here resembles
[7, Corollary 3.42], where the case of elastic high-contrast inclusions is
discussed. We assume that in (2.20) we have on the right hand side the surface
loads:

$\mathcal{G}^{h}(\boldsymbol{G}^{h})(\boldsymbol{v})=\int_{\omega\times\\{-1/2,1/2\\}}\boldsymbol{G}^{h}\boldsymbol{v}\,d\hat{x},\qquad\boldsymbol{v}\in
H^{1}_{\\#}(\Omega;{\mathbb{R}}^{3}),$

and

$\displaystyle\pi_{h}\boldsymbol{G}^{h}\subset
H^{1}(0,T,L^{2}\bigl{(}\omega\times\bigl{\\{}-1/2,1/2\bigr{\\}};{\mathbb{R}}^{3}\bigr{)})\textrm{
is bounded},$
$\displaystyle\pi_{h}\boldsymbol{G}^{h}\xrightharpoonup{t,dr-2,2}\boldsymbol{G}\in
H^{1}\Bigl{(}0,T;L^{2}\bigl{(}\omega\times\bigl{\\{}-1/2,1/2\bigr{\\}};{\mathbb{R}}^{3}\bigr{)}\Bigr{)}.$

Instead of condition (2.32) we demand that for every $t>0$ we have

$\int_{\Omega}\boldsymbol{F}^{h}(t)\,dx+\int_{\omega\times\\{-1/2,1/2\\}}\boldsymbol{G}\,d\hat{x}=0.$

In the limit equations we then obtain on the right hand side of (3.45) the
terms:

$\int_{\omega}\boldsymbol{G}\cdot\boldsymbol{\theta}\,d\hat{x}+\int_{\omega}\left(\boldsymbol{G}(\cdot,-1/2)-\boldsymbol{G}(\cdot,1/2)\right)\cdot\nabla_{\hat{x}}\theta_{3}\,d\hat{x}.$

Under the assumption on strong convergence also the strong convergence of the
solutions in the spirit of Section 3.5 can be proved. Note also that the
existence and uniqueness from Section 3.4 can be used.

####  3.6.2 Fluid part intersects the longitudinal boundary
$\partial\omega\times\\{-1/2,1/2\\}\\}$

In the case when condition (2.1) is not satisfied one needs to impose boundary
condition for the fluid part of the longitudinal boundary (it is not a problem
again to add surface loads on the elastic part). Different boundary conditions
and appropriate equations in the weak formulation for the Stokes problem are
discussed in [10]. It is not possible to impose non-homogeneous Dirichlet
boundary condition for fluid on the longitudinal boundary. The reason for that
is in order to obtain the limit equations one uses the special scaling which
has the consequence that $\boldsymbol{u}_{f}$ is of order $h^{2}$. Here
$\boldsymbol{u}_{f}$ is not the position of the fluid (the position is given
by $\boldsymbol{u}=\hat{\boldsymbol{u}}+\boldsymbol{u}^{f})$ and the position
of the fluid is scaled differently (of order one in the third component). Thus
one can only impose artificial Dirichlet boundary condition on the
longitudinal boundary of the form:

$\boldsymbol{u}^{f}=h^{2}\boldsymbol{g}\textrm{ on
}\partial\omega\times\\{-1/2,1/2\\}\cap\Omega^{h}_{f},\quad\boldsymbol{g}\in
H^{1}(0,T;V^{h})\cap H^{2}(0,T;L^{2}(\Omega;\mathbb{R}^{3})).$

As a consequence of the derivation (3.30) one would obtain the additional term
of the form
$\int_{\omega}\left(\boldsymbol{g}(\cdot,1/2)\cdot\varphi(\cdot,1/2)-g_{3}(\cdot,-1/2)\cdot\varphi(\cdot,-1/2)\right)\,d\hat{x}.$
on the right hand side of (3.46). Again Section 3.4 covers the existence
result for this part. The other boundary conditions discussed in [10] include
again part of the fluid velocity (i.e. position) of the fluid and pressure
where one faces the similar problem as in the case of Dirichlet boundary
condition. However if these conditions are imposed on $\boldsymbol{u}_{f}$ one
can also deal with the following condition:

$\displaystyle(u_{f})_{\alpha}=h^{2}g_{\alpha},\quad p=p_{l},\textrm{ on
}\partial\omega\times\\{-1/2,1/2\\}\cap\Omega^{h}_{f},$ (3.91)

The consequence of condition 3.91 on the weak formulation is discussed in [10]
and includes additional terms on the right hand side of (2.20) which would
again have as a consequence appearance of the additional term on the right
hand side of (3.45). We leave the details to the interested reader. Next we
discuss the Neumann boundary condition (recall (2.11)):

$\sigma^{f,h}\boldsymbol{e}^{3}=\boldsymbol{G}^{h}\in
H^{1}(0,T;L^{2}(\omega\times\\{-1/2,1/2\\};\mathbb{R}^{3}))\textrm{ on
}\partial\omega\times\\{-1/2,1/2\\}\cap\Omega^{h}_{f}.$

This boundary condition has the consequence that in the weak formulation
(2.20) we have the following term

$\int_{\omega\times\\{-1/2,1/2\\}\cap\partial\Omega^{h}_{f}}\boldsymbol{G}^{h}\boldsymbol{v}\,d\hat{x},\qquad\boldsymbol{v}\in
H^{1}_{\\#}(\Omega;{\mathbb{R}}^{3}).$

The consequence of the appearance of this term is the same as for the
situation of surface loads on the elastic part.

##  4 The problem with inertial term

We look the equation (2.20) with $\eta=\eta(h)=1$. We follow the same
structure as in Section 3, but we skip the analogous proofs. In Section 4.1 we
state the compactness result and write the limit problem, in Section 4.2 we
prove the existence and uniqueness as well as the energy type equality for the
limit problem, while in Section 4.3 we state the strong convergence result. We
denote by $L=\dot{H}_{\\#}^{1}(\omega;\mathbb{R}^{2})$,
$N=H^{2}_{\\#}(\omega)$, $M=L^{2}(\omega;H^{1}(I))$ and by
$L^{\prime},N^{\prime},M^{\prime}$ their duals.

###  4.1 Compactness and limit equations

The following theorem can be proved in the same way as Theorem 3.1 and Theorem
3.2. We will state it without proof.

######  Theorem 4.1.

Let assumptions (2.3) and (2.32) be satisfied. We also assume that

$\pi_{h}\boldsymbol{F}^{h}\xrightharpoonup{t,dr-2,2}(\boldsymbol{F}_{*},F_{3}),$

where $\boldsymbol{F}_{*}\in
H^{1}(0,T;L^{2}(\Omega\times\mathcal{Y};\mathbb{R}^{2}))$, $F_{3}\in
L^{2}(0,T;L^{2}(\Omega\times\mathcal{Y}))$. The following statements hold: Let
$(\boldsymbol{u}^{h},p^{h})$ be the variational solution of (2.20) with
initial condition (2.21). Then there exist limits
$\mathfrak{a},\mathfrak{b},\boldsymbol{w},\boldsymbol{g},p$ such that they
satisfy (3.1)-(3.4) and $p\in L^{\infty}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;M)$
such that the convergences (3.6)-(3.11) are satisfied and

$\partial_{t}\boldsymbol{u}^{h}\xrightharpoonup{L^{\infty}(0,T;L^{2}(\Omega))}(0,0,\partial_{t}\mathfrak{b}),\quad\kappa^{h}\partial_{t}\boldsymbol{u}^{h}\xrightharpoonup{L^{\infty}(0,T;L^{2}(\Omega))}\bar{\kappa}(0,0,\partial_{t}\mathfrak{b}).$
(4.1)

The following limit equations are satisfied:

$\displaystyle\mathfrak{b}(0)=0,$ (4.2)
$\displaystyle-\int_{0}^{T}\int_{\omega}\bar{\kappa}(\hat{x})\partial_{t}\mathfrak{b}\partial_{t}\theta_{3}\,dt+\displaystyle\int_{0}^{T}\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b}):(e_{\hat{x}}(\boldsymbol{\theta}_{*}),\nabla^{2}_{\hat{x}}\theta_{3})d\hat{x}\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)pIdx_{3}:\left[\iota(e_{\hat{x}}(\boldsymbol{\theta}_{*}))\right]\,d\hat{x}\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}pIdx_{3}:\left[\iota(\nabla^{2}_{\hat{x}}\theta_{3})\right]\,d\hat{x}\,dt$
$\displaystyle=\displaystyle\int_{0}^{T}\int_{\omega}\overline{\langle\boldsymbol{F}\rangle_{Y}}\cdot(\boldsymbol{\theta}_{*},\theta_{3})\,d\hat{x}\,dt-\displaystyle\int_{0}^{T}\int_{\omega}\overline{\langle
x_{3}\boldsymbol{F}_{*}\rangle_{Y}}\cdot\nabla_{\hat{x}}\theta_{3}\,d\hat{x}\,dt,$
$\displaystyle\quad\forall\left(\boldsymbol{\theta}_{*},{\theta}_{3}\right)\in
L^{2}(0,T;L)\times H^{1}(0,T;N),\textrm{ such that }{\theta}_{3}(T)=0,$ (4.3)
$\displaystyle\displaystyle-\int_{0}^{T}\int_{\Omega}M_{0}(x)p\,\partial_{t}\varphi
dx\,dt+\displaystyle\int_{0}^{T}\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}p\,\partial_{3}\varphi
dx\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))\partial_{t}\varphi
Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}\right)\right)d\hat{x}\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\partial_{t}\varphi
Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}\right)\,d\hat{x}\,dt=0,$
$\displaystyle\forall\varphi\in L^{2}(0,T;M)\cap
H^{1}(0,T;L^{2}(\Omega))\textrm{ such that }\varphi(T)=0.$ (4.4)

###  4.2 Analysis of the limit equations

We again equip the system (4.2)-(4.4) with non-zero initial conditions and
substitute the loads. For $\mathfrak{b}_{0}\in N$, $\mathfrak{b}_{1}\in
L^{2}(\omega)$, $t_{0}\in L^{2}(\Omega)$, $\boldsymbol{F}_{1}\in
H^{1}(0,T;L^{\prime})$, $\boldsymbol{F}_{2}\in L^{2}(0,T;N^{\prime})$,
$\boldsymbol{G}\in L^{2}(0,T;M^{\prime})$, we define the following problem:
Find $\mathfrak{a}\in L^{2}(0,T;L)$, $\mathfrak{b}\in L^{2}(0,T;N)\cap
H^{1}(0,T;L^{2}(\omega))$, $p\in L^{2}(0,T;M)$ such that

$\displaystyle\mathfrak{b}(0)=\mathfrak{b}_{0},$ (4.5)
$\displaystyle-\int_{0}^{T}\int_{\omega}\bar{\kappa}(\hat{x})\partial_{t}\mathfrak{b}\partial_{t}\theta_{3}\,dt+\displaystyle\int_{0}^{T}\int_{\omega}\mathbb{A}^{hom}(\hat{x})(e_{\hat{x}}(\mathfrak{a}),\nabla^{2}_{\hat{x}}\mathfrak{b}):(e_{\hat{x}}(\boldsymbol{\theta}_{*}),\nabla^{2}_{\hat{x}}\theta_{3})d\hat{x}\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)pIdx_{3}:\left[\iota(e_{\hat{x}}(\boldsymbol{\theta}_{*}))\right]\,d\hat{x}\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}pIdx_{3}:\left[\iota(\nabla^{2}_{\hat{x}}\theta_{3})\right]\,d\hat{x}\,dt$
$\displaystyle=\displaystyle\int_{0}^{T}{{}_{L^{\prime}}}\langle\boldsymbol{F}_{1},\boldsymbol{\theta}_{*}\rangle_{L}\,dt\,+\int_{0}^{T}{{}_{N^{\prime}}}\langle\boldsymbol{F}_{2},\theta_{3}\rangle_{N}dt+\int_{\omega}\mathfrak{b}_{1}\cdot\theta_{3}(0)\,d\hat{x},$
$\displaystyle\quad\forall\left(\boldsymbol{\theta}_{*},{\theta}_{3}\right)\in
L^{2}(0,T;L)\times H^{1}(0,T;N),\textrm{ such that }{\theta}_{3}(T)=0,$ (4.6)
$\displaystyle\displaystyle-\int_{0}^{T}\int_{\Omega}M_{0}(x)p\,\partial_{t}\varphi
dx\,dt+\displaystyle\int_{0}^{T}\int_{\Omega}\mathbb{K}_{33}(x)\partial_{3}p\,\partial_{3}\varphi
dx\,dt$
$\displaystyle-\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x))\partial_{t}\varphi
Idx_{3}:\iota\left(e_{\hat{x}}\left(\mathfrak{a}\right)\right)d\hat{x}\,dt$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\omega}\int_{I}\left(|\mathcal{Y}_{f}(x)|I-\mathbb{B}^{H}(x)\right)x_{3}\partial_{t}\varphi
Idx_{3}:\iota\left(\nabla^{2}_{\hat{x}}\mathfrak{b}\right)\,d\hat{x}\,dt$
$\displaystyle=\int_{0}^{T}{{}_{M^{\prime}}}\langle\boldsymbol{G},\varphi\rangle_{M}\,dt+\int_{\Omega}t_{0}\cdot\varphi(0),$
$\displaystyle\forall\varphi\in L^{2}(0,T;M)\cap
H^{1}(0,T;L^{2}(\Omega))\textrm{ such that }\varphi(T)=0.$ (4.7)

This section discusses existence, uniqueness and regularity of the solution of
(4.5)-(4.7) as well as the energy equality. The following theorem gives us
uniqueness of solution of (4.5)-(4.7).

######  Theorem 4.2.

Let $\mathfrak{b}_{0}\in N$, $\mathfrak{b}_{1}\in L^{2}(\omega)$, $t_{0}\in
L^{2}(\Omega)$, $\boldsymbol{F}_{1}\in H^{1}(0,T;L^{\prime})$,
$\boldsymbol{F}_{2}\in L^{2}(0,T;N^{\prime})$, $\boldsymbol{G}\in
L^{2}(0,T;M^{\prime})$. The solution of (4.5)-(4.7), if it exists, is unique.

###### Proof.

We assume that $\boldsymbol{F}_{\alpha}=\boldsymbol{G}=0$, for $\alpha=1,2$,
$\mathfrak{b}_{0}=\mathfrak{b}_{1}=0$, $t_{0}=0$ in (4.5)-(4.7) and for fixed
$0<T^{\prime}<T$ take the following test functions in (4.4)

$\theta_{3}=\begin{cases}\int_{T^{\prime}}^{t}\mathfrak{b}(s)\,ds,&\textrm{
for all }0\leq t<T^{\prime},\\\ 0,&\textrm{ for all }T^{\prime}\leq t\leq
T.\end{cases}\quad\boldsymbol{\theta}_{*}=\begin{cases}\int_{T^{\prime}}^{t}\mathfrak{a}(s)\,ds,&\textrm{
for all }0\leq t<T^{\prime}\\\ 0,&\textrm{ for all }T^{\prime}\leq t\leq
T.\end{cases},$

and in (4.4)

$\varphi(t)=\begin{cases}\int_{T^{\prime}}^{t}\int_{0}^{s}p(r)\,dr\,ds,&\textrm{
for all }0\leq t<T^{\prime},\\\ 0,&\textrm{ for all }T^{\prime}\leq t\leq
T.\end{cases}$

We have that