# Homogenization Theory of Ion Transportation in Multicellular Tissue

Chun Xiao 111School of Mathematical Sciences, Soochow University, Suzhou
215006, China; School of Mathematics and Statistics, Lingnan Normal
University, Zhanjiang 524048, China (cxiao77xc@gmail.com). Xingye
Yue333Department of Mathematics, Soochow University, Suzhou 215006, China
(xyyue@suda.edu.cn). Huaxiong Huang444Research Center for Mathematics,
Advanced Institute of Natural Sciences, Beijing Normal University (Zhuhai),
Guangdong, China; Department of Mathematics and Statistics, York University,
Toronto, ON, Canada (hhuang@uic.edu.cn). Shixin Xu 222Corresponding author. Zu
Chongzhi Center for Mathematics and Computational Sciences (CMCS), Duke
Kunshan University, Kunshan 215316, China (shixin.xu@dukekunshan.edu.cn).

###### Abstract

Ion transport in biological tissues is crucial in the study of many biological
and pathological problems. Some multi-cellular structures, like smooth muscles
on the vessel walls, could be treated as periodic bi-domain structures, which
consist of intracellular space and extracellular space with semipermeable
membranes in between. With the aid of two-scale homogenization theory, macro-
scale models are proposed based on an electro-neutral (EN) microscale model
with nonlinear interface conditions, where membranes are treated as
combinations of capacitors and resistors. The connectivity of intracellular
space is also taken into consideration. If the intracellular space is fully
connected and forms a syncytium, then the macroscale model is a bidomain
nonlinear coupled partial differential equations system. Otherwise, when the
intracellular cells are not connected, the macroscale model for intracellular
space is an ordinary differential system with source/sink terms from the
connected extracellular space.

Keywords: Ion transport, Two-scale homogenization, Bi-domain model,
Connectivity

##  Introduction

Ions in human body play vital roles in many aspects such as helping the
transport of nerve impulses, maintaining the proper functions of muscles,
activating various enzymes, helping blood coagulation and so on. Studying ion
transport in biological tissues can help us understand the mechanisms of many
physiological phenomenon and gain some insight about how to treat certain
diseases. The Poisson-Nernst-Planck (PNP) system is one of the most popular
mathematical models that describe the ion transport under the influence of
both ionic concentration gradient and electric field. PNP system has extensive
and successful applications in biological systems, particularly in ion
channels on cell membrane [10, 12, 17, 24, 27, 37]. Due to the capacitance of
membranes, there are thin boundary layers (BLs) near the interfaces formed by
excessive charges accumulation. BLs requires extra computation cost during
numerical simulations in order to resolve the fast change behaviors of
solution inside the layers and attain certain accuracy [8, 44]. A lot of
efforts are put in order to get rid of this constrain, like Mori [33]. By
using asymptotic analysis, Song et al. [43, 42] proposed effective interface
conditions by introducing a time dependent capacitance on the membrane. In
this paper, we take the linearization of the effective boundary conditions and
propose a microscale EN PNP system with interface conditions describing the
membrane fluxes and capacitor effect.

Due to the existence of pumps on membranes, ion concentrations across the
membrane are discontinuous, for example potassium is $140\sim 150~{}mM$ in the
intracelluar space and $3\sim 5~{}mM$ in the extracellular space. However, the
flux across the membrane is continuous and determined by conductance and the
difference between membrane potential and Nernst Potential. When studying
biological tissues, there are thousands of cells in the system and the
solution is highly oscillatory. The obtained mircoscale model requires
significant computational resources to solve numerically. In order to simulate
ion transport in biological tissues more efficiently, an effective macroscale
model is demanded. One of the most popular ways to derive the macroscale
models is through homogenization by deriving an “average” or ”effective”
homogenized system, which extracts macro information from micro structures.
Specific methods of homogenization include oscillating test function method
[45], asymptotic expansion method [6], two-scale method [36, 1] and unfolding
method [15, 13]. The homogenization theory for system with jump solution is
first developed by Monsurro and Donato [30, 32, 21, 18] to a linear elliptic
model for heat conductors problem by using extension operators. Results for
linear parabolic and hyperbolic equations can be found in [28, 20, 48, 47,
19]. Another factor for bi-domain homogenization is whether two sub-domains
are all connected domain, respectively. The linear problems mentioned above
only considered the case when one sub-domain is embedded in the other sub-
domain and disconnected. In [9], a linear diffusion problem in a bi-domain
with flux jump at the interface is discussed. The authors considered both when
one sub-domain is connected and disconnected, and explained the reasons for
the difference in the resulting homogenization systems for the two cases,
which is mainly due to the different spaces of the test functions. For
nonlinear bi-domain homogenization problems, the main difficulty is the strong
convergence requirement both in domain and on the interface. By two-scale
convergence and unfolding operator, Gahn et al. [25, 23] proposed
homogenization results for a nonlinear problem in a bi-domain for calcium
dynamics (connected sub-domains) and diffusion-reaction system (one
disconnected sub-domain).

For the homogenization theory of PNP system, which is a nonlinear coupled
system, asymptotic expansion method [29] is first used to derive the
homogenized system for the linearized Navier-Stokes-PNP model in porous media
in [38]. Later, a rigorous derivation by using two-scale method is proposed by
Alliare et al. [4]. Similar method is extended to study ion transport through
deformable porous media [2]. Homogenization of a full nonlinear PNP model in
porous media is discussed in [46, 41] by using unfolding method and two-scale
method, where nonflux boundary condition is used on the interface for ion
concentraion. Most of the homogenization research for PNP model and ion
transport model are established in porous media without considering the
electric-diffusion of ions in the intracellular region, and sometimes
linearization technique is used to simplify the process. In this paper, by
using unfolding operator and two-scale convergence, we develop the
homogenization theory for the fully nonlinear EN bi-domain model in the whole
domain with nonlinear interface flux boundary conditions which depends on the
jumps of ion concentrations, electric potential, and the time derivative of
potential jump. Two different scenarios, connected and disconnected
intracellular regions, are both taken into consideration and lead to different
macroscopic models.

The remainder of this paper is organized as follows: The microscale EN ion
transport model is given in Section 2; Section 3 is devoted to proving the a
priori estimates; Then convergence results and homogenization process are
presented in Section 4 according to different connectivity conditions of
intracellular region; we draw conclusions in the last section.

##  Setting of the mathematical model

In this section, a microscale EN bi-domain ion transport model in
multicellular tissues is introduced. Consider a domain $\Omega=(0,1)^{d}$
which consists of two components: $\Omega_{I}^{\varepsilon}$ and
$\Omega_{E}^{\varepsilon}$ (see Figure 1). Let $Y=(0,1)^{d}$ and $Y_{I},Y_{E}$
are two disjoint subsets of $Y$, such that

$\overline{Y}=\overline{Y_{E}}\cup\overline{Y_{I}}.$

And $\Gamma=\partial Y_{I}\bigcap\partial Y_{E}$ is Lipschitz continuous. Let
$\vec{n}_{1}$ be the normal direction of $\Gamma$, pointing from $Y_{I}$ to
$Y_{E}$. Let $\beta_{s},s=I,E$ be the characteristic function on
$Y_{s},s=I,E$, which extend periodically to $\mathbb{R}^{d}$. For any
$k\in\mathbb{Z}^{d}$, let

$Y^{k}=k+Y,~{}\Gamma^{k}=k+\Gamma,~{}Y_{s}^{k}=k+Y_{s},$

where $k=\left(k_{1},\cdots,k_{d}\right)$, $s=I,E$. For any $\varepsilon>0$
and $1/\varepsilon\in\mathbb{N}^{+}$, let
$K_{\varepsilon}=\left\\{k\in\mathbb{Z}^{d}|\varepsilon
Y_{s}^{k}\cap\Omega\neq\emptyset,s=I,E\right\\}$. Denote the two disjoint
subsets of $\Omega$ and the interface between them as:

$\Omega_{I}^{\varepsilon}=\bigcup_{k\in K_{\varepsilon}}\varepsilon
Y_{I}^{k},\quad\Omega_{E}^{\varepsilon}=\Omega\backslash\overline{\Omega_{I}^{\varepsilon}},\quad\Gamma_{\varepsilon}=\bigcup_{k\in
K_{\varepsilon}}\varepsilon\Gamma^{k}.$

Figure 1: Schematic of the domain $\Omega$.

So
$\Omega=\Omega_{I}^{\varepsilon}\bigcup\Omega_{E}^{\varepsilon}\bigcup\Gamma_{\varepsilon}$
and $\Omega$ is a union of $\varepsilon Y$-periodic sets. Assume both
$\Omega_{E}^{\varepsilon}$ and $\Omega_{I}^{\varepsilon}$ have Lipschitz
boundaries, especially $\Gamma_{\varepsilon}$ is a Lipschitz boundary.

Consider three different ion species in $\Omega$:
$\mathrm{Na^{+},K^{+},Cl^{-}}$, their concentrations and valences are denoted
by $C_{i,s}^{\varepsilon},i=1,2,3,s=I,E$ and $z_{i}$ respectively, the
electric potential is $\phi_{s}^{\varepsilon}$, subscript $s$ represent
variables in $\Omega_{s}^{\varepsilon},s=I,E$.

The assumptions are summarized as follows:

  * •

Connectivity: $\Omega_{E}^{\varepsilon}$ is connected and
$\Omega_{I}^{\varepsilon}$ can be connected or not (see Figure 1).

  * •

Positivity:

$\displaystyle 0<C_{d}\leq C_{i,s}^{\varepsilon}\leq
C_{u}~{}~{}~{}~{}\text{on}~{}\Gamma_{\varepsilon},~{}~{}i=1,2,3,s=I,E,$ (2.1)
$\displaystyle
0<C_{l}<\sum\limits_{i=1}^{3}z_{i}^{2}C_{i,s}^{\varepsilon}(t,x)~{}~{}~{}~{}\text{in}~{}\Omega_{s}^{\varepsilon},~{}~{}s=I,E,$
(2.2)

where $C_{d},C_{u},C_{l}$ are positive constants.

  * •

Electric Neutrality (EN):

$\sum\limits_{i=1}^{3}z_{i}C_{i,s}^{\varepsilon}(t,x)=0,~{}~{}s=I,E.$ (2.3)

  * •

Diffusion Constant: diffusion constants of ions are the same and denoted by
$D$.

When $\Omega_{I}^{\varepsilon}$ is not connected, we call this case
“connected-disconnected” case. In this case, $\overline{Y_{I}}\subset Y$ and
$\Omega_{I}^{\varepsilon}$ is a disconnected union of $\varepsilon Y$-periodic
sets of size $\varepsilon Y_{I}$. When $\Omega_{I}^{\varepsilon}$ is
connected, we call this case “connected-connected” case. In this case both
$Y_{I}$ and $Y_{E}$ reach $\partial Y$, thus both $\Omega_{I}^{\varepsilon}$
and $\Omega_{E}^{\varepsilon}$ reach $\partial\Omega$.

The EN bi-domain ion transport model is given as follow: for $i=1,2,3,s=I,E$,

$\displaystyle\partial_{t}C_{i,s}^{\varepsilon}=-\nabla\cdot{J}_{i,s}=\nabla\cdot[D(\nabla
C_{i,s}^{\varepsilon}+z_{i}C_{i,s}^{\varepsilon}\nabla\phi_{s}^{\varepsilon})]~{}~{}$
$\displaystyle in~{}(0,T)\times\Omega^{\varepsilon}_{s},$ (2.4a)
$\displaystyle z_{i}{J}_{i,s}\cdot\vec{n}=\varepsilon
G_{i}\big{(}\llbracket{\phi^{\varepsilon}}\rrbracket+\dfrac{1}{z_{i}}\llbracket{\ln
C_{i}^{\varepsilon}}\rrbracket\big{)}+\varepsilon
P_{i}^{\varepsilon}+\varepsilon\lambda_{i}P_{m}\partial_{t}\llbracket{\phi^{\varepsilon}}\rrbracket~{}~{}$
$\displaystyle on~{}(0,T)\times\Gamma_{\varepsilon},$ (2.4b)
$\displaystyle{J}_{i,E}\cdot\vec{\nu}=0~{}~{}$ $\displaystyle
on~{}(0,T)\times\partial\Omega,$ (2.4c) $\displaystyle
C_{i,s}^{\varepsilon}|_{t=0}=C_{i,s}^{0}~{}~{}$ $\displaystyle
in~{}\Omega^{\varepsilon}_{s}.$ (2.4d)

and

$\displaystyle-\nabla\cdot(D\sigma_{s}^{\varepsilon}\nabla\phi_{s}^{\varepsilon})=0~{}~{}$
$\displaystyle in~{}(0,T)\times\Omega^{\varepsilon}_{s},$ (2.5a)
$\displaystyle-D\sigma_{s}^{\varepsilon}\nabla\phi_{s}^{\varepsilon}\cdot\vec{n}=\sum\limits_{i=1}^{3}\varepsilon
G_{i}\llbracket{\phi^{\varepsilon}}\rrbracket+\sum\limits_{i=1}^{3}\varepsilon
G_{i}\dfrac{1}{z_{i}}\llbracket{\ln C_{i}^{\varepsilon}}\rrbracket+\varepsilon
I_{P}^{\varepsilon}+\varepsilon
P_{m}\partial_{t}\llbracket{\phi^{\varepsilon}}\rrbracket~{}~{}$
$\displaystyle on~{}(0,T)\times\Gamma_{\varepsilon},$ (2.5b)
$\displaystyle-D\sigma_{E}^{\varepsilon}\nabla\phi_{E}^{\varepsilon}\cdot\vec{\nu}=0~{}~{}$
$\displaystyle on~{}(0,T)\times\partial\Omega,$ (2.5c)
$\displaystyle\llbracket{\phi^{\varepsilon}}\rrbracket|_{t=0}=\phi^{0}~{}~{}$
$\displaystyle on~{}\Gamma_{\varepsilon}.$ (2.5d)

where $\phi_{s}^{\epsilon}$ is the electric potential in domain $\Omega_{s}$,
$C_{i,s}^{\epsilon}$ is the concentration of $i_{th}$ ion in domain
$\Omega_{s}$.

(2.4a) is the usual Nernst-Planck equation, (2.5) is a direct result of (2.4)
and EN assumption (2.3) with

$\sigma_{s}^{\varepsilon}=\sum\limits_{i=1}^{3}z_{i}^{2}C_{i,s}^{\varepsilon},~{}~{}s=I,E.$
(2.6)

(2.4b) is the interface flux condition for ion concentration which consists of
three parts.

  * •

The first part is the current induced by passive ion channel which is modeled
by Ohm’s Law. Here $G_{i}$ is the conductance of $i_{th}$ ion on the membrane,
$\llbracket{\phi^{\varepsilon}}\rrbracket=\phi_{I}-\phi_{E}$ is the membrane
potential and $E_{i}\triangleq-\frac{1}{z_{i}}\llbracket{\ln
C_{i}^{\varepsilon}}\rrbracket=\frac{1}{z_{i}}(\ln C_{i,E}^{\varepsilon}-\ln
C_{i,I}^{\varepsilon})$ is the Nernst potential.

  * •

The second part is the current induced by the active pumps on the membrane.
Here we only consider the $Na^{+}/K^{+}$ pumps [49].

$P_{1}^{\varepsilon}=3I_{p}^{\varepsilon}~{}~{},P_{2}^{\varepsilon}=-2I_{p}^{\varepsilon},~{}~{}P_{3}^{\varepsilon}=0,$

with

$\displaystyle I_{p}^{\varepsilon}$ $\displaystyle=$ $\displaystyle I_{\max
1}\left(\frac{C_{1,I}^{\varepsilon}}{C_{1,I}^{\varepsilon}+K_{Na1}}\right)^{3}\left(\frac{C_{2,E}^{\varepsilon}}{C_{2,E}^{\varepsilon}+K_{K1}}\right)^{2}$
(2.7) $\displaystyle+$ $\displaystyle I_{\max
2}\left(\frac{C_{1,I}^{\varepsilon}}{C_{1,I}^{\varepsilon}+K_{Na2}}\right)^{3}\left(\frac{C_{2,E}^{\varepsilon}}{C_{2,E}^{\varepsilon}+K_{K2}}\right)^{2},$

where $I_{max1},I_{max2}$ are maximum $Na^{+}/K^{+}$ pump current density and
$K_{K1},K_{K2}$, $K_{Na1}$, $K_{Na2}$ are threshold constants for
$I_{p}^{\varepsilon}$.

  * •

The last term is the current induced by the capacitor effect of membrane with
capacitance $P_{m}$ . $\lambda_{i}$ reflects the effect of capacitor on the
$i_{th}$ ion species and $\sum\limits_{i=1}^{3}\lambda_{i}=1$. Here we take
$\lambda_{i}$ as constant [33] which is simplification and linearization
version of models in [43, 42].

######  _R_ emark 2.1.

In (2.4), (2.5) we only consider the case when $\Omega_{I}^{\varepsilon}$ is
not connected, and this would not affect the main idea because when
$\Omega_{I}^{\varepsilon}$ is also connected and reaches the exterior boundary
$\partial\Omega$, we only need to add another exterior boundary condition for
the functions in $\Omega_{I}^{\varepsilon}$. And the connectivity condition of
$\Omega_{I}^{\varepsilon}$ would not affect the a priori estimates (Theorem
3.1) either.

The homogenization results (Theorem 4.1 and Theorem 4.2) show that the
interfacial fluxes become source terms in the homogenized equations, and the
connectivity of intracellular region will affect the homogenization results.
When the intracellular region is connected, the homogenized one is a
diffusion-reaction equation; when cells are isolated with each other, the
homogenized equation is a reaction equation. This is consistent with relevant
results in [9]. And the EN condition still holds for the homogenized equations
both when intracellular region is connected and not connected.

##  A priori estimate

In order to derive the convergence results, we need a priori estimates for the
solutions of (2.4), (2.5). In the rest of the paper, $A$ (with or without
subscript) represent a generic positive constant independent of $\varepsilon$
(in different equations, the value of $A$ may be different).

The weak form of (2.4) and (2.5) is: for $i=1,2,3,s=I,E$, find
$C_{i,s}^{\varepsilon}\in
L^{2}(0,T;~{}H^{1}(\Omega_{s}^{\varepsilon})),~{}\phi_{s}^{\varepsilon}\in
L^{2}(0,T;~{}H^{1}(\Omega_{s}^{\varepsilon}))$, such that

$\displaystyle\int_{\Omega_{s}^{\varepsilon}}\partial_{t}C_{i,s}^{\varepsilon}v_{s}~{}dx+\int_{\Omega_{s}^{\varepsilon}}D\nabla
C_{i,s}^{\varepsilon}\nabla
v_{s}~{}dx+z_{i}\int_{\Omega_{s}^{\varepsilon}}C_{i,s}^{\varepsilon}D\nabla\phi_{s}^{\varepsilon}\nabla
v_{s}~{}dx$
$\displaystyle\pm\varepsilon\int_{\Gamma_{\varepsilon}}\big{(}\dfrac{G_{i}}{z_{i}}\llbracket{\phi^{\varepsilon}}\rrbracket
v_{s}+\dfrac{G_{i}}{z_{i}^{2}}\llbracket{\ln C_{i}^{\varepsilon}}\rrbracket
v_{s}+\dfrac{P_{i}^{\varepsilon}}{z_{i}}v_{s}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}\llbracket{\phi^{\varepsilon}}\rrbracket
v_{s}\big{)}~{}dS=0,$ (3.1a)
$\displaystyle\int_{\Omega_{s}^{\varepsilon}}\sigma_{s}^{\varepsilon}D\nabla\phi_{s}^{\varepsilon}\nabla\varphi_{s}~{}dx\pm\varepsilon\int_{\Gamma_{\varepsilon}}\big{(}\sum\limits_{k=1}^{3}G_{k}\llbracket{\phi^{\varepsilon}}\rrbracket\varphi_{s}+\sum\limits_{k=1}^{3}\dfrac{G_{k}}{z_{k}}\llbracket{\ln
C_{k}^{\varepsilon}}\rrbracket\varphi_{s}+I_{p}^{\varepsilon}\varphi_{s}+P_{m}\partial_{t}\llbracket{\phi^{\varepsilon}}\rrbracket\varphi_{s}\big{)}~{}dS=0,$
(3.1b)

for $\forall~{}v_{s}\in L^{2}(0,T;~{}H^{1}(\Omega_{s}^{\varepsilon}))$ and
$\varphi_{s}\in L^{2}(0,T;~{}H^{1}(\Omega_{s}^{\varepsilon}))$, where “$\pm$”
takes “$+$” for $s=I$ and “$-$” for $s=E$. Next we derive a priori estimates
for $C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon},~{}i=1,2,3,s=I,E$.

We also impose the following condition for $\phi_{E}^{\varepsilon}$ to ensure
the uniqueness,

$\int_{\Omega_{E}^{\varepsilon}}\phi_{E}^{\varepsilon}~{}dx=0.$ (3.2)

In the rest of the paper, for a subset $\omega\in\mathbb{R}^{d}$, we simply
denote $\|f\|_{L^{2}(\omega)}$ as $\|f\|_{\omega}$. And for a function
$u^{\varepsilon}$ defined both in $\Omega_{I}^{\varepsilon}$ and
$\Omega_{E}^{\varepsilon}$, denote
$\|u^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}\triangleq\|u_{I}^{\varepsilon}\|^{2}_{\Omega_{I}^{\varepsilon}}+\|u_{E}^{\varepsilon}\|^{2}_{\Omega_{E}^{\varepsilon}}$.
Then a priori estimates are summarized in the following theorem.

######  _T_ heorem 3.1.

Let $C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon},i=1,2,3,s=I,E$ be the
solutions of (3.1) and suppose assumption (3.2) hold, then the following
estimates are valid, for $i=1,2,3,s=I,E$,

$\displaystyle\|C_{i,s}^{\varepsilon}\|_{L^{\infty}(0,T;L^{2}(\Omega_{s}^{\varepsilon}))}\leq
A,$ (3.3a)
$\displaystyle\|C_{i,s}^{\varepsilon}\|_{L^{2}(0,T;H^{1}(\Omega_{s}^{\varepsilon}))}\leq
A,$ (3.3b)
$\displaystyle\sqrt{\varepsilon}\|C_{i,s}^{\varepsilon}\|_{L^{2}(0,T;L^{2}(\Gamma^{\varepsilon}))}\leq
A,$ (3.3c)
$\displaystyle\|\partial_{t}C^{\varepsilon}_{i,s}\|_{L^{2}(0,T;H^{-1}(\Omega^{\varepsilon}_{s}))}\leq
A,$ (3.3d)
$\displaystyle\sqrt{\varepsilon}\|\phi^{\varepsilon}\|_{L^{2}(0,T;L^{2}(\Gamma^{\varepsilon}))}\leq
A,$ (3.3e)
$\displaystyle\|\phi_{s}^{\varepsilon}\|_{L^{2}(0,T;H^{1}(\Omega_{s}^{\varepsilon}))}\leq
A.$ (3.3f)

Proof: If let
$v_{s}=C_{i,s}^{\varepsilon},~{}\varphi_{s}=\phi_{s}^{\varepsilon},~{}s=I,E$
in (3.1), we have

$\displaystyle\sum\limits_{i=1}^{3}\dfrac{1}{2}\partial_{t}\|C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\sum\limits_{i=1}^{3}D\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\sum\limits_{i=1}^{3}\int_{\Omega^{\varepsilon}}z_{i}C_{i}^{\varepsilon}D\nabla\phi^{\varepsilon}\nabla
C_{i}^{\varepsilon}~{}dx+\sum\limits_{i=1}^{3}\int_{\Gamma^{\varepsilon}}\varepsilon\dfrac{P_{i}^{\varepsilon}}{z_{i}}\llbracket{C_{i}^{\varepsilon}}\rrbracket~{}dS$
$\displaystyle+\sum\limits_{i=1}^{3}\int_{\Gamma^{\varepsilon}}\varepsilon\dfrac{G_{i}}{z_{i}}\llbracket{\phi^{\varepsilon}}\rrbracket\llbracket{C_{i}^{\varepsilon}}\rrbracket~{}dS+\sum\limits_{i=1}^{3}\int_{\Gamma^{\varepsilon}}\varepsilon\dfrac{G_{i}}{z_{i}^{2}}\llbracket{\ln
C_{i}^{\varepsilon}}\rrbracket\llbracket{C_{i}^{\varepsilon}}\rrbracket~{}dS$
$\displaystyle+\sum\limits_{i=1}^{3}\int_{\Gamma^{\varepsilon}}\varepsilon\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}\llbracket{\phi^{\varepsilon}}\rrbracket\llbracket{C_{i}^{\varepsilon}}\rrbracket~{}dS=0,$
(3.4a)
$\displaystyle\int_{\Omega^{\varepsilon}}\sigma^{\varepsilon}D\nabla\phi^{\varepsilon}\nabla\phi^{\varepsilon}~{}dx+\sum\limits_{i=1}^{3}\varepsilon
G_{i}\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+\varepsilon\dfrac{P_{m}}{2}\partial_{t}\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}$
$\displaystyle=-\sum\limits_{i=1}^{3}\int_{\Gamma^{\varepsilon}}\varepsilon\dfrac{G_{i}}{z_{i}}\llbracket{\ln
C_{i}^{\varepsilon}}\rrbracket\llbracket{\phi^{\varepsilon}}\rrbracket~{}dS-\int_{\Gamma^{\varepsilon}}\varepsilon
I_{p}^{\varepsilon}\llbracket{\phi^{\varepsilon}}\rrbracket~{}dS.$ (3.4b)

From assumption (2.1), we have

$\displaystyle\begin{cases}\llbracket{\ln
C_{i}^{\varepsilon}}\rrbracket\llbracket{C_{i}^{\varepsilon}}\rrbracket=(\ln
C_{i,I}^{\varepsilon}-\ln
C_{i,E}^{\varepsilon})(C_{i,I}^{\varepsilon}-C_{i,E}^{\varepsilon})=\dfrac{1}{\xi}(C_{i,I}^{\varepsilon}-C_{i,E}^{\varepsilon})^{2}\geq\dfrac{1}{C_{u}}(C_{i,I}^{\varepsilon}-C_{i,E}^{\varepsilon})^{2}\\\
|\llbracket{\ln C_{i}^{\varepsilon}}\rrbracket|=|\ln C_{i,I}^{\varepsilon}-\ln
C_{i,E}^{\varepsilon}|=|\dfrac{1}{\xi}(C_{i,I}^{\varepsilon}-C_{i,E}^{\varepsilon})|\leq\dfrac{1}{C_{d}}|C_{i,I}^{\varepsilon}-C_{i,E}^{\varepsilon}|,\end{cases}$
(3.5)

where $\xi$ is between $C_{i,I}^{\varepsilon}$ and $C_{i,E}^{\varepsilon}$.
Then let $\varphi_{s}=\frac{\lambda_{i}}{z_{i}}C_{i,s}^{\varepsilon},~{}s=I,E$
in (3.1b), we have

$\displaystyle\varepsilon\int_{\Gamma^{\varepsilon}}\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}\llbracket{\phi^{\varepsilon}}\rrbracket\llbracket{C_{i}^{\varepsilon}}\rrbracket~{}dS$
$\displaystyle=$
$\displaystyle-\dfrac{\lambda_{i}}{z_{i}}\int_{\Omega^{\varepsilon}}\sigma^{\varepsilon}D\nabla\phi^{\varepsilon}\nabla
C_{i}^{\varepsilon}~{}dx-\varepsilon\dfrac{\lambda_{i}}{z_{i}}\int_{\Gamma^{\varepsilon}}\big{(}\sum\limits_{k=1}^{3}G_{k}\llbracket{\phi^{\varepsilon}}\rrbracket\llbracket{C_{i}^{\varepsilon}}\rrbracket+\sum\limits_{k=1}^{3}\dfrac{G_{k}}{z_{k}}\llbracket{\ln
C_{k}^{\varepsilon}}\rrbracket\llbracket{C_{i}^{\varepsilon}}\rrbracket+I_{p}^{\varepsilon}\llbracket{C_{i}^{\varepsilon}}\rrbracket\big{)}~{}dS$

Combining the above equaiton, (3.5), (2.1) and (3.4a) yields

$\displaystyle\sum\limits_{i=1}^{3}(\partial_{t}\|C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\varepsilon\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}})$
(3.6) $\displaystyle\leq$ $\displaystyle
A_{1}\left(K_{1}\|\nabla\phi^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\dfrac{1}{K_{1}}\sum\limits_{i=1}^{3}\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}\right)+A_{2}\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+A_{3}\varepsilon\sum\limits_{i=1}^{3}\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+A_{4}.$

where we used the fact that $I_{P}^{\varepsilon}$ is bounded.

From assumption (2.2), we can multiply (3.4b) with
$\frac{2A_{1}K_{1}}{DC_{l}}$ and add it to (3.6) to get

$\displaystyle\sum\limits_{i=1}^{3}(\partial_{t}\|C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\varepsilon\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}})+A_{1}K_{1}\|\nabla\phi^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+A_{2}\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+A_{3}\varepsilon\partial_{t}\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}$
$\displaystyle\leq$
$\displaystyle\dfrac{A_{4}}{K_{1}}\sum\limits_{i=1}^{3}\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+A_{5}\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+A_{6}\varepsilon\sum\limits_{i=1}^{3}\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+A_{7}$
$\displaystyle\leq$
$\displaystyle\dfrac{A_{4}}{K_{1}}\sum\limits_{i=1}^{3}\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+A_{5}\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+A_{8}\varepsilon\sum\limits_{i=1}^{3}\left(\varepsilon^{-1}K_{2}\|C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\dfrac{1}{K_{2}}\varepsilon\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}\right)+A_{7},$

where $K_{1},K_{2}$ are two positive constants, and the trace theorem [16, 31]
has been used

$\left\|v_{s}\right\|_{L^{2}\left(\Gamma^{\epsilon}\right)}\leq
A\left(\delta\varepsilon^{-\frac{1}{2}}\left\|v_{s}\right\|_{L^{2}\left(\Omega_{s}^{\varepsilon}\right)}+\dfrac{1}{\delta}\varepsilon^{\frac{1}{2}}\left\|\nabla
v_{s}\right\|_{L^{2}\left(\Omega_{s}^{\varepsilon}\right)}\right),$ (3.7)

for $\forall~{}v_{s}\in H^{1}(\Omega_{s}^{\varepsilon}),~{}s=I,E$.

Choosing $K_{1},K_{2}$ big enough, we have

$\displaystyle\sum\limits_{i=1}^{3}(\partial_{t}\|C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\varepsilon\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}})+\|\nabla\phi^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}+\varepsilon\partial_{t}\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}$
(3.8) $\displaystyle\leq$ $\displaystyle
A_{1}\bigg{(}\sum\limits_{i=1}^{3}\|C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|_{\Gamma^{\varepsilon}}^{2}\bigg{)}+A_{2}.$

Then by using Gronwall inequality, we obtain

$\displaystyle\sup_{t\in[0,T]}(\sum\limits_{i=1}^{3}\|C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}+\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|_{\Gamma^{\varepsilon}}^{2})+\sum\limits_{i=1}^{3}\int_{0}^{T}\|\nabla
C_{i}^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}~{}dt$ (3.9) $\displaystyle+$
$\displaystyle\int_{0}^{T}\|\nabla\phi^{\varepsilon}\|^{2}_{\Omega^{\varepsilon}}~{}dt+\sum\limits_{i=1}^{3}\int_{0}^{T}\varepsilon\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|_{\Gamma^{\varepsilon}}^{2}~{}dt+\int_{0}^{T}\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|_{\Gamma^{\varepsilon}}^{2}~{}dt\leq
A.$

This leads to (3.3a) and (3.3b).

For the $L^{2}$ estimates of $\phi^{\varepsilon}$, by assumption (3.2), Lemma
A.5 and (3.9), we obtain

$\int_{0}^{T}\big{(}\|\phi_{I}^{\varepsilon}\|^{2}_{\Omega_{I}^{\varepsilon}}+\|\phi_{E}^{\varepsilon}\|^{2}_{\Omega_{E}^{\varepsilon}}\big{)}~{}dt\leq
A_{1}\int_{0}^{T}\big{(}\|\nabla\phi_{I}^{\varepsilon}\|^{2}_{\Omega_{I}^{\varepsilon}}+\|\nabla\phi_{E}^{\varepsilon}\|^{2}_{\Omega_{E}^{\varepsilon}}+\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}\big{)}~{}dt\leq
A_{2},$

(3.3f) is obtained by combining this with (3.9).

Using again the inequality (3.7), (3.3c) follows from (3.3b) and (3.3e)
follows from (3.3f).

Lastly we show
$\|\partial_{t}C^{\varepsilon}_{i,s}\|_{L^{2}(0,T;H^{-1}(\Omega^{\varepsilon}_{s}))},i=1,2,3,s=I,E$
are bounded, we only consider $s=I$ since other cases can be discussed
similarly. From (3.1b) we have

$\displaystyle\dfrac{\varepsilon\lambda_{i}}{z_{i}}P_{m}\int_{\Gamma^{\varepsilon}}\partial_{t}\llbracket{\phi^{\varepsilon}}\rrbracket
v_{I}~{}dS$ (3.10) $\displaystyle=$
$\displaystyle-\dfrac{\lambda_{i}}{z_{i}}\int_{\Omega_{I}^{\varepsilon}}\sigma_{I}^{\varepsilon}D\nabla\phi_{I}^{\varepsilon}\nabla
v_{I}~{}dx-\dfrac{\varepsilon\lambda_{i}}{z_{i}}\int_{\Gamma^{\varepsilon}}\sum\limits_{k=1}^{3}\left(G_{k}\llbracket{\phi^{\varepsilon}}\rrbracket
v_{I}+\dfrac{G_{k}}{z_{k}}\llbracket{\ln C_{k}^{\varepsilon}}\rrbracket
v_{I}+I_{p}^{\varepsilon}v_{I}\right)~{}dS.$

Substituting the above equation into (3.1a) leads to

$\displaystyle\|\partial_{t}C^{\varepsilon}_{1,I}\|_{H^{-1}(\Omega^{\varepsilon}_{I})}$
$\displaystyle\leq$ $\displaystyle
A_{1}\sup\limits_{\begin{subarray}{c}v_{1}\in
H^{1}(\Omega^{\varepsilon}_{I}),\\\
\|v_{1}\|_{H^{1}(\Omega^{\varepsilon}_{I})}\leq
1\end{subarray}}\left((\|\nabla
C^{\varepsilon}_{1,I}\|_{\Omega^{\varepsilon}_{I}}+\|\nabla\phi^{\varepsilon}_{I}\|_{\Omega^{\varepsilon}_{I}})\|\nabla
v_{1}\|_{\Omega^{\varepsilon}_{I}}+\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|_{\Gamma_{\varepsilon}}\|v_{1}\|_{\Gamma_{\varepsilon}}\right.$
$\displaystyle\left.+\varepsilon\sum\limits_{i=1}^{3}\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|_{\Gamma_{\varepsilon}}\|v_{1}\|_{\Gamma_{\varepsilon}}+\sqrt{\varepsilon}\|v_{1}\|_{\Gamma_{\varepsilon}}\right)$
$\displaystyle\leq$ $\displaystyle A_{2}\big{(}\|\nabla
C^{\varepsilon}_{1,I}\|_{\Omega^{\varepsilon}_{I}}+\|\nabla\phi^{\varepsilon}_{I}\|_{\Omega^{\varepsilon}_{I}}+\sqrt{\varepsilon}\|\llbracket{\phi^{\varepsilon}}\rrbracket\|_{\Gamma_{\varepsilon}}+\sqrt{\varepsilon}\|\llbracket{C_{1}^{\varepsilon}}\rrbracket\|_{\Gamma_{\varepsilon}}\big{)}+A_{3},$

where we used $\|v_{1}\|_{\Gamma_{\varepsilon}}=O(\varepsilon^{-1/2})$.
Finally, combining above equation with (3.9) yields

$\displaystyle\|\partial_{t}C^{\varepsilon}_{1,I}\|^{2}_{L^{2}(0,T;H^{-1}(\Omega^{\varepsilon}_{I}))}$
$\displaystyle\leq$ $\displaystyle A_{1}\int_{0}^{T}\left(\|\nabla
C^{\varepsilon}_{1,I}\|^{2}_{\Omega^{\varepsilon}_{I}}+\|\nabla\phi^{\varepsilon}_{I}\|_{\Omega^{\varepsilon}_{I}}+\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma_{\varepsilon}}+\varepsilon\sum\limits_{i=1}^{3}\|\llbracket{C_{i}^{\varepsilon}}\rrbracket\|^{2}_{\Gamma_{\varepsilon}}\right)~{}dt+A_{2}$
$\displaystyle\leq$ $\displaystyle A_{3}.$

$\square$

##  Homogenization

In this section, the homogenization theories for the EN model (2.4) and (2.5)
are developed with different connectivity. The homogenization process for
“connected-disconnected” case is shown in subsection 4.1, and then the
“connected-connected” case is shown in subsection 4.2.

###  “Connected-disconnected” case

In order to use compactness results from unfolding operators and two-scale
theory, we extend the functions
$C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon},i=1,2,3,s=I,E$ in a suitable way
to the whole domain $\Omega$. Since $\Omega_{E}^{\varepsilon}$ is connected
and has a Lipschitz boundary, by [26, Theorem 2.2], there exists a linear and
bounded extension operator
$\mathcal{L}^{\varepsilon}:~{}L^{2}\left((0,T);H^{1}\left(\Omega_{E}^{\epsilon}\right)\right)\rightarrow
L^{2}\left((0,T);H^{1}(\Omega)\right)$, we simply denote

$\widetilde{C}_{i,E}^{\varepsilon}=\mathcal{L}^{\varepsilon}C_{i,E}^{\varepsilon},~{}\widetilde{\phi}_{E}^{\varepsilon}=\mathcal{L}^{\varepsilon}\phi_{E}^{\varepsilon}.$
(4.1)

The a priori estimates (3.3a), (3.3b) and (3.3f) remain valid for the
extensions $\widetilde{C}_{i,E}^{\varepsilon}$ and
$\widetilde{\phi}_{E}^{\varepsilon}$. Such extensions can not be applied to
the functions in $\Omega_{I}^{\varepsilon}$ since it is not connected [31].
Zero extensions is used instead. The zero extension of a function defined on
$\Omega_{s}^{\varepsilon}$ for $s=I,E$ will be denoted by
$\overline{C}_{i,I}^{\varepsilon}$ and $\overline{\phi}_{I}^{\varepsilon}$. It
is obvious that the time derivative of $\overline{C}_{i,I}^{\varepsilon}$
satisfies

$\displaystyle\left\langle\partial_{t}\overline{C}_{i,s}^{\varepsilon},\psi\right\rangle_{\Omega}=\left\langle\partial_{t}{C}_{i,s}^{\varepsilon},\psi\right\rangle_{\Omega_{s}^{\epsilon}}\text{
for all }\psi\in H^{1}(\Omega)\text{ and a.e. }t\in(0,T),$ (4.2)
$\displaystyle\left\|\partial_{t}\overline{C}_{i,s}^{\varepsilon}\right\|_{L^{2}\left((0,T);H^{-1}(\Omega)\right)}\leq\left\|\partial_{t}{C}_{i,s}^{\varepsilon}\right\|_{L^{2}((0,T);H^{-1}(\Omega_{s}^{\epsilon}))}\leq
A,$ (4.3)

where $\langle\cdot,\cdot\rangle_{G}$ represent the duality paring
$\langle\cdot,\cdot\rangle_{H^{-1}(G)\times H^{1}(G)}$ for an arbitrary
measurable set $G$. Since unfolding operators and two-scale convergence method
are used in our proof, we give some definitions and properties for them in the
Appendix. First we give a lemma.

######  _L_ emma 4.1.

Suppose $a_{\varepsilon},b_{\varepsilon},c_{\varepsilon}\in L^{2}(\omega)$ be
three series such that

$a_{\varepsilon}\rightarrow a~{}~{}\text{strongly
in}~{}L^{2}(\omega),~{}~{}b_{\varepsilon}\rightarrow b~{}~{}\text{strongly
in}~{}L^{2}(\omega),~{}~{}c_{\varepsilon}\rightharpoonup c~{}~{}\text{weakly
in}~{}L^{2}(\omega),$

and $\|a_{\varepsilon}\|_{L^{\infty}(\omega)}\leq
A,~{}\|a\|_{L^{\infty}(\omega)}\leq
A,~{}\|b_{\varepsilon}\|_{L^{\infty}(\omega)}\leq
A,~{}\|b\|_{L^{\infty}(\omega)}\leq A$ for some open subset
$\omega\in\mathbb{R}^{d}$, then the following convergence result holds:

$\displaystyle\int_{\omega}a_{\varepsilon}b_{\varepsilon}c_{\varepsilon}~{}dx\rightarrow\int_{\omega}abc~{}dx.$
(4.4)

Proof: We have

$\displaystyle\int_{\omega}a_{\varepsilon}b_{\varepsilon}c_{\varepsilon}~{}dx=\int_{\omega}abc~{}dx+\int_{\omega}ab(c_{\varepsilon}-c)~{}dx+\int_{\omega}a(b_{\varepsilon}-b)c_{\varepsilon}~{}dx+\int_{\omega}(a_{\varepsilon}-a)b_{\varepsilon}c_{\varepsilon}~{}dx,$

it is obvious that the limits of the last three terms on the right hand side
are zero. For example, for the last term on the right hand side, we have

$\left|\int_{\omega}(a_{\varepsilon}-a)b_{\varepsilon}c_{\varepsilon}~{}dx\right|\leq
A\int_{\omega}|(a_{\varepsilon}-a)c_{\varepsilon}|~{}dx\rightarrow 0,$

the assertion is proved.

The following theorem is the convergence and homogenization result for
“connected-disconnected” case:

######  _T_ heorem 4.1.

Suppose the intracellular region $\Omega_{I}^{\varepsilon}$ is disconnected
and assumption (3.2) holds, let
$C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon},~{}i=1,2,3,s=I,E$ be the
solutions of (2.4) and (2.5), then there exist subsequences of
$C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon}$, still denoted as
$C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon}$, and $C_{i,E}\in
L^{2}(0,T;H^{1}(\Omega)),~{}\widehat{C}_{i,E}\in
L^{2}((0,T)\times\Omega;H^{1}_{per}(Y))$, $C_{i,I}\in
L^{2}((0,T)\times\Omega),~{}\widehat{C}_{i,I}\in
L^{2}((0,T)\times\Omega;H^{1}(Y))$ and $\phi_{E}\in
L^{2}(0,T;H^{1}(\Omega)),~{}\widehat{\phi}_{E}\in~{}L^{2}((0,T)\times\Omega;H^{1}_{per}(Y)),~{}\phi_{I}\in
L^{2}((0,T)\times\Omega),~{}\widehat{\phi}_{I}\in~{}L^{2}((0,T)\times\Omega;H^{1}(Y))$,
such that, for $i=1,2,3$

$\displaystyle\mathcal{T}^{\varepsilon}(\widetilde{C}_{i,E}^{\varepsilon})\rightarrow
C_{i,E}~{}~{}$ $\displaystyle\text{strongly
in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.5a)
$\displaystyle\mathcal{T}^{\varepsilon}(\nabla\widetilde{C}_{i,E}^{\varepsilon})\rightharpoonup\nabla_{x}C_{i,E}+\nabla_{y}\widehat{C}_{i,E}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.5b)
$\displaystyle\mathcal{T}_{b}^{\varepsilon}(\widetilde{C}_{i,E}^{\varepsilon})\rightarrow
C_{i,E}~{}~{}$ $\displaystyle\text{strongly
in}~{}L^{2}((0,T)\times\Omega\times\Gamma),$ (4.5c)
$\displaystyle\partial_{t}\overline{C}_{i,E}^{\varepsilon}\rightharpoonup|Y_{E}|\partial_{t}C_{i,E}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}(0,T;H^{-1}(\Omega)).$ (4.5d)
$\displaystyle\mathcal{T}^{\varepsilon}(\overline{C}_{i,I}^{\varepsilon})\rightarrow\beta_{I}C_{i,I}~{}~{}$
$\displaystyle\text{strongly in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.6a)
$\displaystyle\mathcal{T}^{\varepsilon}(\nabla\overline{C_{i,I}^{\varepsilon}})\rightharpoonup\beta_{I}\nabla_{y}\widehat{C}_{i,I}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.6b)
$\displaystyle\mathcal{T}_{b}^{\varepsilon}(\overline{C}_{i,I}^{\varepsilon})\rightarrow
C_{i,I}~{}~{}$ $\displaystyle\text{strongly
in}~{}L^{2}((0,T)\times\Omega\times\Gamma),$ (4.6c)
$\displaystyle\partial_{t}\overline{C}_{i,I}^{\varepsilon}\rightharpoonup|Y_{I}|\partial_{t}C_{i,I}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}(0,T;H^{-1}(\Omega)).$ (4.6d)
$\displaystyle\mathcal{T}^{\varepsilon}(\nabla\widetilde{\phi}_{E}^{\varepsilon})\rightharpoonup\nabla_{x}\phi_{E}+\nabla_{y}\widehat{\phi}_{E}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.7a)
$\displaystyle\mathcal{T}_{b}^{\varepsilon}(\widetilde{\phi}_{E}^{\varepsilon})\rightharpoonup\phi_{E}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times\Gamma),$ (4.7b)
$\displaystyle\mathcal{T}^{\varepsilon}(\overline{\nabla\phi_{I}^{\varepsilon}})\rightharpoonup\beta_{I}\nabla_{y}\widehat{\phi}_{I}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.7c)
$\displaystyle\mathcal{T}_{b}^{\varepsilon}({\phi}_{I}^{\varepsilon})\rightharpoonup\phi_{I}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times\Gamma).$ (4.7d)

And $C_{i,s},\phi_{s},~{}i=1,2,3,s=I,E$ are the solutions of the following
equations

$\displaystyle\begin{cases}|Y_{E}|\partial_{t}C_{i,E}-\nabla\cdot(D_{E}^{\star}\nabla
C_{i,E}+z_{i}C_{i,E}D_{E}^{\star}\nabla\phi_{E})\\\
=|\Gamma|\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}\ln\dfrac{C_{i,I}}{C_{i,E}}+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}&in~{}(0,T)\times\Omega,\\\
-(D_{E}^{\star}\nabla
C_{i,E}+z_{i}C_{i,E}D_{E}^{\star}\nabla\phi_{E})\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega,\\\
|Y_{I}|\partial_{t}C_{i,I}\\\
=-|\Gamma|\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}\ln\dfrac{C_{i,I}}{C_{i,E}}+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}&in~{}(0,T)\times\Omega,\\\
C_{i,I}|_{t=0}=C_{i,I}^{0},~{}~{}C_{i,E}|_{t=0}=C_{i,E}^{0}&in~{}\Omega,\end{cases}$
(4.8)
$\displaystyle\begin{cases}-\nabla\cdot\big{(}\sigma_{E}D_{E}^{\star}\nabla\phi_{E}\big{)}=0&in~{}(0,T)\times\Omega,\\\
-\sigma_{E}D_{E}^{\star}\nabla\phi_{E}\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega,\\\
P_{m}\partial_{t}(\phi_{I}-\phi_{E})=-\big{(}\sum\limits_{i=1}^{3}G_{i}(\phi_{I}-\phi_{E})+\sum\limits_{i=1}^{3}\dfrac{G_{i}}{z_{i}}\ln\dfrac{C_{i,I}}{C_{i,E}}+I_{p}^{0}\big{)}&in~{}(0,T)\times\Omega,\\\
(\phi_{I}-\phi_{E})|_{t=0}=\phi^{0}&in~{}\Omega,\end{cases}$ (4.9)

where $\sigma_{E}=\sum\limits_{i=1}^{3}z_{i}^{2}C_{i,E}$,
$D_{E}^{\star}=\int_{Y_{E}}D(I+\nabla_{y}\chi_{E})~{}dy$ and
$\chi_{E}=(\chi_{E}^{1},\chi_{E}^{2},\cdots,\chi_{E}^{d})$ is the solution of
(4.19), $P_{1}^{0}=3I_{p}^{0},P_{2}^{0}=-2I_{p}^{0},P_{3}^{0}=0$, and
$I_{p}^{0}$ has the same expression as in (2.7) but the concentrations are
replaced by the corresponding limits. And $C_{i,s}$ are electro-neutral, i.e.
$\sum\limits_{i=1}^{3}z_{i}C_{i,s}=0$ for $s=I,E$.

Proof: Convergence (4.5), (4.6) and (4.7) are the results of a priori estimate
Theorem 3.1 and relavent convergence results in [23]. Note that
$C_{E},~{}C_{I},~{}\phi_{E},~{}\phi_{I}$ are independent of $y$. This property
will be used in the following derivation process.

First, in (3.1b), let

$\varphi_{I}=\varepsilon u(x)v_{1}(x/\varepsilon)w(t),$ (4.10)

where $u\in C_{0}^{\infty}(\Omega),~{}v_{1}\in C_{per}^{\infty}(Y_{I}),~{}w\in
C_{0}^{\infty}(0,T)$. Then integrating (3.1b) with respect to time gives

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\int_{0}^{T}\int_{\Omega_{I}^{\varepsilon}}\sigma_{I}^{\varepsilon}D\nabla\phi_{I}^{\varepsilon}(\varepsilon
v_{1}w\nabla
u+uw\nabla_{y}v_{1})~{}dxdt+\int_{0}^{T}\int_{\Gamma_{\varepsilon}}\varepsilon^{2}\sum\limits_{i=1}^{3}G_{i}\llbracket{\phi^{\varepsilon}}\rrbracket
uv_{1}w~{}dSdt$ (4.11)
$\displaystyle+\int_{0}^{T}\\!\\!\\!\int_{\Gamma_{\varepsilon}}\varepsilon^{2}\sum\limits_{i=1}^{3}\dfrac{G_{i}}{z_{i}}\llbracket{\ln
C_{i}^{\varepsilon}}\rrbracket
uv_{1}w~{}dSdt-\int_{0}^{T}\\!\\!\\!\int_{\Gamma_{\varepsilon}}\varepsilon^{2}P_{m}\llbracket{\phi^{\varepsilon}}\rrbracket
uv_{1}\partial_{t}w~{}dSdt$
$\displaystyle+\int_{0}^{T}\\!\\!\\!\int_{\Gamma_{\varepsilon}}\varepsilon^{2}I_{p}uv_{1}w~{}dSdt$
$\displaystyle=$ $\displaystyle M_{1}+M_{2}+M_{3}-M_{4}+M_{5}.$

For $M_{1}$, using the property of unfolding operator Lemma A.2, we deduce

$\displaystyle M_{1}=\int_{0}^{T}\int_{\Omega\times
Y_{I}}\mathcal{T}^{\varepsilon}\left(\overline{\sigma}_{I}^{\varepsilon}\right)D~{}\mathcal{T}^{\varepsilon}\left(\overline{\nabla\phi_{I}^{\varepsilon}}\right)~{}(\varepsilon
v_{1}w\mathcal{T}^{\varepsilon}\left(\nabla
u\right)+\mathcal{T}^{\varepsilon}\left(u\right)w\nabla_{y}v_{1})~{}dxdydt.$
(4.12)

By (4.6a) and the relation (2.6):
$\sigma_{I}^{\varepsilon}=\sum\limits_{i=1}^{3}z_{i}^{2}C_{i,I}^{\varepsilon}$,
it follows that

$\displaystyle\mathcal{T}^{\varepsilon}\left(\overline{\sigma}_{I}^{\varepsilon}\right)\rightarrow\beta_{I}\sigma_{I}~{}~{}~{}\text{strongly
in}~{}L^{2}((0,T)\times\Omega\times Y_{I}),$ (4.13)

where $\sigma_{I}=\sum\limits_{i=1}^{3}z_{i}^{2}C_{i,I}$. From Lemma A.4, it
is obvious that, as $\varepsilon\rightarrow 0$

$\displaystyle(\varepsilon v_{1}w\mathcal{T}^{\varepsilon}\left(\nabla
u\right)+\mathcal{T}^{\varepsilon}\left(u\right)w\nabla_{y}v_{1})\rightarrow
uw\nabla_{y}v_{1}~{}~{}~{}\text{strongly in}~{}L^{2}((0,T)\times\Omega\times
Y_{I}).$ (4.14)

From (4.14), (4.13), (4.7c), and Lemma 4.1, passing to the limit for (4.12) as
$\varepsilon\rightarrow 0$, while noticing that
$C_{0}^{\infty}((0,T)\times\Omega)\bigotimes C_{per}^{\infty}(Y_{I})$ is dense
in $L^{2}((0,T)\times\Omega;H^{1}(Y_{I}))$, we deduce

$\displaystyle M_{1}\rightarrow\int_{0}^{T}\int_{\Omega\times
Y_{I}}\sigma_{I}D\nabla_{y}\widehat{\phi}_{I}\nabla_{y}\Phi_{I}~{}dxdydt.$

for $\forall~{}\Phi_{I}\in L^{2}((0,T)\times\Omega;H^{1}(Y_{I}))$. For
$M_{2}$, from Lemma A.2 we have

$\displaystyle
M_{2}=\int_{0}^{T}\int_{\Omega\times\Gamma}\sum\limits_{i=1}^{3}G_{i}\big{(}\mathcal{T}_{b}^{\varepsilon}\left(\phi_{I}^{\varepsilon}\right)-\mathcal{T}_{b}^{\varepsilon}(\widetilde{\phi}_{E}^{\varepsilon})\big{)}\varepsilon\mathcal{T}_{b}^{\varepsilon}\left(u\right)v_{1}w~{}dxdS_{y}dt,$

Obviously,

$\varepsilon\mathcal{T}_{b}^{\varepsilon}\left(u\right)v_{1}w~{}\rightarrow~{}0~{}~{}~{}\text{strongly
in}~{}L^{2}((0,T)\times\Omega\times\Gamma),$

by (4.7b) and (4.7d) it follows that $M_{2}\rightarrow 0$. For $M_{4}$, the
result is similar, i.e. $M_{4}\rightarrow 0$.

Then, for $M_{3}$, noticing that $\mathcal{T}_{b}^{\varepsilon}(\ln
C_{i,s}^{\varepsilon})=\ln\mathcal{T}_{b}^{\varepsilon}(C_{i,s}^{\varepsilon})$,
from Lemma A.2 it follows

$M_{3}=\int_{0}^{T}\int_{\Omega\times\Gamma}\sum\limits_{i=1}^{3}(\ln\mathcal{T}_{b}^{\varepsilon}(\overline{C}_{i,I}^{\varepsilon})-\ln\mathcal{T}_{b}^{\varepsilon}(\widetilde{C}_{i,E}^{\varepsilon}))\varepsilon\mathcal{T}_{b}^{\varepsilon}(u)v_{1}w~{}dxdS_{y}dt.$

The results in [23, Corrolary 15] yield that
$\ln(\mathcal{T}_{b}^{\varepsilon}(\widetilde{C}_{i,E}^{\varepsilon}))$ and
$\mathcal{T}_{b}^{\varepsilon}(I_{p}^{\varepsilon})$ converges weakly in
$L^{2}((0,T)\times\Omega\times\Gamma)$ to $\ln(C_{i,E})$ and $I_{p}^{0}$
respectively. Then we have $\varepsilon\mathcal{T}_{b}^{\varepsilon}(u)v_{1}w$
converges to zero strongly in $L^{2}((0,T)\times\Omega\times\Gamma)$. So the
limits of $M_{3}$ and $M_{5}$ are zero as $\varepsilon\rightarrow 0$.

Summarizing the above convergence results for $M_{1},M_{2},M_{3},M_{4},M_{5}$,
we deduce that the limit equation for (4.11) as $\varepsilon\rightarrow 0$ is

$\int_{0}^{T}\int_{\Omega\times
Y_{I}}\sigma_{I}D\nabla_{y}\widehat{\phi}_{I}\nabla_{y}\Phi_{I}~{}dxdydt=0,$
(4.15)

for $\forall~{}\Phi_{I}\in L^{2}((0,T)\times\Omega;H^{1}(Y_{I}))$. Let
$\Phi_{I}=\widehat{\phi}_{I}$ and noticing $\sigma_{I}>0$, we obtain

$\nabla_{y}\widehat{\phi}_{I}=0.$ (4.16)

Similar results were derived in, for example [23, Proposition 11] and [22,
Theorem 3.3].

Then let $\varphi_{I}=\varphi_{1}(t,x)$ in (3.1b), where $\varphi_{1}\in
C_{0}^{\infty}((0,T)\times\Omega)$. Using similar arguments as for
$M_{1},M_{2},M_{3},M_{4},M_{5}$ while noticing (4.16), we obtain, as
$\varepsilon\rightarrow 0$,

$\displaystyle\int_{0}^{T}\int_{\Omega\times\Gamma}\bigg{[}\sum\limits_{i=1}^{3}G_{i}(\phi_{I}-\phi_{E})+\sum\limits_{i=1}^{3}\dfrac{G_{i}}{z_{i}}(\ln
C_{i,I}-\ln
C_{i,E})+I_{p}^{0}+P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{]}\varphi_{1}~{}dxdS_{y}dt=0.$

Since $\phi_{I},~{}\phi_{E}$ and $C_{i,I},~{}C_{i,E}$ are independent of $y$,
so

$\displaystyle\sum\limits_{i=1}^{3}G_{i}(\phi_{I}-\phi_{E})+\sum\limits_{i=1}^{3}\dfrac{G_{i}}{z_{i}}(\ln
C_{i,I}-\ln C_{i,E})+I_{p}^{0}+P_{m}\partial_{t}(\phi_{I}-\phi_{E})=0.$ (4.17)

If we let $\varphi_{I}=u(x)w_{1}(t)$ in (3.1b) and pass to the limit as
$\varepsilon\rightarrow 0$, where $w_{1}\in C^{\infty}(0,T),w_{1}(T)=0$, then
the initial condition for $\phi_{I}-\phi_{E}$ can be derived as

$\displaystyle(\phi_{I}-\phi_{E})|_{t=0}=\phi^{0}.$

Then in (3.1b), let $\varphi_{E}=\varepsilon u(x)v_{2}(x/\varepsilon)w(t)$,
where $v_{2}\in C_{per}^{\infty}(Y_{E})$ and $u,w$ are the same as in (4.10).
Using similar discussion as for (4.11), from (4.7a) and (4.4) we have, after
passing to the limit for (3.1b) as $\varepsilon\rightarrow 0$,

$2\int_{0}^{T}\int_{\Omega\times
Y_{E}}\sigma_{E}D(\nabla_{x}\phi_{E}+\nabla_{y}\widehat{\phi}_{E})\nabla_{y}\Phi_{E}~{}dxdydt=0,$
(4.18)

for $\forall~{}\Phi_{E}\in L^{2}((0,T)\times\Omega;H_{per}^{1}(Y_{E}))$.

Above equation yields the following representation for $\widehat{\phi}_{E}$

$\widehat{\phi}_{E}=\nabla_{x}\phi_{E}~{}\cdot\chi_{E},$

where $\chi_{E}=(\chi_{E}^{1},\chi_{E}^{2},\cdots,\chi_{E}^{d})$, satisfying
the following auxiliary problem

$\displaystyle\begin{cases}-\nabla_{y}\cdot\big{(}D\nabla_{y}(y_{j}+\chi_{E}^{j})\big{)}=0~{}~{}~{}~{}~{}~{}in~{}Y_{E},\\\
-D\nabla_{y}(y_{j}+\chi_{E}^{j})\cdot\vec{n}_{1}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}on~{}\Gamma,\\\
\chi_{E}^{j}~{}is~{}Y-periodic,~{}~{}~{}~{}\int_{Y_{E}}\chi_{E}^{j}=0.\end{cases}$
(4.19)

If we let $\varphi_{E}=\varphi_{1}(t,x)$ in (3.1b), from (4.17) and (4.4),
using density property, it yields the limit for (3.1b) as
$\varepsilon\rightarrow 0$

$2\int_{0}^{T}\int_{\Omega}\sigma_{E}D_{E}^{\star}\nabla\phi_{E}\nabla\Phi_{1}~{}dxdt=0,$

for $\forall~{}\Phi_{1}\in L^{2}((0,T);H^{1}(\Omega))$, where
$D_{E}^{\star}=\int_{Y_{E}}D(I+\nabla_{y}\chi_{E})~{}dy$. The strong form of
the above equation is

$\displaystyle\begin{cases}-\nabla\cdot\big{(}\sigma_{E}D_{E}^{\star}\nabla\phi_{E}\big{)}=0&in~{}(0,T)\times\Omega,\\\
-\sigma_{E}D_{E}^{\star}\nabla\phi_{E}\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega.\end{cases}$
(4.20)

Now consider the equations for $C_{i}^{\varepsilon}$. Let $v_{I}=\varepsilon
u(x)v_{1}(x/\varepsilon)w(t)$ in (3.1a), where $u,v_{1},w$ is the same as in
(4.10). We deal with the interface integral similarly as for (4.11). From
(4.6b), (4.4), (4.16) and density property, passing to the limit for (3.1a) as
$\varepsilon\rightarrow 0$ leads to

$\int_{0}^{T}\int_{\Omega\times
Y_{I}}D\nabla_{y}\widehat{C}_{i,I}\nabla_{y}\Phi_{I}~{}dxdydt=0,$ (4.21)

for $\forall~{}\Phi_{I}\in L^{2}((0,T)\times\Omega;H^{1}(Y_{I}))$. Let
$\Phi_{I}=\widehat{C}_{i,I}$ in the above equation, we have

$\nabla_{y}\widehat{C}_{i,I}=0.$ (4.22)

If let $v_{I}=\varphi_{1}(t,x)$ in (3.1a), using (4.6d), (4.4), (4.16), (4.22)
and passing to the limit for (3.1a) as $\varepsilon\rightarrow 0$ yields

$\displaystyle|Y_{I}|\int_{0}^{T}\int_{\Omega}\partial_{t}C_{i,I}uw_{1}~{}dxdt$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}\int_{\Omega\times\Gamma}\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}(\ln
C_{i,I}-\ln
C_{i,E})+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}uw_{1}~{}dxdS_{y}dt.$

The integrand in the above equation is independent of $y$, so it is equivalent
to

$|Y_{I}|\partial_{t}C_{i,I}=-|\Gamma|\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}(\ln
C_{i,I}-\ln
C_{i,E})+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}.$
(4.23)

To derive the initial conditions for $C_{i,I}$, if we let $v_{I}=u(x)w_{1}(t)$
in (3.1a) and pass to the limit as $\varepsilon\rightarrow 0$, it is easy to
show that $C_{i,I}|_{t=0}=C_{i,I}^{0}$.

For $C_{i,E}^{\varepsilon}$, let $v_{E}=\varepsilon
u(x)v_{2}(x/\varepsilon)w(t)$ in (3.1b). From (4.4), passing to the limit as
$\varepsilon\rightarrow 0$ leads to

$\displaystyle\int_{0}^{T}\int_{\Omega\times Y_{E}}\left(D(\nabla
C_{i,E}+\nabla_{y}\widehat{C}_{i,E})+z_{i}C_{i,E}D(\nabla_{x}\phi_{E}+\nabla_{y}\widehat{\phi}_{E})\right)\nabla_{y}\Phi_{E}~{}dxdydt$
(4.24)

for $\forall~{}\Phi_{E}\in L^{2}((0,T)\times\Omega;H_{per}^{1}(Y_{E}))$.

Thanks to (4.18), the second term on the left hand side is zero, and the
following representation holds,

$\widehat{C}_{i,E}=\nabla_{x}C_{i,E}~{}\cdot\chi_{E}.$

Lastly, if let $v_{E}=\varphi_{1}(t,x)$ in (3.1a) and pass to the limit as
$\varepsilon\rightarrow 0$, we have by density property that

$\displaystyle\int_{0}^{T}\int_{\Omega}\left(|Y_{E}|\partial_{t}C_{i,E}\Phi_{1}+D_{E}^{\star}\nabla
C_{i,E}\nabla\Phi_{1}+z_{i}C_{i,E}D_{E}^{\star}\nabla\phi_{E}\nabla\Phi_{1}\right)~{}dxdt$
$\displaystyle-$
$\displaystyle\int_{0}^{T}\int_{\Omega\times\Gamma}\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}(\ln
C_{i,I}-\ln
C_{i,E})+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}\Phi_{1}~{}dxdS_{y}dt=0,$

for $\forall~{}\Phi_{1}\in L^{2}((0,T);H^{1}(\Omega))$.

The derivation of initial condition for $C_{i,E}$ is the same as for
$C_{i,I}$, so we omit it. The strong form for the above equation is

$\displaystyle\begin{cases}|Y_{E}|\partial_{t}C_{i,E}-\nabla\cdot(D_{E}^{\star}\nabla
C_{i,E}+z_{i}C_{i,E}D_{E}^{\star}\nabla\phi_{E})=|\Gamma|\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\\\
\dfrac{G_{i}}{z^{2}_{i}}(\ln C_{i,I}-\ln
C_{i,E})+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}&in~{}(0,T)\times\Omega,\\\
-(D_{E}^{\star}\nabla
C_{i,E}+z_{i}C_{i,E}D_{E}^{\star}\nabla\phi_{E})\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega,\\\
C_{i,E}|_{t=0}=C_{i,E}^{0}&in~{}\Omega.\end{cases}$ (4.25)

Now we have derived all the homogenization equations, i.e. (4.25), (4.23),
(4.17), (4.20), which are (4.8) and (4.9). The electro-neutrality condition
for $C_{i,s}$ is obvious since the initial value of $C_{i,s}$ is electro-
neutral. The theorem is proved. $\square$

###  “Connected-connected” case

Now we consider the “connected-connected” case. In this case, the status of
$\Omega_{I}^{\varepsilon}$ and $\Omega_{E}^{\varepsilon}$ are “equivalent”,
which means the homogenized equations for both regions are in the same form.
Noticing that $\Omega_{I}^{\varepsilon}$ also reaches the boundary of
$\partial\Omega$, we need to add boundary conditions for
$C_{I}^{\varepsilon},~{}\phi_{I}^{\varepsilon}$ on $\partial\Omega$, i.e.
$J_{i,I}\cdot\vec{\nu}=0,~{}\text{and}~{}D\sigma_{I}^{\varepsilon}\nabla\phi_{I}^{\varepsilon}\cdot\vec{\nu}=0,~{}~{}on~{}\partial\Omega$.

Since now both $\Omega_{I}^{\varepsilon}$ and $\Omega_{E}^{\varepsilon}$ are
connected, by [26, Theorem 2.2], there exist linear and bounded extension
operators
$\mathcal{L}_{s}^{\varepsilon}:~{}L^{2}\left((0,T),H^{1}\left(\Omega_{s}^{\epsilon}\right)\right)\rightarrow
L^{2}\left((0,T),H^{1}(\Omega)\right),s=I,E$. Simply denote

$\widetilde{C}_{i,s}^{\varepsilon}=\mathcal{L}_{s}^{\varepsilon}C_{i,s}^{\varepsilon},~{}\widetilde{\phi}_{s}^{\varepsilon}=\mathcal{L}_{s}^{\varepsilon}\phi_{s}^{\varepsilon},~{}~{}~{}s=I,E.$
(4.26)

The priori estimates (3.3a), (3.3b) and (3.3f) remain valid for the extensions
$\widetilde{C}_{i,s}^{\varepsilon}$ and $\widetilde{\phi}_{s}^{\varepsilon}$.
The convergence and homogenization results for “connected-connected” case is
summarized in the following theorem.

######  _T_ heorem 4.2.

Suppose $\Omega_{I}^{\varepsilon}$ is connected and assumption (3.2) holds,
let $C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon},~{}i=1,2,3,s=I,E$ be
solutions of (2.4) and (2.5), then there exist subsequences of
$C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon}$, still denoted as
$C_{i,s}^{\varepsilon},\phi_{s}^{\varepsilon}$, and $C_{i,s},\phi_{s}\in
L^{2}(0,T;H^{1}(\Omega)),~{}\widehat{C}_{i,s},\widehat{\phi}_{s}\in
L^{2}((0,T)\times\Omega;H^{1}_{per}(Y))$, such that for $i=1,2,3,s=I,E$,

$\displaystyle\mathcal{T}^{\varepsilon}(\widetilde{C}_{i,s}^{\varepsilon})\rightarrow
C_{i,s}~{}~{}$ $\displaystyle\text{strongly
in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.27a)
$\displaystyle\mathcal{T}^{\varepsilon}(\nabla\widetilde{C}_{i,s}^{\varepsilon})\rightharpoonup\nabla_{x}C_{i,s}+\nabla_{y}\widehat{C}_{i,s}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.27b)
$\displaystyle\mathcal{T}_{b}^{\varepsilon}(\widetilde{C}_{i,s}^{\varepsilon})\rightarrow
C_{i,s}~{}~{}$ $\displaystyle\text{strongly
in}~{}L^{2}((0,T)\times\Omega\times\Gamma),$ (4.27c)
$\displaystyle\partial_{t}\overline{C}_{i,s}^{\varepsilon}\rightharpoonup|Y_{s}|\partial_{t}C_{i,s}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}(0,T;H^{-1}(\Omega)).$ (4.27d)
$\displaystyle\mathcal{T}^{\varepsilon}(\nabla\widetilde{\phi}_{s}^{\varepsilon})\rightharpoonup\nabla_{x}\phi_{s}+\nabla_{y}\widehat{\phi}_{s}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times Y),$ (4.28a)
$\displaystyle\mathcal{T}_{b}^{\varepsilon}(\widetilde{\phi}_{s}^{\varepsilon})\rightharpoonup\phi_{s}~{}~{}$
$\displaystyle\text{weakly in}~{}L^{2}((0,T)\times\Omega\times\Gamma).$
(4.28b)

$C_{i,s},\phi_{s},~{}i=1,2,3,s=I,E$ are the solutions of the following
equations

$\displaystyle\begin{cases}|Y_{s}|\partial_{t}C_{i,s}-\nabla\cdot(D_{s}^{\star}\nabla
C_{i,s}+z_{i}C_{i,s}D_{s}^{\star}\nabla\phi_{s})\\\
=\pm|\Gamma|\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}\ln\dfrac{C_{i,I}}{C_{i,E}}+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}&in~{}(0,T)\times\Omega,\\\
(D_{s}^{\star}\nabla
C_{i,s}+z_{i}C_{i,s}D_{s}^{\star}\nabla\phi_{s})\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega,\\\
C_{i,s}|_{t=0}=C_{i,s}^{0}&in~{}\Omega,\end{cases}$ (4.29)
$\displaystyle\begin{cases}-\nabla\cdot\big{(}\sigma_{s}D_{s}^{\star}\nabla\phi_{s}\big{)}\\\
=\pm\bigg{(}\sum\limits_{i=1}^{3}G_{i}(\phi_{I}-\phi_{E})+\sum\limits_{i=1}^{3}\dfrac{G_{i}}{z_{i}}\ln\dfrac{C_{i,I}}{C_{i,E}}+I_{p}^{0}+P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}&in~{}(0,T)\times\Omega,\\\
\sigma_{s}D_{s}^{\star}\nabla\phi_{s}\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega,\\\
(\phi_{I}-\phi_{E})|_{t=0}=\phi^{0}&in~{}\Omega,\end{cases}$ (4.30)

where “$\pm$” takes “$+$” for $s=E$ and “$-$” for $s=I$.
$D_{s}^{\star}=\int_{Y_{s}}D(I+\nabla_{y}\chi_{s})~{}dy$,
$\sigma_{s}=\sum\limits_{i=1}^{3}z_{i}^{2}C_{i,s}$ and
$\chi_{s}=(\chi_{s}^{1},\chi_{s}^{2},\cdots,\chi_{s}^{d}),s=I,E$ are the
solutions of (4.19) and (4.32), $P_{i}^{0},I_{p}^{0}$ have the same meaning as
in theorem 4.1. And $C_{i,s}$ are electro-neutral, i.e.
$\sum\limits_{i=1}^{3}z_{i}C_{i,s}=0$ for $s=I,E$.

Proof: The proof of the convergence results (4.27) and (4.28) are the same as
the proof of the convergence results for
$\widetilde{C}_{i,E}^{\varepsilon},\widetilde{\phi}_{E}^{\varepsilon}$ in
Theorem 4.1, so we omit it. Since the status of $\Omega_{I}^{\varepsilon}$ and
$\Omega_{E}^{\varepsilon}$ are “equivalent” in “connected-connected” case, we
can see that the convergence results are the same for intracellular functions
and extracellular functions. The homogenized equations (4.29) and (4.30) are
different from the results in Theorem 4.1, but the proof are quite similar, so
we only give a simplified proof and point out the differences. As the
derivation of homogenized equations in $\Omega_{I}^{\varepsilon}$ and
$\Omega_{E}^{\varepsilon}$ are the same, we only consider equations in
$\Omega_{I}^{\varepsilon}$.

First consider $\phi_{I}^{\varepsilon}$. Let $\varphi_{I}=\varepsilon
u(x)v_{1}(x/\varepsilon)w(t)$ in (3.1b), where $u\in
C_{0}^{\infty}(\Omega),~{}v_{1}\in C_{per}^{\infty}(Y_{I}),~{}w\in
C_{0}^{\infty}(0,T)$. By convergence result (4.27) and (4.28), we can use
similar argument as for (4.11) to get that, for $\forall~{}\Phi_{I}\in
L^{2}((0,T)\times\Omega;H_{per}^{1}(Y_{I}))$,

$\int_{0}^{T}\int_{\Omega\times
Y_{I}}\sigma_{I}D(\nabla_{x}\phi_{I}+\nabla_{y}\widehat{\phi}_{I})\nabla_{y}\Phi_{I}~{}dxdydt=0.$
(4.31)

This result is different from (4.15) in last subsection, which is due to
convergence result (4.28a) for $s=I$.

From (4.31) we have the following representation

$\widehat{\phi}_{I}=\nabla_{x}\phi_{I}~{}\cdot\chi_{I},$

where $\chi_{I}=(\chi_{I}^{1},\chi_{I}^{2},\cdots,\chi_{I}^{d})$, satisfying

$\displaystyle\begin{cases}-\nabla_{y}\cdot\big{(}D\nabla_{y}(y_{j}+\chi_{I}^{j})\big{)}=0~{}~{}~{}~{}~{}~{}in~{}Y_{I},\\\
-D\nabla_{y}(y_{j}+\chi_{I}^{j})\cdot\vec{n}_{1}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}on~{}\Gamma.\\\
\chi_{I}^{j}~{}is~{}Y-periodic,~{}~{}~{}~{}\int_{Y_{I}}\chi_{I}^{j}=0\end{cases}$
(4.32)

Then let $\varphi_{I}=\varphi_{1}(t,x)$ in (3.1b), where $\varphi_{1}\in
C_{0}^{\infty}((0,T)\times\Omega)$, from convergence result (4.27) and (4.28),
by density property, we can pass to the limit for (3.1b) as
$\varepsilon\rightarrow 0$ to get

$\displaystyle\int_{0}^{T}\int_{\Omega}\sigma_{I}D_{I}^{\star}\nabla\phi_{I}\nabla\Phi_{1}~{}dxdt$
$\displaystyle-$
$\displaystyle\int_{0}^{T}\int_{\Omega\times\Gamma}\bigg{[}\sum\limits_{i=1}^{3}G_{i}(\phi_{I}-\phi_{E})+\sum\limits_{i=1}^{3}\dfrac{G_{i}}{z_{i}}\ln\dfrac{C_{i,I}}{C_{i,E}}+I_{p}^{0}+P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{]}\Phi_{1}~{}dxdS_{y}dt=0,$

for $\forall\Phi_{1}\in L^{2}((0,T);H^{1}(\Omega))$, where
$D_{I}^{\star}=\int_{Y_{I}}D(I+\nabla_{y}\chi_{I})~{}dy$. The corresponding
strong form is

$\displaystyle\begin{cases}-\nabla\cdot\big{(}\sigma_{I}D_{I}^{\star}\nabla\phi_{I}\big{)}\\\
=-\bigg{(}\sum\limits_{i=1}^{3}G_{i}(\phi_{I}-\phi_{E})+\sum\limits_{i=1}^{3}\dfrac{G_{i}}{z_{i}}\ln\dfrac{C_{i,I}}{C_{i,E}}+I_{p}^{0}+P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}&in~{}(0,T)\times\Omega,\\\
\sigma_{I}D_{I}^{\star}\nabla\phi_{I}\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega,\end{cases}$
(4.33)

which is (4.30) for $s=I$.

Next we consider $C_{i,I}^{\varepsilon}$. Let $v_{I}=\varepsilon
u(x)v_{1}(x/\varepsilon)w(t)$ in (3.1a), passing to the limit as
$\varepsilon\rightarrow 0$ leads to

$\int_{0}^{T}\int_{\Omega\times Y_{I}}\left(D(\nabla
C_{i,I}+\nabla_{y}\widehat{C}_{i,I})+z_{i}C_{i,I}D(\nabla_{x}\phi_{I}+\nabla_{y}\widehat{\phi}_{I})\right)\nabla_{y}\Phi_{I}~{}dxdydt=0,$

for $\forall~{}\Phi_{I}\in L^{2}((0,T)\times\Omega;H_{per}^{1}(Y_{I}))$. This
is different from (4.21) in last subsection, which is due to the convergence
result (4.28a) and (4.27b) for $s=I$.

According to (4.31), the second term on the left hand side of the above
equation is zero, so we have

$\widehat{C}_{i,I}=\nabla_{x}C_{i,I}~{}\cdot\chi_{I},$

where $\chi_{I}$ is the solution of (4.32). Then, let $v_{I}=\varphi_{1}(t,x)$
in (3.1a), passing to the limit as $\varepsilon\rightarrow 0$ yields

$\displaystyle\int_{0}^{T}\int_{\Omega}\left(|Y_{I}|\partial_{t}C_{i,I}u_{1}w+D_{I}^{\star}\nabla
C_{i,I}\nabla u_{1}w+z_{i}C_{i,I}D_{I}^{\star}\nabla\phi_{I}\nabla
u_{1}w\right)~{}dxdt$ $\displaystyle+$
$\displaystyle\int_{0}^{T}\int_{\Omega\times\Gamma}\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}\dfrac{\ln
C_{i,I}}{\ln
C_{i,E}}+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}u_{1}w~{}dxdS_{y}dt=0.$

The corresponding strong form is

$\displaystyle\begin{cases}|Y_{I}|\partial_{t}C_{i,I}-\nabla\cdot(D_{I}^{\star}\nabla
C_{i,I}+z_{i}C_{i,I}D_{I}^{\star}\nabla\phi_{I})\\\
=-|\Gamma|\bigg{(}\dfrac{G_{i}}{z_{i}}(\phi_{I}-\phi_{E})+\dfrac{G_{i}}{z^{2}_{i}}\dfrac{\ln
C_{i,I}}{\ln
C_{i,E}}+\dfrac{P_{i}^{0}}{z_{i}}+\dfrac{\lambda_{i}}{z_{i}}P_{m}\partial_{t}(\phi_{I}-\phi_{E})\bigg{)}&in~{}(0,T)\times\Omega,\\\
-(D_{I}^{\star}\nabla
C_{i,I}+z_{i}C_{i,I}D_{I}^{\star}\nabla\phi_{I})\cdot\vec{\nu}=0&on~{}(0,T)\times\partial\Omega.\end{cases}$
(4.34)

which is (4.29) for $s=I$. The initial condition for $C_{i,I}$ can be obtained
by similar arguments as in the proof of Theorem 4.1. Summarizing the above
results, we have (4.29), (4.30). The electro-neutrality condition for
$C_{i,s}$ is a result of the electro-neutrality of $C_{i,s}^{0}$. The theorem
is proved. $\square$

##  Conclusion

In this paper, a micro-scale EN bi-domain ion transport model is first
proposed to model the ion transportation on tissues. On the membrane, we
considered the passive currents induced by ion channels, active currents
induced by pumps and currents induced by the capacitance property of membrane.
Then the homogenization theory for the nonlinear coupled system is derived
rigorously by using unfolding operator and two-scale convergence method.
Different connectivity conditions for intracellular region
$\Omega_{I}^{\varepsilon}$ leads to different homogenized equations. When both
the intracelluar region and extracellular region are connected to form a
syncytium, repectively, the obtained system is a diffusion-convection-reaction
system. While, when the intracellulars are disconnected, the macroscale
effective equation of intracellular region is only a reaction equation. In
this case, the intracellular could indirectly communicate through the
connected extracellular space. Our macroscale model could be used to study
some diseases induced by ion micro-circulation disorder, like spreading
depression [11] and lens problems [49].

##  Appendix A  Two-scale convergence and unfolding operator

We present in this Appendix some homogenization theory and results about two-
scale convergence and unfolding operator. The two-scale convergence theory in
periodic domains was first establish by Nguetseng [36] and further developed
by Allaire [1]. Later two-scale convergence was extended to periodic
interfaces [3, 34]. First we give the definition of two-scale convergence with
time “t” and strong two-scale convergence. Let $\omega$ be an open set in
$\mathbb{R}^{d}$, and $Y=(0,1)^{d}$.

######  _D_ efinition A.1.

Let $u_{\varepsilon}\subset L^{2}((0,T)\times\omega),~{}u_{0}\in
L^{2}((0,T)\times\omega\times Y)$, $u_{\varepsilon}$ is said to converge
(weakly) in the two-scale sense to $u_{0}$, if for $\forall\phi\in
C\left([0,T]\times\overline{\omega},C_{per}(\overline{Y})\right)$ the
following relation holds:

$\lim_{\varepsilon\rightarrow
0}\int_{0}^{T}\int_{\omega}u_{\varepsilon}(t,x)\phi\left(t,x,\frac{x}{\varepsilon}\right)~{}dxdt=\int_{0}^{T}\int_{\omega}u_{0}(t,x,y)\phi(t,x,y)~{}dydxdt.$
(1.1)

If, in addition to (1.1), the following relation holds:

$\lim_{\varepsilon\rightarrow
0}\left\|u_{\varepsilon}\right\|_{L^{2}((0,T)\times\omega)}=\left\|u_{0}\right\|_{L^{2}((0,T)\times\omega\times
Y)},$ (1.2)

then $u_{\varepsilon}$ is said to converge strongly in the two-scale sense to
$u_{0}$.

The following compactness result is an extension of the stationary case in [1]
to the case with time “t”. The time variable can be treated as an independent
parameter and all the proof remain valid.

######  _L_ emma A.1.

(i) Every bounded sequence in $L^{2}((0,T)\times\omega)$ has a two-scale
convergent subsequence.
(ii) Let $u_{\varepsilon}$ be a bounded sequence in
$L^{2}\left((0,T),H^{1}(\omega)\right)$. Then there exists
$u_{0}~{}\in~{}L^{2}$
$\left((0,T),H^{1}(\omega)\right)$, $u_{1}\in
L^{2}\left((0,T)\times\omega,H_{per}^{1}(Y)\right)$ and a subsequence
$u_{\varepsilon}$ still denoted by $u_{\varepsilon}$, such that

$u_{\varepsilon}\rightarrow u_{0},~{}~{}~{}\nabla
u_{\varepsilon}\rightarrow\nabla_{x}u_{0}+\nabla_{y}u_{1},~{}~{}~{}~{}\text{in
the two-scale sense}.$

Next we introduce unfolding operator. Unfolding operator was first constructed
in [5] and further studied in detail in [13, 14]. There are some equivalence
results between two-scale convergence and unfolding operator. Let $\Gamma$ be
a (n-1)-dimensional Lipschitz manifold, compactly included in $Y$, and
$\Gamma_{\varepsilon}=\bigcup\limits_{k\in\mathbb{Z}^{n}}\varepsilon(\Gamma+k),~{}\Gamma_{\varepsilon}~{}\subset~{}\omega$.
Denote $[x]$ as the largest integer not greater than $x$, and let
$\\{x\\}:=x-[x]$, then we have
$[x]\in\mathbb{Z}^{n},~{}\\{x\\}\in\overline{Y}$. Let $Y^{*}\subset Y$ and
$\omega_{\varepsilon}$ be the internal of
$\omega\cap\bigcup\limits_{k\in\mathbb{Z}^{n}}\varepsilon(k+\overline{Y^{*}})$.

######  _D_ efinition A.2.

(i) Let $u_{\varepsilon}\in L^{2}((0,T)\times\omega)$, unfolding operator
$\mathcal{T}^{\varepsilon}:L^{2}((0,T)\times\omega)\rightarrow
L^{2}((0,T)\times\omega\times Y)$ is defined by

$\left(\mathcal{T}^{\varepsilon}u_{\varepsilon}\right)(t,x,y):=u_{\varepsilon}\left(t,\varepsilon\left(\left[\frac{x}{\varepsilon}\right]+y\right)\right).$

(ii) Similarly, unfolding operator in $\omega_{\varepsilon}$ is defined by
$\mathcal{T}^{\varepsilon}:L^{2}\left((0,T)\times\omega_{\varepsilon}\right)\rightarrow
L^{2}((0,T)$
$\times\omega\times Y^{*})$, and unfolding operator on $\Gamma_{\varepsilon}$
is defined by
$\mathcal{T}^{\varepsilon}_{b}:L^{2}((0,T)\times\left.\Gamma_{\varepsilon}\right)\rightarrow
L^{2}((0,T)\times\omega\times\Gamma)\text{ on }\Gamma_{\varepsilon}$.

From the above definition it is clear that $\mathcal{T}^{\varepsilon}$ and
$\mathcal{T}^{\varepsilon}_{b}$ are bounded linear operators. Some properties
of unfolding operator are given in the following lemma

######  _L_ emma A.2.

[13, 14, 15] (i) Let $u_{\varepsilon},v_{\varepsilon}\in
L^{2}\left((0,T)\times\omega_{\varepsilon}\right)$, then we have

$\left(\mathcal{T}^{\varepsilon}u_{\varepsilon},\mathcal{T}^{\varepsilon}v_{\varepsilon}\right)_{L^{2}((0,T)\times\Omega\times
Y^{\star})}=\left(u_{\varepsilon},v_{\varepsilon}\right)_{L^{2}\left((0,T)\times\omega_{\varepsilon}\right)},$

and for $u_{\varepsilon}\in
L^{2}\left((0,T),H^{1}\left(\omega_{\varepsilon}\right)\right)$ we have
$\nabla_{y}\mathcal{T}^{\varepsilon}u_{\varepsilon}=\varepsilon\mathcal{T}^{\varepsilon}\left(\nabla_{x}u_{\varepsilon}\right)$.
(ii) Let $u_{\varepsilon},v_{\varepsilon}\in
L^{2}\left((0,T)\times\Gamma_{\varepsilon}\right)$, then we have

$\displaystyle\left(\mathcal{T}^{\varepsilon}_{b}u_{\varepsilon},\mathcal{T}^{\varepsilon}_{b}v_{\varepsilon}\right)_{L^{2}((0,T)\times\Omega\times\Gamma)}$
$\displaystyle=\varepsilon\left(u_{\varepsilon},v_{\varepsilon}\right)_{L^{2}\left((0,T)\times\Gamma_{\varepsilon}\right)},$
$\displaystyle\left\|\mathcal{T}^{\varepsilon}_{b}u_{\varepsilon}\right\|_{L^{2}((0,T)\times\Omega\times\Gamma)}$
$\displaystyle=\sqrt{\varepsilon}\left\|u_{\varepsilon}\right\|_{L^{2}\left((0,T)\times\Gamma_{\varepsilon}\right)},$

where $(\cdot,\cdot)_{L^{2}(G)}$ is the inner product of $L^{2}(G)$ for an
arbitrary measurable set $G$. The relation between unfolding operator and two-
scale convergence is given below (see [7, Proposition 4.6] and [35,
Proposition 4.7]).

######  _L_ emma A.3.

For bounded sequences in $L^{2}((0,T)\times\omega)$, the following two
statements are equivalent
(i) $u_{\varepsilon}\rightarrow u_{0}$ weakly/strongly in the two-scale sense,
(ii) $\mathcal{T}^{\varepsilon}u_{\varepsilon}\rightarrow u_{0}$
weakly/strongly in $L^{2}((0,T)\times\Omega\times Y)$.
The same result holds for $u_{\varepsilon}$ in
$L^{2}\left((0,T)\times\Gamma_{\varepsilon}\right)$ with
$\sqrt{\varepsilon}\left\|u_{\varepsilon}\right\|_{L^{2}\left((0,T)\times\Gamma_{\varepsilon}\right)}\leq
A$ and bounded unfolding operator $\mathcal{T}^{\varepsilon}_{b}$.

For the unfolding operator in $\omega_{\varepsilon}$, we have the following
convergence result [48, Proposition 2.6].

######  _L_ emma A.4.

Let
$\mathcal{T}^{\varepsilon}:L^{2}\left((0,T)\times\omega_{\varepsilon}\right)\rightarrow
L^{2}\left((0,T)\times\omega\times Y^{*}\right)$ and $u\in
L^{2}\left((0,T)\times\omega\right)$, then we have

$\mathcal{T}^{\varepsilon}(u)\rightarrow u~{}~{}~{}\text{strongly in
}L^{2}\left((0,T)\times\omega\times Y^{*}\right).$

We also state a lemma which is used in the proof of the a priori estimate
(3.3).

######  _L_ emma A.5.

Let $\phi_{s}^{\varepsilon},s=I,E$ be the solutions of (3.1), if
$\int_{\Omega_{E}^{\varepsilon}}\phi_{E}^{\varepsilon}=0$, then we have

$\int_{0}^{T}\big{(}\|\phi_{I}^{\varepsilon}\|^{2}_{\Omega_{I}^{\varepsilon}}+\|\phi_{E}^{\varepsilon}\|^{2}_{\Omega_{E}^{\varepsilon}}\big{)}~{}dt\leq
A\int_{0}^{T}\big{(}\|\nabla\phi_{I}^{\varepsilon}\|^{2}_{\Omega_{I}^{\varepsilon}}+\|\nabla\phi_{E}^{\varepsilon}\|^{2}_{\Omega_{E}^{\varepsilon}}+\varepsilon\|\llbracket{\phi^{\varepsilon}}\rrbracket\|^{2}_{\Gamma^{\varepsilon}}\big{)}~{}dt$
(1.3)

Proof: Using poincaré inequality for functions with zero mean [39, 40], we can
prove (1.3) along the same line as the proof of [31, Lemma 2.8]. $\square$

Acknowledgment This work was partially supported by the NSFC (grant numbers
11971342, 12071190), and NSERC (CA). The first author is grateful to the
financial support by China Scholarship Council (CSC) during his visit to York
University in Toronto, where this work is done.

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