Homogenization of a stochastic model of a single phase flow in partially fissured media

Abstract

In this paper, we present new homogenization results of a stochastic model for flow of a single-phase fluid through a partially fissured porous medium. The model is a double-porosity model with two flow fields, one associated with the system of fissures and the other associated with the porous system. This model is mathematically described by a system of nonlinear stochastic partial differential equations defined on perforated domain. The main tools to derive the homogenized stochastic model are the Nguetseng’s two-scale convergence, tightness of constructed probability measures, Prokhorov and Skorokhod compactness process and Minty’s monotonicity method.

Keywords

Homogenization single phase flow partially fissured media two-scale convergence Minty’s monotonicity method probabilistic compactness results stochastic calculus

1. Introduction

A fissured or fractured medium is a material made up of permeable and porous blocks interwoven by a system of fissures, the porous blocks make up the matrix of the media. Fissured media are differentiated by the extent to which the system of fissures are developed within the medium. Bulk of the fluid transport takes place through the system of fissures while significant fluid storage occurs in the porous blocks.

Flow in fissured porous domains was first investigated by reservoir engineers in the petroleum industry because many petroleum reservoirs are in fractured rock formations made up of porous blocks of rocks surrounded by fractures. The blocks have low permeability but the porosity and consequently the storage capacity of fluids is high, which led to an overestimation in well production and capacity. Scientists and engineers have been studying this subject. Hence there are many articles and professional literature in multiple fields including hydrology, geology and environmental engineering.

There are certain characteristics of fissured media, namely, that transport occurs through the fissures while fluid storage occurs in the pore system. There are two cases of fissured media, the totally fissured media and the partially fissured media. In the case of a totally fissured medium, the matrix is assumed to be broken down into individual porous blocks by well developed system of fissures, resulting in no flow in the porous matrix and only through the system of fissures. In the case of a partially fissured medium, the system of fissures are less developed and the porous blocks may be connected, resulting in some amount of flow within the porous matrix. See Fig. 1.

Fig. 1.

An illustration of a partially fissured material.

A mathematical model that describes the flow of a fluid through a fissured porous medium domain can be stated for every point in the phase considered and on the matrix-fissure interface. This description is said to be at the microscopic scale. Due to the difficulty in measuring values of variables within a phase and determining the parameters of the model, a complete description of a model at the microscopic level is difficult and a solution to a said model is almost impossible. To bypass these difficulties, a macroscopic model is derived as the limit of the microscopic model, this process can be done using various homogenization methods, see for instance [9,15] (Chapter 9) and [29].

Fluid flows through fissured media as if it has two pore systems, one for the porous matrix and the other for the system of fissures, giving rise to the concept of double porosity. The flow of a fluid through totally fissured medium can be modeled using two flow fields, one representing the porous matrix and the other representing the system of fissures. These systems are coupled to form a system of equations over the flow domain, this type of model was introduced by Barenblatt, Zheltov and Kochina in [3]; see also [2,32].

Coeffield and Spagnuolo in [11] considered a model for single-phase flow through a totally fractured layered medium, where the fractures are horizontal and the matrix blocks are stacked vertically. The structure considered in [11] is assumed to be periodic only in one direction (vertically). In [12], Douglas, Peszyńska and Showalter extended the model for single phase flow in totally fissured media to that of a single phase fluid through periodic partially fissured media in the deterministic case; the model was constructed following [3,33] and the macroscopic model was derived using the method of asymptotic expansion. In the microscopic model for partially fissured medium in [12], there are two flows in the matrix; a global flow within the matrix and a flow that leads to local storage. The model for partially fissured media in [12] was extended by Clark and Showalter [10] to a quasi-linear version still in the deterministic case and the corresponding macroscopic model was derived using Nguetseng’s two-scale convergence. Nguetseng, Showalter and Woukeng [25] considered a general deterministic version of the problem in [12] beyond the periodic setting using Sigma convergence.

In geological formations, such as oil reservoirs, there are many factors that affect the flows within the domain, leading to uncertainties in estimating or predicting the flow in this type of formations which could lead to overestimates in well capacity. A stochastic process or random process is used to quantify uncertainties associated with physical or chemical processes since it provides a natural method for evaluating uncertainties. In [34], Wright reformulated the model in [12] for a randomly fissured media and used stochastic two-scale convergence in the mean (introduced by Bourgeat, Mikelić and himself in [6]) for the homogenization process. The homogenized problem obtained is a stochastic analog of the homogenized problem obtained in [12]. Here, we model the influence of random fluctuations on a single-phase flow through a random force driven by the Wiener process. This leads to the flow in the partially fissured media being governed by a system of stochastic partial differential equations (SPDEs) of nonlinear diffusion type involving oscillating coefficients. Since SPDEs are more advanced and more efficient tools in modelling random fluctuations on evolution systems arising in applied sciences, our model is naturally more elaborate than Wright’s in [34] which captures random influence through the random perturbation of the coefficient of a partial differential equation which does not involve random forces.

In this paper, we study the homogenization of a stochastic nonlinear evolution problem. We use the two-scale convergence together with Ito’s stochastic calculus and the probabilistic compactness process of Prokhorov and Skorokhod. For general account of homogenization of SPDEs, we refer to [19–23,31] and the references therein. In Section 2, we state the assumptions on the geometry of the partially fissured porous medium under consideration, some monotone operator properties including Browder–Minty theorem and the function spaces relevant to our study. In Section 3, we introduce the microscopic model and assumptions on the model. Section 4 contains the existence result of the governing stochastic diffusion equation and the a priori estimates for the solution of the equation. Section 5 is devoted to the tightness property for a family of probability measures linked to the sequence of the solution to the stochastic diffusion equation and the driving Wiener process, it also contains the Prokhorov and Skorokhod compactness procedure, see [5]. In Section 6, we prove the convergence of the microscopic problem to the macroscopic problem using Nguetseng’s two-scale convergence [1,24] and use Minty’s trick (monotonicity method) [14,17,18], to identify the weak limit. For more on Minty’s monotonicity method, we refer to [13].

2. Setting of the problem and preliminaries

Let us consider ϵ to be a positive parameter taking its values in a sequence which tends to zero and $[0, T]$ , a time interval with $T \in (0, \infty)$ .

Let Q be a bounded domain in $R^{n}$ consisting of two sub-domains; one representing the fissures and the other representing the matrix.

Let $Y = {[0, 1]}^{n}$ denote the unit cell of measure $| Y | = 1$ consisting of two disjoint parts, $Y_{1}$ and $Y_{2}$ representing the local structure of the fissure and the matrix respectively. We let $χ_{i} (y)$ denote the characteristic function of $Y_{i}$ for $i = 1, 2$ such that $χ_{1} (y) + χ_{2} (y) = 1$ . We assume that the sets ${y \in R^{n}; χ_{i} (y) = 1}$ for $i = 1, 2$ are smooth, $Y_{i}$ -periodic and extended to all of $R^{n}$ periodically.

For a given scale factor $ϵ > 0$ , the sub-domains $Q_{1}^{ϵ}$ and $Q_{2}^{ϵ}$ of Q represent the fissures and the matrix respectively with $\begin{matrix} Q_{i}^{ϵ} = {x \in Q; χ_{i} (\frac{x}{ϵ}) = 1}, i = 1, 2 . \end{matrix}$ Let us denote the interface of $Q_{1}^{ϵ}$ and $Q_{2}^{ϵ}$ lying within Q as $Γ_{1, 2}^{ϵ} = \partial Q_{1}^{ϵ} \cap \partial Q_{2}^{ϵ} \cap Q$ and let $Γ_{1, 2} = \partial Y_{1} \cap \partial Y_{2} \cap Y$ be the interface in the representative cell Y, and $Γ_{2, 2} = \bar{Y_{2}} \cap \partial Y$ we denote by ${\vec{ν}}_{i}$ the outer normal on $\partial Q_{i}^{ϵ}$ for $i = 1, 2$ .

2.1. Functional spaces

We introduce some functional spaces needed throughout the paper. Let Q be an open bounded set in $R^{n}$ . We define $W_{per}^{1, p}$ as the closure of $C^{\infty} (Y)$ for the $W^{1, p}$ -norm.

For $ϵ > 0$ , we define the following spaces $\begin{matrix} H^{ϵ} = L^{2} (Q_{1}^{ϵ}) \times L^{2} (Q_{2}^{ϵ}) \times L^{2} (Q_{2}^{ϵ}), \end{matrix}$ equipped with the inner product $\begin{matrix} {([v_{1}, v_{2}, v_{3}], [ψ_{1}, ψ_{2}, ψ_{3}])}_{H^{ϵ}} & = \int_{Q_{1}^{ϵ}} c_{1}^{ϵ} (x) v_{1} (x) ψ_{1} (x) d x + \int_{Q_{2}^{ϵ}} c_{2}^{ϵ} (x) v_{2} (x) ψ_{2} (x) d x \\ + \int_{Q_{2}^{ϵ}} c_{3}^{ϵ} (x) v_{3} (x) ψ_{3} (x) d x, \end{matrix}$ where $c_{i}^{ϵ} (x) = c_{i} (\frac{x}{ϵ}) \in L^{\infty} (Y)$ , for $i = 1, 2, 3$ .

We write $\begin{matrix} {(v_{i}, ψ_{i})}_{H_{i}^{ϵ}} = \int_{Q_{i}^{ϵ}} c_{i}^{ϵ} (x) v_{i} (x) ψ_{i} (x) d x, i = 1, 2, {(v_{3}, ψ_{3})}_{H_{2}^{ϵ}} = \int_{Q_{2}^{ϵ}} c_{3}^{ϵ} (x) v_{3} (x) ψ_{3} (x) d x, \end{matrix}$ and $\begin{matrix} ‖ ψ_{i} ‖_{H_{i}^{ϵ}}^{2} = \int_{Q_{i}^{ϵ}} c_{i}^{ϵ} (x) {| ψ_{i} (x) |}^{2} d x, i = 1, 2, ‖ ψ_{3} ‖_{H_{2}^{ϵ}}^{2} = \int_{Q_{2}^{ϵ}} c_{3}^{ϵ} (x) {| ψ_{3} (x) |}^{2} d x . \end{matrix}$ Let $γ_{i}^{ϵ} : W^{1, p} (Q_{i}^{ϵ}) \to L^{p} (\partial Q_{i}^{ϵ})$ for $i = 1, 2$ be the usual trace map. We define the space $\begin{matrix} G^{ϵ} = {[v_{1}, v_{2}, v_{3}] \in W^{1, p} (Q_{1}^{ϵ}) \times W^{1, p} (Q_{2}^{ϵ}) \times W^{1, p} (Q_{2}^{ϵ}) : γ_{1}^{ϵ} v_{1} = α γ_{2}^{ϵ} v_{2} + β γ_{2}^{ϵ} v_{3} on Γ_{1, 2}^{ϵ}}, \end{matrix}$ where $β + α = 1$ with $β > 0$ and $α ⩾ 0$ and $\begin{matrix} H^{ϵ} = H^{ϵ} \cap G^{ϵ} . \end{matrix}$ $H^{ϵ}$ is a Banach space equipped with the norm $\begin{matrix} {‖ [v_{1}, v_{2}, v_{3}] ‖}_{H^{ϵ}} = & ‖ v_{1} ‖_{L^{2} (Q_{1}^{ϵ})} + ‖ v_{2} ‖_{L^{2} (Q_{2}^{ϵ})} + ‖ v_{3} ‖_{L^{2} (Q_{2}^{ϵ})} + ‖ \nabla v_{1} ‖_{L^{p} (Q_{1}^{ϵ})} \\ + ‖ \nabla v_{2} ‖_{L^{p} (Q_{2}^{ϵ})} + ‖ \nabla v_{3} ‖_{L^{p} (Q_{2}^{ϵ})} . \end{matrix}$ Let X be a Banach space and $(Ω, F, P)$ be a probability space. For $1 ⩽ p, q ⩽ \infty$ , the space $L^{q} (Ω, F, P, L^{p} (0, T; X))$ is a probability space with filtration ${F}_{t \in [0, T]}$ consisting of all stochastic processes $ϕ : (ω, t) \in Ω \times [0, T] \to ϕ (ω, t, \cdot) \in X$ such that $ϕ (ω, t, \cdot)$ is progressively measurable with respect to $(ω, t)$ . $E$ is the corresponding mathematical expectation.

For $1 ⩽ p < \infty$ , we endow this space with the norm $\begin{matrix} ‖ ϕ ‖_{L^{q} (Ω, F, P, L^{p} (0, T; X))} = {(E ‖ ϕ ‖_{L^{p} (0, T; X)}^{q})}^{\frac{1}{q}} . \end{matrix}$ For $p = \infty$ , the norm in the space $L^{q} (Ω, F, P, L^{\infty} (0, T; X))$ is given by $\begin{matrix} ‖ ϕ ‖_{L^{q} (Ω, F, P, L^{\infty} (0, T; X))} = {(E ‖ ϕ ‖_{L^{\infty} (0, T; X)}^{q})}^{\frac{1}{q}} . \end{matrix}$

2.2. Minty’s trick

In this subsection we recall the classical result of Minty (Minty’s trick) [18]. This result will be used in Section 6 in the construction of the homogenized problem for a sequence of SPDEs involving monotone operators. For proofs of the results, we refer to [13,35].

Let X be a real, reflexive Banach space and $X^{*}$ its dual. Let us denote the inner product $(g, p)$ by $g (p)$ for $g \in X^{*}$ , $p \in X$ .

Definition 2.1.
A mapping $E : X \to X^{}$ is said to be bounded if it maps bounded subsets of X to bounded subsets of $X^{}$ .

$E : X \to X^{}$ is continuous if $\forall p \in X$ , $\begin{matrix} {‖ E (p) - E (q) ‖}_{X^{}} \to 0 whenever ‖ p - q ‖_{X} \to 0, \end{matrix}$ and E is hemicontinuous if $\forall p, q, r \in X$ the map $\begin{matrix} t \mapsto (E (p + t q), r) \end{matrix}$ is continuous.
Lemma 2.2.
[Minty’s trick] Let $E : X \to X^{}$ be monotone and hemicontinuous on a real Banach space X and let* $\begin{matrix} ⟨ g - E (q), p - q ⟩ ⩾ 0, \forall q \in X . \end{matrix}$ Then $\begin{matrix} g = E (p) . \end{matrix}$

3. The micro-model

Now we develop a microscopic model for a single phase flow in a partially fissured medium.

In the fissures $Q_{1}^{ϵ}$ , we shall denote the flow potential of the fluid by $u_{1}^{ϵ} (t, x)$ and $- μ_{1} (\frac{x}{ϵ}, \nabla u_{1}^{ϵ})$ its corresponding flux. On the matrix $Q_{2}^{ϵ}$ , we account for the global diffusion through the pore system in the matrix denoted by $u_{2}^{ϵ} (t, x)$ with flux $- μ_{2} (\frac{x}{ϵ}, \nabla u_{2}^{ϵ})$ and the very high frequency spatial variation which leads to local storage in the matrix which we shall denote by $u_{3}^{ϵ} (t, x)$ with flux $- μ_{3} (\frac{x}{ϵ}, ϵ \nabla u_{3}^{ϵ})$ . We specify two coefficients β and α which correspond to the proportion of the local and global phases of the total flow potential in the matrix $Q_{2}^{ϵ}$ as measured on the interface $Γ_{1, 2}^{ϵ}$ . Here, we take $β + α = 1$ with $β > 0$ and $α ⩾ 0$ .

Let $μ_{i} : R^{n} \times R^{n} \to R^{n}$ , ( $i = 1, 2, 3$ ), we make the following assumptions;

A (1)

$μ_{i} (\cdot, \vec{ξ})$ is measurable and Y-periodic for every $\vec{ξ} \in R^{n}$ ,

A (2)

$μ_{i} (y, \cdot)$ is continuous for a.e. $y \in Y$ ,

A (3)

there are positive constants k, C, $C^{'}$ , $C_{0}$ and $2 < p < \infty$ such that for every $\vec{ξ}, \vec{ζ} \in R^{n}$ and a.e. $y \in Y$ ;

$A (3.1)$

$| μ_{i} (y, \vec{ξ}) | ⩽ C | \vec{ξ} |^{p - 1} + k$ ,

A (3.2)

$(μ_{i} (y, \vec{ξ}) - μ_{i} (y, \vec{ζ})) \cdot (\vec{ξ} - \vec{ζ}) ⩾ C^{'} | \vec{ξ} - \vec{ζ} |^{p}$ ,

A (3.2)

$μ_{i} (y, \vec{ξ}) \cdot \vec{ξ} ⩾ C_{0} | \vec{ξ} |^{p} - k$ .

Let $c_{i} \in C_{per} (Y)$ for $i = 1, 2, 3$ be given such that $\begin{matrix} (3.1) & 0 < c_{0} ⩽ c_{i} (y) ⩽ C, c_{i}, c_{i}^{- 1} \in L^{\infty} (Q) . \end{matrix}$ For $i = 1, 2, 3$ , we can define the corresponding scaled coefficient at $x \in Q_{i}^{ϵ}$ , $\vec{ξ} \in R^{n}$ by $\begin{matrix} c_{i}^{ϵ} (x) = c_{i} (\frac{x}{ϵ}), μ_{i}^{ϵ} (x, \vec{ξ}) = μ_{i} (\frac{x}{ϵ}, \vec{ξ}) . \end{matrix}$ The micro-model for diffusion in a partially fissured medium driven by a random force is given by $(P^{ϵ}) \begin{matrix} c_{1}^{ϵ} d u_{1}^{ϵ} = \nabla \cdot μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (t, x)) d t + f_{1}^{ϵ} (t, x) d B_{1} (t) in Q_{1}^{ϵ}, \\ c_{2}^{ϵ} d u_{2}^{ϵ} = \nabla \cdot μ_{2}^{ϵ} (x, \nabla u_{2}^{ϵ} (t, x)) d t + f_{2}^{ϵ} (t, x) d B_{2} (t) in Q_{2}^{ϵ}, \\ c_{3}^{ϵ} d u_{3}^{ϵ} = ϵ \nabla \cdot μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ} (t, x)) d t + f_{3}^{ϵ} (t, x) d B_{3} (t) in Q_{2}^{ϵ}, \\ u_{1}^{ϵ} = α u_{2}^{ϵ} + β u_{3}^{ϵ} on Γ_{1, 2}^{ϵ}, \\ α μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (t, x)) \cdot {\vec{ν}}_{1} = μ_{2}^{ϵ} (x, \nabla u_{2}^{ϵ} (t, x)) \cdot {\vec{ν}}_{1} on Γ_{1, 2}^{ϵ}, \\ β μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (t, x)) \cdot {\vec{ν}}_{1} = ϵ μ_{3}^{ϵ} (x, ϵ \nabla u_{2}^{ϵ} (t, x)) \cdot {\vec{ν}}_{1} on Γ_{1, 2}^{ϵ}, \end{matrix}$ where $t \in [0, T]$ , $T \in (0, \infty)$ . The first equation is the conservation of mass defined in the fissures, with $u_{1}^{ϵ} (t, x)$ representing the flow potential in the fissures. We have two components of flow potential in the matrix; $u_{2}^{ϵ} (t, x)$ represents the usual flow through the matrix and $u_{3}^{ϵ} (t, x)$ scaled by ϵ represents the very high frequency variation in the flow resulting from the relatively low permeability of the matrix, $f_{i}^{ϵ} (t, x)$ ( $i = 1, 2, 3$ ) is the intensity of the noise. These flows are assumed to satisfy corresponding conservation equations. ${(B_{i} (t))}_{0 ⩽ t ⩽ T}$ ( $i = 1, 2, 3$ ) are mutually independent standard 1-dimensional Wiener processes defined on a given filtered probability space $(Ω, F, P, {(F)}_{0 ⩽ t ⩽ T})$ .

We assume that

A (4)

$f_{i}^{ϵ} (t, x) = f_{i} (t, \frac{x}{ϵ}) \in L^{4} ((0, T) \times Q_{i}^{ϵ})$ for ( $i = 1, 2$ ) and $f_{3}^{ϵ} (t, x) = f_{3} (t, \frac{x}{ϵ}) \in L^{4} ((0, T) \times Q_{2}^{ϵ})$ is such that $f_{i}^{ϵ}$ ( $i = 1, 2, 3$ ) is uniformly bounded.

$(P^{ϵ})$ is a transmission problem of stochastic partial differential equations due to the prescribed transmission boundary conditions on the interface $Γ_{1, 2}^{ϵ}$ .

Recall that on the matrix $Q_{2}^{ϵ}$ , α and β denote the corresponding partitions for the flow potentials $u_{2}^{ϵ}$ and $u_{3}^{ϵ}$ respectively. The coupling on the interface is a vital element in the system. The continuity of the flow potential is represented in the first interface condition (the fourth relation in $(P^{ϵ})$ ), with prescribed partitions corresponding to the global and local phases in the matrix. The fifth and sixth relations describe the flux across the interface $Γ_{1, 2}^{ϵ}$ between the flow potential in the fissures and the total flow potential in the matrix. The external boundary conditions (on $\partial Q$ ) will play no role so we assume the following homogeneous Dirichlet boundary conditions; $\begin{matrix} (3.2) & \begin{matrix} u_{1}^{ϵ} (t, x) = 0, x \in \partial Q_{1}^{ϵ} \cap \partial Q, \\ u_{2}^{ϵ} (t, x) = 0, x \in \partial Q_{2}^{ϵ} \cap \partial Q and \\ u_{3}^{ϵ} (t, x) = 0, x \in \partial Q_{2}^{ϵ} \cap \partial Q, \end{matrix} \end{matrix}$ with initial conditions $\begin{matrix} (3.3) & u_{1}^{ϵ} (0, \cdot) = u_{1}^{0} (\cdot), u_{2}^{ϵ} (0, \cdot) = u_{2}^{0} (\cdot), u_{3}^{ϵ} (0, \cdot) = u_{3}^{0} (\cdot) in H^{ϵ} . \end{matrix}$ The aim of the paper is to show that the sequence ${\vec{u}}^{ϵ} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ converges in suitable topologies to the stochastic process $\vec{u} = [u_{1}, u_{2}, u_{3}]$ which is a solution to the following SPDEs: $\begin{array}{l} (3.4) & \begin{matrix} d \int_{Y_{1}} c_{1} (y) u_{1} (t, x) d y + \frac{1}{β} d \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) d y d t \\ = \nabla \cdot \int_{Y_{1}} μ_{1} (y, \nabla u_{1} (t, x) + \nabla_{y} U_{1} (t, x, y)) d y d t \\ + \int_{Y_{1}} f_{1} (t, x, y) d y d {\tilde{B}}_{1} (t) + \int_{Y_{2}} \frac{1}{β} f_{3} (t, x, y) d y d {\tilde{B}}_{3} (t), \end{matrix} \\ t \in (0, T), x \in Q, y \in Y_{i}, i = 1, 2, \\ (3.5) & \begin{matrix} d \int_{Q} \int_{Y_{2}} c_{2} (y) u_{2} (t, x) d y d x - \frac{α}{β} d \int_{Q} \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) d y d x \\ = \nabla \cdot (\int_{Q} \int_{Y_{2}} μ_{2} (y, \nabla u_{2} (t, x) + \nabla_{y} U_{2} (t, x, y)) d y d x) d t \\ + \int_{Q} \int_{Y_{2}} f_{2} (t, x, y) d y d x d {\tilde{B}}_{2} (t) - \frac{α}{β} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) d y d x d {\tilde{B}}_{3} (t), \end{matrix} \\ t \in (0, T), x \in Q, y \in Y_{2}, \\ (3.6) & \begin{matrix} d \int_{Q} \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) d y d x & = \nabla_{y} \cdot \int_{Q} \int_{Y_{2}} μ_{3} (y, \nabla_{y} U_{3} (t, x, y)) d y d x d t \\ + \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) d y d x d {\tilde{B}}_{3} (t), y \in Y_{2}, \end{matrix} \\ U_{3} (t, x, y) and \nabla_{y} \cdot μ_{3} (y, \nabla_{y} U_{3} (t, x, y)) \cdot ν are Y -periodic on Γ_{2, 2}, \\ β U_{3} = u_{1} - α u_{2} on Γ_{1, 2} \end{array}$ with initial conditions $\begin{array}{l} u_{i} (0, x) = u_{i}^{0} (x) for i = 1, 2, \\ U_{3} (0, x, y) = u_{3}^{0} (x), \end{array}$ where $\begin{matrix} U_{i} \in D ((0, T) \times Q; W_{0}^{1, p} (Y)), i = 1, 2, 3 \end{matrix}$ and $\tilde{B} = ({\tilde{B}}_{1}, {\tilde{B}}_{2}, {\tilde{B}}_{3})$ is an appropriate Wiener process which is the result of the Prokhorov–Skorokhod process.

4. Existence and uniqueness

We define the notion of solution of problem $(P^{ϵ})$ that is of interest to us.

Definition 4.1.
For fixed $ϵ > 0$ , we define a strong probabilistic solution of problem $(P^{ϵ})$ as a stochastic process $\vec{u} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ such that
${\vec{u}}^{ϵ} \in L^{2} (Ω, F, P, L^{\infty} (0, T; H^{ϵ})) \cap L^{2} (Ω, F, P, L^{2} (0, T; G^{ϵ}))$ .

For all $t \in [0, T]$ , ${\vec{u}}^{ϵ}$ satisfies $\begin{array}{l} \begin{aligned} \sum_{i = 1}^{2} (c_{i}^{ϵ} (\cdot) u_{i}^{ϵ} (t, \cdot), ϕ_{i}) + (c_{3}^{ϵ} (\cdot) u_{3}^{ϵ} (t, \cdot), ϕ_{3}) \\ = \sum_{i = 1}^{2} (c_{i}^{ϵ} (\cdot) u_{i}^{ϵ} (0, \cdot), ϕ_{i}) + (c_{3}^{ϵ} (\cdot) u_{3}^{ϵ} (0, \cdot), ϕ_{3}) \\ + \sum_{i = 1}^{2} \int_{0}^{t} (μ_{i}^{ϵ} (\cdot, \nabla u_{i}^{ϵ} (s, \cdot)), \nabla ϕ_{i}) d s + \int_{0}^{t} (μ_{3}^{ϵ} (\cdot, ϵ \nabla u_{3}^{ϵ} (s, \cdot)), ϵ \nabla ϕ_{3}) d s \\ = \sum_{i = 1}^{2} \int_{0}^{t} (f_{i}^{ϵ} (s, \cdot), ϕ_{i}) d B_{i} (s) + \int_{0}^{t} (f_{3}^{ϵ} (s, \cdot), ϕ_{3}) d B_{3} (s), P -a.s., \end{aligned} \\ u_{i}^{ϵ} (0, \cdot) = u_{i}^{0} (\cdot) (i = 1, 2) and u_{3}^{ϵ} (0, \cdot) = u_{3}^{0} (\cdot), \forall ϕ_{1}, ϕ_{2}, ϕ_{3} \in H^{ϵ} . \end{array}$

Theorem 4.2.
For each $ϵ > 0$ , under assumptions $A (1) - A (4)$ there exists a unique solution of problem $(P^{ϵ})$ in the sense of Definition 4.1 .

Theorem 4.2 has essentially been proven by Pardoux in [27] and Krylov and Rozovskii in [16] using monotonicity method.

If we weaken the condition $A (3.2)$ to the usual monotonicity condition i.e $\begin{matrix} (μ_{i} (\vec{ξ}) - μ_{i} (\vec{ζ})) \cdot (\vec{ξ} - \vec{ζ}) ⩾ 0, \end{matrix}$ for all $\vec{ξ}, \vec{ζ} \in R^{n}$ , we lose uniqueness of the solution but the existence still holds in a weaker sense. Indeed Bensoussan established in [4] the existence of a weak probabilistic solution in the case of one equation involving monotone operators.
4.1. A priori estimates

We now establish essential a priori estimates for problem $(P^{ϵ})$ .

Lemma 4.3.
We assume that ϵ is a fixed positive number, under the assumptions $A (1) - A (4)$ , the solution $[u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ of $(P^{ϵ})$ satisfies the following estimate; $\begin{matrix} (4.1) & \begin{matrix} E sup_{0 ⩽ t ⩽ T} {‖ u_{1}^{ϵ} ‖}_{H_{1}^{ϵ}}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{2}^{ϵ} ‖}_{H_{2}^{ϵ}}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{3}^{ϵ} ‖}_{H_{2}^{ϵ}}^{2} \\ + E \sum_{i = 1}^{2} \int_{0}^{T} {‖ \nabla u_{i}^{ϵ} ‖}_{L^{p} (Q_{i}^{ϵ})}^{p} d t + \int_{0}^{T} {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{p} d t \\ ⩽ C . \end{matrix} \end{matrix}$
Proof.
Using Ito’s formula on the first equation of $(P^{ϵ})$ gives $\begin{matrix} (4.2) & \begin{matrix} {‖ u_{1}^{ϵ} (t) ‖}_{H_{1}^{ϵ}}^{2} & = {‖ u_{1}^{ϵ} (0) ‖}_{H_{1}^{ϵ}}^{2} + 2 \int_{0}^{t} {(\nabla \cdot μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)), u_{1}^{ϵ} (s))}_{L^{2} (Q_{1}^{ϵ})} d s \\ + 2 \int_{0}^{t} {(f_{1}^{ϵ} (s), u_{1}^{ϵ} (s))}_{L^{2} (Q_{1}^{ϵ})} d B_{1} (s) + \int_{0}^{t} {‖ f_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} d s . \end{matrix} \end{matrix}$ Integrating by parts on the second term on the right hand side of (4.2) yields $\begin{matrix} 2 \int_{0}^{t} {(\nabla \cdot μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)), u_{1}^{ϵ} (s))}_{L^{2} (Q_{1}^{ϵ})} d s & = - 2 \int_{0}^{t} {(μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)), \nabla u_{1}^{ϵ} (s))}_{L^{2} (Q_{1}^{ϵ})} d s \\ + 2 \int_{0}^{t} {(μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{1}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s, \end{matrix}$ using the identity $u_{1}^{ϵ} = α u_{2}^{ϵ} + β u_{3}^{ϵ}$ , on $Γ_{1, 2}^{ϵ}$ we get $\begin{array}{l} {‖ u_{1}^{ϵ} (t) ‖}_{H_{1}^{ϵ}}^{2} + 2 \int_{0}^{t} {(μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)), \nabla u_{1}^{ϵ} (s))}_{L^{2} (Q_{1}^{ϵ})} d s \\ = {‖ u_{1}^{ϵ} (0) ‖}_{H_{1}^{ϵ}}^{2} \\ + 2 \int_{0}^{t} {(α μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{2}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s \\ + 2 \int_{0}^{t} {(β μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{3}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s \\ (4.3) & + 2 \int_{0}^{t} {(f_{1}^{ϵ} (s), u_{1}^{ϵ} (s))}_{L^{2} (Q_{1}^{ϵ})} d B_{1} (s) + \int_{0}^{t} {‖ f_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} d s . \end{array}$ Ito’s formula on the second equation in $(P^{ϵ})$ gives $\begin{matrix} (4.4) & \begin{matrix} {‖ u_{2}^{ϵ} (t) ‖}_{H_{2}^{ϵ}}^{2} + 2 \int_{0}^{t} {(μ_{2}^{ϵ} (x, \nabla u_{2}^{ϵ} (s)), \nabla u_{2}^{ϵ} (s))}_{L^{2} (Q_{2}^{ϵ})} d s \\ = {‖ u_{2}^{ϵ} (0) ‖}_{H_{2}^{ϵ}}^{2} \\ - 2 \int_{0}^{t} {(μ_{2}^{ϵ} (x, \nabla u_{2}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{2}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s + 2 \int_{0}^{t} {(f_{2}^{ϵ} (s), u_{2}^{ϵ} (s))}_{L^{2} (Q_{2}^{ϵ})} d B_{2} (s) \\ + \int_{0}^{t} {‖ f_{2}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d s, \end{matrix} \end{matrix}$ where we have used the relation ${\vec{ν}}_{1} = - {\vec{ν}}_{2}$ on $Γ_{1, 2}^{ϵ}$ .

Lastly, Ito’s formula on the third equation on $(P^{ϵ})$ and the relation $\vec{ν_{1}} = - \vec{ν_{2}}$ on $Γ_{1, 2}^{ϵ}$ gives $\begin{matrix} (4.5) & \begin{matrix} {‖ u_{3}^{ϵ} (t) ‖}_{H_{2}^{ϵ}}^{2} + 2 \int_{0}^{t} {(μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ} (s)), ϵ \nabla u_{3}^{ϵ} (s))}_{L^{2} (Q_{2}^{ϵ})} d s \\ = {‖ u_{3}^{ϵ} (0) ‖}_{H_{2}^{ϵ}}^{2} \\ - 2 \int_{0}^{t} {(ϵ μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{3}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s + 2 \int_{0}^{t} {(f_{3}^{ϵ} (s), u_{3}^{ϵ} (s))}_{L^{2} (Q_{2}^{ϵ})} d B_{3} (s) \\ + \int_{0}^{t} {‖ f_{3}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d s . \end{matrix} \end{matrix}$ Summing (4.3), (4.4) and (4.5), we get $\begin{array}{l} \sum_{i = 1}^{2} {‖ u_{i}^{ϵ} (t) ‖}_{H_{i}^{ϵ}}^{2} + {‖ u_{3}^{ϵ} (t) ‖}_{H_{2}^{ϵ}}^{2} + \sum_{i = 1}^{2} 2 \int_{0}^{t} {(μ_{i}^{ϵ} (x, \nabla u_{i}^{ϵ} (s)), \nabla u_{i}^{ϵ} (s))}_{L^{2} (Q_{i}^{ϵ})} d s \\ + 2 \int_{0}^{t} {(μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ} (s)), ϵ \nabla u_{3}^{ϵ} (s))}_{L^{2} (Q_{2}^{ϵ})} d s \\ = \sum_{i = 1}^{2} {‖ u_{i}^{ϵ} (0) ‖}_{H_{i}^{ϵ}}^{2} + {‖ u_{3}^{ϵ} (0) ‖}_{H_{2}^{ϵ}}^{2} \\ + 2 \int_{0}^{t} {(α μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{2}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s \\ + 2 \int_{0}^{t} {(β μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{3}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s \\ - 2 \int_{0}^{t} {(μ_{2}^{ϵ} (x, \nabla u_{2}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{2}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s \\ - 2 \int_{0}^{t} {(ϵ μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ} (s)) \cdot {\vec{ν}}_{1}, u_{3}^{ϵ} (s))}_{L^{2} (Γ_{1, 2}^{ϵ})} d s \\ + 2 \int_{0}^{t} {(f_{1}^{ϵ}, u_{1}^{ϵ})}_{L^{2} (Q_{1}^{ϵ})} d s + 2 \int_{0}^{t} {(f_{2}^{ϵ}, u_{2}^{ϵ})}_{L^{2} (Q_{2}^{ϵ})} d s + 2 \int_{0}^{t} {(f_{3}^{ϵ}, u_{3}^{ϵ})}_{L^{2} (Q_{2}^{ϵ})} d s \\ + \sum_{i = 1}^{2} \int_{0}^{t} {‖ f_{i}^{ϵ} ‖}_{L^{2} (Q_{i}^{ϵ})}^{2} d s + \int_{0}^{t} {‖ f_{3}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d s . \end{array}$ The boundary terms mutually cancel out thanks to the fifth and sixth relations in $(P^{ϵ})$ . Using $A (3.2)$ in the resulting relation we have $\begin{matrix} \sum_{i = 1}^{2} {‖ u_{i}^{ϵ} (t) ‖}_{H_{i}^{ϵ}}^{2} + {‖ u_{3}^{ϵ} (t) ‖}_{H_{2}^{ϵ}}^{2} + 2 C_{0} \sum_{i = 1}^{2} \int_{0}^{t} {‖ \nabla u_{i}^{ϵ} ‖}_{L^{2} (Q_{i}^{ϵ})}^{p} d s \\ + 2 C_{0} \int_{0}^{t} {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{2} \\ ⩽ \sum_{i = 1}^{2} {‖ u_{i}^{ϵ} (0) ‖}_{H_{i}^{ϵ}}^{2} + {‖ u_{3}^{ϵ} (0) ‖}_{H_{2}^{ϵ}}^{2} \\ + 2 \sum_{i = 1}^{2} \int_{0}^{t} {(f_{i}^{ϵ}, u_{i}^{ϵ})}_{L^{2} (Q_{i}^{ϵ})} d B_{i} (s) + 2 \int_{0}^{t} {(f_{3}^{ϵ}, u_{3}^{ϵ})}_{L^{2} (Q_{2}^{ϵ})} d B_{3} (s) \\ + \sum_{i = 1}^{2} \int_{0}^{t} {‖ f_{i}^{ϵ} ‖}_{L^{2} (Q_{i}^{ϵ})}^{2} d s + \int_{0}^{t} {‖ f_{3}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d s + t | k | . \end{matrix}$ Taking the supremum over $[0, T]$ followed by the expectation in both sides we get $\begin{array}{l} E sup_{0 ⩽ t ⩽ T} \sum_{i = 1}^{2} {‖ u_{i}^{ϵ} (t) ‖}_{H_{i}^{ϵ}}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{3}^{ϵ} (t) ‖}_{H_{2}^{ϵ}}^{2} + 2 C_{0} E \sum_{i = 1}^{2} \int_{0}^{T} {‖ \nabla u_{i}^{ϵ} ‖}_{L^{2} (Q_{i}^{ϵ})}^{p} d s \\ + 2 C_{0} E \int_{0}^{T} {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{2} \\ ⩽ \sum_{i = 1}^{2} {‖ u_{i}^{ϵ} (0) ‖}_{H_{i}^{ϵ}}^{2} + {‖ u_{3}^{ϵ} (0) ‖}_{H_{2}^{ϵ}}^{2} \\ + E sup_{0 ⩽ t ⩽ T} [2 \sum_{i = 1}^{2} \int_{0}^{t} {(f_{i}^{ϵ}, u_{i}^{ϵ})}_{L^{2} (Q_{i}^{ϵ})} d B_{i} (s) + \int_{0}^{t} {(f_{3}^{ϵ}, u_{3}^{ϵ})}_{L^{2} (Q_{2}^{ϵ})} d B_{3} (s)] \\ + E \sum_{i = 1}^{2} \int_{0}^{T} {‖ f_{i}^{ϵ} ‖}_{L^{2} (Q_{i}^{ϵ})}^{2} d s + \int_{0}^{T} {‖ f_{3}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d s + T | k | . \end{array}$ Thanks to Burkhölder–Davis–Gundy inequality, Cauchy–Schwarz’s and Young’s inequalities, we get $\begin{matrix} E sup_{0 ⩽ t ⩽ T} 2 | \int_{0}^{t} {(f_{1}^{ϵ}, u_{1}^{ϵ})}_{L^{2} (Q_{1}^{ϵ})} d B_{1} (s) | & ⩽ E C {(\int_{0}^{T} {(f_{1}^{ϵ}, u_{1}^{ϵ})}_{L^{2} (Q_{1}^{ϵ})}^{2} d t)}^{\frac{1}{2}} \\ ⩽ E C {(\int_{0}^{T} {‖ f_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} {‖ u_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} d t)}^{\frac{1}{2}} \\ ⩽ E sup_{0 ⩽ t ⩽ T} {‖ u_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})} {(\int_{0}^{T} {‖ f_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} d t)}^{\frac{1}{2}} \\ ⩽ ϖ E sup_{0 ⩽ t ⩽ T} {‖ u_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} + c (ϖ) E \int_{0}^{T} {‖ f_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} d t, \end{matrix}$ where ϖ is an arbitrary positive number. Similarly, $\begin{matrix} E sup_{0 ⩽ t ⩽ T} 2 | \int_{0}^{t} {(f_{2}^{ϵ}, u_{2}^{ϵ})}_{L^{2} (Q_{2}^{ϵ})} d B_{2} (s) | ⩽ ϖ E sup_{0 ⩽ t ⩽ T} {‖ u_{2}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} + C (ϖ) E \int_{0}^{T} {‖ f_{2}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d t, \end{matrix}$ and $\begin{matrix} E sup_{0 ⩽ t ⩽ T} 2 | \int_{0}^{t} {(f_{3}^{ϵ}, u_{3}^{ϵ})}_{L^{2} (Q_{2}^{ϵ})} d B_{3} (s) | ⩽ ϖ E sup_{0 ⩽ t ⩽ T} {‖ u_{3}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} + C (ϖ) E \int_{0}^{T} {‖ f_{3}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d t . \end{matrix}$ For ϖ sufficiently small, we have $\begin{matrix} E sup_{0 ⩽ t ⩽ T} {‖ u_{1}^{ϵ} ‖}_{H_{1}^{ϵ}}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{2}^{ϵ} ‖}_{H_{2}^{ϵ}}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{3}^{ϵ} ‖}_{H_{2}^{ϵ}}^{2} \\ + E C [\int_{0}^{T} {‖ \nabla u_{1}^{ϵ} ‖}_{L^{p} (Q_{1}^{ϵ})}^{p} + {‖ \nabla u_{2}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{p} + {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{p} d t] \\ ⩽ E [{‖ u_{1}^{0} (x) ‖}_{H_{1}^{ϵ}}^{2} + {‖ u_{2}^{0} (x) ‖}_{H_{2}^{ϵ}}^{2} + {‖ u_{3}^{0} (x) ‖}_{H_{2}^{ϵ}}^{2}] \\ + C E \int_{0}^{T} {‖ f_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} + {‖ f_{2}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} + {‖ f_{3}^{ϵ} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d t + T k . \end{matrix}$ Based on the assumptions on $u_{1}^{0}$ , $u_{2}^{0}$ , $u_{3}^{0}$ and on $f_{1}^{ϵ}$ , $f_{2}^{ϵ}$ , $f_{3}^{ϵ}$ we get $\begin{matrix} E sup_{0 ⩽ t ⩽ T} \sum_{i = 1}^{2} {‖ u_{i}^{ϵ} (t) ‖}_{H_{i}^{ϵ}}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ u_{3}^{ϵ} (t) ‖}_{H_{2}^{ϵ}}^{2} + \sum_{i = 1}^{2} E \int_{0}^{T} {‖ \nabla u_{i}^{ϵ} ‖}_{L^{p} (Q_{i}^{ϵ})}^{p} d t \\ + E \int_{0}^{T} {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{p} d t ⩽ C . \end{matrix}$ □

Next we establish a key estimate of the finite difference of ${\vec{u}}^{ϵ}$ in the dual of $H^{ϵ}$ . It plays an important role in the implementation of the compactness result. It should be noted that such an estimate was not required in the deterministic case considered in [10].
Lemma 4.4.
Under the assumptions of Lemma 4.3 , the solution ${\vec{u}}^{ϵ} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ of problem $(P^{ϵ})$ satisfies the estimate $\begin{matrix} E sup_{| h | ⩽ δ} \int_{0}^{T - h} {‖ {\vec{u}}^{ϵ} (t + h) - {\vec{u}}^{ϵ} (t) ‖}_{{(H^{ϵ})}^{'}}^{p^{'}} d t ⩽ C max {δ^{\frac{1}{p - 1}}, δ^{\frac{p^{'}}{4}}}, \end{matrix}$ for any h such that $t + h \in [0, T]$ and $\forall δ ⩽ 1$ .
Proof.
Let $\vec{ψ} = [ψ_{1}, ψ_{2}, ψ_{3}] \in H^{ϵ}$ , such that $‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1$ , we have $\begin{array}{l} {‖ {\vec{u}}^{ϵ} (t + h) - {\vec{u}}^{ϵ} (t) ‖}_{{(H^{ϵ})}^{'}} \\ = sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} | {⟨ {\vec{u}}^{ϵ} (t + h) - {\vec{u}}^{ϵ} (t), \vec{ψ} ⟩}_{{(H^{ϵ})}^{'}, H^{ϵ}} | \\ = sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} | ⟨ [u_{1}^{ϵ} (t + h) - u_{1}^{ϵ} (t), u_{2}^{ϵ} (t + h) - u_{2}^{ϵ} (t), u_{3}^{ϵ} (t + h) - u_{3}^{ϵ} (t)], \\ ψ_{1}, ψ_{2}, ψ_{3} ⟩_{{(H^{ϵ})}^{'}, H^{ϵ}} | \\ ⩽ sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} | [\sum_{i = 1}^{2} \int_{t}^{t + h} \int_{Q_{i}^{ϵ}} \nabla \cdot μ_{i}^{ϵ} (x, \nabla u_{i}^{ϵ}) ψ_{i} d x d s \\ + \int_{t}^{t + h} \int_{Q_{2}^{ϵ}} ϵ \nabla \cdot μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ}) ψ_{3} d x d s] | \\ + | | \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s), \int_{t}^{t + h} f_{2}^{ϵ} d B_{2} (s), \int_{t}^{t + h} f_{3}^{ϵ} d B_{3} (s) | |_{{(H^{ϵ})}^{'}} \\ ⩽ sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} | [- \sum_{i = 1}^{2} \int_{t}^{t + h} \int_{Q_{i}^{ϵ}} μ_{i}^{ϵ} (x, \nabla u_{i}^{ϵ}) \nabla ψ_{i} d x d s - \int_{t}^{t + h} \int_{Q_{2}^{ϵ}} μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ}) ϵ \nabla ψ_{3} d x d s \\ + \int_{t}^{t + h} \int_{Γ_{1, 2}^{ϵ}} μ_{1}^{ϵ} (x, \nabla u_{1}^{ϵ}) \cdot {\vec{ν}}_{1} ψ_{1} d x d s + \int_{t}^{t + h} \int_{Γ_{1, 2}^{ϵ}} μ_{2}^{ϵ} (x, \nabla u_{2}^{ϵ}) \cdot {\vec{ν}}_{2} ψ_{2} d x d s \\ + \int_{t}^{t + h} \int_{Γ_{1, 2}^{ϵ}} μ_{3}^{ϵ} (x, \nabla u_{3}^{ϵ}) \cdot {\vec{ν}}_{2} ψ_{3} d x d s] | \\ + | | \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s), \int_{t}^{t + h} f_{2}^{ϵ} d B_{2} (s), \int_{t}^{t + h} f_{3}^{ϵ} d B_{3} (s) | |_{{(H^{ϵ})}^{'}} . \end{array}$ Since $ψ_{1}, ψ_{2}, ψ_{3} \in H^{ϵ}$ , we have $ψ_{1} = α ψ_{2} + β ψ_{3}$ on $Γ_{1, 2}^{ϵ}$ and the terms on $Γ_{1, 2}^{ϵ}$ cancel out due to the fifth and sixth relations in $(P^{ϵ})$ , so we get $\begin{matrix} (4.6) & \begin{matrix} {‖ {\vec{u}}^{ϵ} (t + h) - {\vec{u}}^{ϵ} (t) ‖}_{{(H^{ϵ})}^{'}} \\ ⩽ sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} | [- \sum_{i = 1}^{2} \int_{t}^{t + h} \int_{Q_{i}^{ϵ}} μ_{i}^{ϵ} (x, \nabla u_{i}^{ϵ}) \nabla ψ_{i} d x d s \\ - \int_{t}^{t + h} \int_{Q_{2}^{ϵ}} μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ}) ϵ \nabla ψ_{3} d x d s] | \\ + | | \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s), \int_{t}^{t + h} f_{2}^{ϵ} d B_{2} (s), \int_{t}^{t + h} f_{3}^{ϵ} d B_{3} (s) | |_{{(H^{ϵ})}^{'}} . \end{matrix} \end{matrix}$ Using assumption $A (3.1)$ on the first two terms on the right hand side of (4.6) gives $\begin{array}{rcl} (4.7) & \begin{matrix} sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} [\sum_{i = 1}^{2} \int_{t}^{t + h} \int_{Q_{i}^{ϵ}} | μ_{i}^{ϵ} (x, \nabla u_{i}^{ϵ}) ‖ \nabla ψ_{i} | d x d s + \int_{t}^{t + h} \int_{Q_{2}^{ϵ}} | μ_{3}^{ϵ} (x, ϵ \nabla u_{3}^{ϵ}) ‖ ϵ \nabla ψ_{3} | d x d s] \\ ⩽ sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} [\sum_{i = 1}^{2} \int_{t}^{t + h} \int_{Q_{i}^{ϵ}} {| \nabla u_{i}^{ϵ} |}^{p - 1} | \nabla ψ_{i} | d x d s + \int_{t}^{t + h} \int_{Q_{2}^{ϵ}} {| ϵ \nabla u_{3}^{ϵ} |}^{p - 1} | ϵ \nabla ψ_{3} | d x d s \\ + \sum_{i = 1}^{2} \int_{t}^{t + h} \int_{Q_{i}^{ϵ}} k | \nabla ψ_{i} | d x d s + \int_{t}^{t + h} \int_{Q_{2}^{ϵ}} k | ϵ \nabla ψ_{3} | d x d s] \\ ⩽ \sum_{i = 1}^{2} \int_{t}^{t + h} {‖ \nabla u_{i}^{ϵ} ‖}_{L^{p} (Q_{i}^{ϵ})}^{\frac{p}{p^{'}}} d s + \int_{t}^{t + h} {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{\frac{p}{p^{'}}} d s + 3 \int_{t}^{t + h} k d s \\ ⩽ h^{\frac{1}{p}} \sum_{i = 1}^{2} {(\int_{t}^{t + h} {‖ \nabla u_{i}^{ϵ} ‖}_{L^{p} (Q_{i}^{ϵ})}^{p} d s)}^{\frac{1}{p^{'}}} + h^{\frac{1}{p}} {(\int_{t}^{t + h} {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{p} d s)}^{\frac{1}{p^{'}}} + 3 \int_{t}^{t + h} k d s . \end{matrix} \end{array}$ Now we estimate the terms involving the stochastic term using the following embedding $\begin{matrix} H^{ϵ} ↪ H^{ϵ}, \end{matrix}$ and since both $H^{ϵ}$ and $H^{ϵ}$ are reflexive spaces, we have $\begin{matrix} {(H^{ϵ})}^{'} ↪ (H^{ϵ}) .^{'} \end{matrix}$ Consequently, $\begin{matrix} {‖ \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s), \int_{t}^{t + h} f_{2}^{ϵ} d B_{2} (s), \int_{t}^{t + h} f_{3}^{ϵ} d B_{3} (s) ‖}_{{(H^{ϵ})}^{'}} \\ ⩽ sup_{\vec{ψ} \in H^{ϵ}, ‖ \vec{ψ} ‖_{H^{ϵ}} ⩽ 1} | {⟨ \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s), ψ_{1} ⟩}_{{(H^{ϵ})}^{'}, H^{ϵ}} + {⟨ \int_{t}^{t + h} f_{2}^{ϵ} d B_{2} (s), ψ_{2} ⟩}_{{(H^{ϵ})}^{'}, H^{ϵ}} \\ + {⟨ \int_{t}^{t + h} f_{3}^{ϵ} d B_{3} (s), ψ_{3} ⟩}_{{(H^{ϵ})}^{'}, H^{ϵ}} | \\ ⩽ C [| | \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s) | |_{H_{1}^{ϵ}} + | | \int_{t}^{t + h} f_{2}^{ϵ} d B_{2} (s) | |_{H_{2}^{ϵ}} + | | \int_{t}^{t + h} f_{3}^{ϵ} d B_{3} (s) | |_{H_{2}^{ϵ}}], \end{matrix}$ where we have used (3.1) on the functions $c_{i}$ .

Estimating each term at a time, we have $\begin{matrix} {‖ \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s) ‖}_{H_{1}^{ϵ}}^{p^{'}} ⩽ {‖ \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s) ‖}_{L^{2} (Q_{1}^{ϵ})}^{p^{'}} . \end{matrix}$ For $h > 0$ , using Hölder’s inequality and Fubini’s theorem we get $\begin{matrix} E sup_{h ⩽ δ} \int_{0}^{T - h} {‖ \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s) ‖}_{L^{2} (Q_{1}^{ϵ})}^{p^{'}} d t & = E sup_{h ⩽ δ} \int_{0}^{T - h} {(\int_{Q_{1}^{ϵ}} {(\int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s))}^{2} d x)}^{\frac{p^{'}}{2}} d t \\ ⩽ C {(E sup_{h ⩽ δ} \int_{0}^{T - h} \int_{Q_{1}^{ϵ}} {(\int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s))}^{2} d x d t)}^{\frac{p^{'}}{2}} \\ ⩽ {(E sup_{h ⩽ δ} \int_{0}^{T} \int_{Q_{1}^{ϵ}} {(\int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s))}^{2} d x d t)}^{\frac{p^{'}}{2}} \\ ⩽ {(\int_{0}^{T} \int_{Q_{1}^{ϵ}} E sup_{h ⩽ δ} {(\int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s))}^{2} d x d t)}^{\frac{p^{'}}{2}} . \end{matrix}$ Recall that $| h | ⩽ δ$ , using Burkhölder–Davis–Gundy’s inequality, we deduce that $\begin{matrix} E sup_{h ⩽ δ} \int_{0}^{T - h} {‖ \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s) ‖}_{L^{2} (Q_{1}^{ϵ})}^{p^{'}} d t & ⩽ {(\int_{0}^{T} \int_{Q_{1}^{ϵ}} E \int_{t}^{t + δ} {| f_{1}^{ϵ} |}^{2} d s d x d t)}^{\frac{p^{'}}{2}} \\ = {(E \int_{0}^{T} \int_{t}^{t + δ} {‖ f_{1}^{ϵ} ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} d s d t)}^{\frac{p^{'}}{2}} . \end{matrix}$ By Hölder’s inequality and the assumption on $f_{1}^{ϵ}$ i.e. $f_{1}^{ϵ} \in L^{4} (0, T; L^{2} (Q_{1}^{ϵ}))$ , we get $\begin{matrix} E sup_{h ⩽ δ} \int_{0}^{T - h} {‖ \int_{t}^{t + h} f_{1}^{ϵ} d B_{1} (s) ‖}_{L^{2} (Q_{1}^{ϵ})}^{2} d t & ⩽ C (T) {(δ E \int_{0}^{T} \int_{t}^{t + δ} {‖ f_{1}^{ϵ} (s) ‖}_{L^{2} (Q_{1}^{ϵ})}^{4} d s d t)}^{\frac{p^{'}}{4}} \\ ⩽ C (T) δ^{\frac{p^{'}}{4}} . \end{matrix}$ Similarly, $\begin{matrix} E sup_{| h | ⩽ δ} \int_{0}^{T - h} {‖ \int_{t}^{t + h} f_{2}^{ϵ} d B_{2} (s) ‖}_{{(H^{ϵ})}^{'}}^{p^{'}} d t ⩽ C (T) δ^{\frac{p^{'}}{4}}, \end{matrix}$ and $\begin{matrix} E sup_{| h | ⩽ δ} \int_{0}^{T - h} {‖ \int_{t}^{t + h} f_{3}^{ϵ} d B_{3} (s) ‖}_{{(H^{ϵ})}^{'}}^{p^{'}} d t ⩽ C (T) δ^{\frac{p^{'}}{4}} . \end{matrix}$ Raising (4.7) to the power of $p^{'}$ , integrating over $[0, T - h]$ , taking the supremum, the expectation and using Lemma 4.3 gives $\begin{matrix} E sup_{| h | ⩽ δ} h^{\frac{p^{'}}{p}} \int_{0}^{T - h} {[{(\int_{t}^{t + h} {‖ \nabla u_{1}^{ϵ} ‖}_{L^{p} (Q_{1}^{ϵ})}^{p} d s)}^{\frac{1}{p^{'}}}]}^{p^{'}} d t \\ ⩽ E sup_{| h | ⩽ δ} h^{\frac{p^{'}}{p}} \int_{0}^{T - h} \int_{t}^{t + h} {‖ \nabla u_{1}^{ϵ} ‖}_{L^{p} (Q_{1}^{ϵ})}^{p} d s d t \\ ⩽ δ^{\frac{p^{'}}{p}} C . \end{matrix}$ Similarly, $\begin{array}{l} E sup_{| h | ⩽ δ} h^{\frac{p^{'}}{p}} \int_{0}^{T - h} {[{(\int_{t}^{t + h} {‖ \nabla u_{2}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{p} d s)}^{\frac{1}{p^{'}}}]}^{p^{'}} d t ⩽ C δ^{\frac{p^{'}}{p}}, \\ E sup_{| h | ⩽ δ} h^{\frac{p^{'}}{p}} \int_{0}^{T - h} {[{(\int_{t}^{t + h} {‖ ϵ \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q_{2}^{ϵ})}^{p} d s)}^{\frac{1}{p^{'}}}]}^{p^{'}} d t ⩽ C δ^{\frac{p^{'}}{p}}, \end{array}$ and $\begin{matrix} E sup_{| h | ⩽ δ} \int_{0}^{T - h} {(3 \int_{t}^{t + h} k d s)}^{p^{'}} d t ⩽ C δ^{p^{'}} . \end{matrix}$ Hence collecting all the above inequalities, we assert that $\begin{matrix} E sup_{| h | ⩽ δ} \int_{0}^{T - h} {‖ {\vec{u}}^{ϵ} (t + h) - {\vec{u}}^{ϵ} (t) ‖}_{{(H^{ϵ})}^{'}}^{p^{'}} d t ⩽ C max {δ^{\frac{1}{p - 1}}, δ^{\frac{p^{'}}{4}}} . \end{matrix}$ □

One of the difficulties encountered in the homogenization of problems in perforated domain is to establish that the sequence of solutions admits a limit in the whole domain, in our case Q. From the estimates in Lemmas 4.3 and 4.4, we cannot extract a convergent subsequence by weak compactness, since each $u_{i}^{ϵ}$ ( $i = 1, 2, 3$ ) is defined on a space which varies in ϵ.

As in [1], we will extend the functions ${\vec{u}}^{ϵ} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ by zero to the whole domain Q. The domain Q has two sub-domains $Q_{1}^{ϵ}$ representing the fissures and $Q_{2}^{ϵ}$ representing the matrix with $Q = Q_{1}^{ϵ} \cup Q_{2}^{ϵ}$ , hence we can assert that the flow potential $u_{1}^{ϵ}$ defined on $Q_{1}^{ϵ}$ equals zero on $Q_{2}^{ϵ}$ and the flow potentials $u_{2}^{ϵ}$ , $u_{3}^{ϵ}$ defined on $Q_{2}^{ϵ}$ are zero on $Q_{1}^{ϵ}$ .

We recall the characteristic function $\begin{matrix} χ_{i}^{ϵ} = χ_{i} (\frac{x}{ϵ}), i = 1, 2 . \end{matrix}$ We use this function to denote the extension by zero of various functions from $Q_{i}^{ϵ}$ to Q and $χ_{i}$ to denote the extension of functions from $Y_{i}$ to Y.

Now we state the estimates for the extension of ${\vec{u}}^{ϵ} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ to all of Q from $Q_{i}^{ϵ}$ , $i = 1, 2$ . $\begin{array}{rcl} (4.8) & \begin{matrix} E sup_{0 ⩽ t ⩽ T} {‖ χ_{1}^{ϵ} u_{1}^{ϵ} ‖}_{L^{2} (Q)}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ χ_{2}^{ϵ} u_{2}^{ϵ} ‖}_{L^{2} (Q)}^{2} + E sup_{0 ⩽ t ⩽ T} {‖ χ_{2}^{ϵ} u_{3}^{ϵ} ‖}_{L^{2} (Q)}^{2} + E \int_{0}^{T} {‖ χ_{1}^{ϵ} \nabla u_{1}^{ϵ} ‖}_{L^{p} (Q)}^{p} d t \\ + E \int_{0}^{T} {‖ χ_{2}^{ϵ} \nabla u_{2}^{ϵ} ‖}_{L^{p} (Q)}^{p} d t + \int_{0}^{T} {‖ ϵ χ_{2}^{ϵ} \nabla u_{3}^{ϵ} ‖}_{L^{p} (Q)}^{p} d t ⩽ C . \end{matrix} \end{array}$ Let us still denote the zero extension of ${\vec{u}}^{ϵ} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ by ${\vec{u}}^{ϵ} = [χ_{1}^{ϵ} u_{1}^{ϵ}, χ_{2}^{ϵ} u_{2}^{ϵ}, χ_{2}^{ϵ} u_{3}^{ϵ}]$ , we state the estimate of the finite difference in the space $V^{'}$ $\begin{matrix} E sup_{| h | ⩽ δ} \int_{0}^{T - h} {‖ {\vec{u}}^{ϵ} (t + h) - {\vec{u}}^{ϵ} (t) ‖}_{V^{'}}^{p^{'}} d t ⩽ C max {δ^{\frac{1}{p - 1}}, δ^{\frac{p^{'}}{4}}} \end{matrix}$ where we define $V$ as $\begin{matrix} V = & {\vec{ψ} = [ψ_{1}, ψ_{2}, ϕ_{3}] \in L^{2} (Q) \times L^{2} (Q) \times L^{2} (Q) \cap (W_{0}^{1, p} (Q) \times W_{0}^{1, p} (Q) \times W_{0}^{1, p} (Q)) \\ : β ϕ_{3} = ψ_{1} - α ψ_{2}, \forall y \in Γ_{1, 2}} . \end{matrix}$
5. Compactness result and tightness property

This section contains some results that are essential in the proof of the tightness property of the probability measures generated by the sequence $(B, {\vec{u}}^{ϵ})$ , where $B = [B_{1}, B_{2}, B_{3}]$ and ${\vec{u}}^{ϵ} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ .

Lemma 5.1 (Prokhorov).

A sequence of probability measures ${(π_{m})}_{m \in N}$ on $(S, B (S))$ is tight if and only if it is relatively compact.

Lemma 5.2 (Skorokhod).

Suppose $S$ is a separable Banach space with $B (S)$ as it’s σ-algebra. Assume that the probability measures ${(μ_{m})}_{m \in N}$ on $(S, B (S))$ weakly converges to a probability measure μ. Then there exists random variables $ξ, ξ_{1}, \dots, ξ_{m}, \dots$ defined on a common probability space $(Ω, F, P)$ such that $L {ξ_{m}} = μ_{m}$ and $L (ξ) = μ$ and $\begin{matrix} lim_{m \to \infty} ξ_{m} = ξ P -a.s., \end{matrix}$ where $L (\cdot)$ stands for the law of ·.

Let us consider the set Z depending on the sequences $ρ_{n}, τ_{n} ⩾ 0$ of numbers such that $ρ_{n}, τ_{n} \to 0$ as $n \to \infty$ and on the constants J, K, L, M, N, R. We define the set Z by $\begin{matrix} Z = & {\vec{ψ} = [ψ_{1}, ψ_{2}, ψ_{3}] : sup_{0 ⩽ t ⩽ T} ‖ ψ_{1} ‖_{L^{2} (Q)} ⩽ J; ‖ ψ_{2} ‖_{L^{2} (Q)} ⩽ K; ‖ ψ_{3} ‖_{L^{2} (Q)} ⩽ L; \\ \sum_{i = 1}^{2} \int_{0}^{T} ‖ \nabla ψ_{i} ‖_{L^{p} (Q)}^{p} d t ⩽ M; \int_{0}^{T} ‖ \nabla ψ_{3} ‖_{L^{p} (Q)}^{p} d t ⩽ N \\ and sup_{| h | ⩽ ρ_{n}} \int_{0}^{T} {‖ \vec{ψ} (t + h) - \vec{ψ} (t) ‖}_{{(H)}^{'}}^{p^{'}} d t ⩽ τ_{n} R, \forall n} . \end{matrix}$

Lemma 5.3.
The set Z is a compact subset of $L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; L^{2} (Q))$ .

The proof of this lemma is similar to the proof found in [4] (Proposition 3.1).

Let the space H be defined as $\begin{matrix} H = L^{2} (Q) \times L^{2} (Q) \times L^{2} (Q), \end{matrix}$ equipped with the inner product $\begin{matrix} {([v_{1}, v_{2}, v_{3}], [ψ_{1}, ψ_{2}, ψ_{3}])}_{H} & = \int_{Q} v_{1} (x) ψ_{1} (x) d x + \int_{Q} v_{2} (x) ψ_{2} (x) d x \\ + \int_{Q} v_{3} (x) ψ_{3} (x) d x . \end{matrix}$ Let ${\vec{u}}^{ϵ} = [χ_{1}^{ϵ} u_{1}^{ϵ}, χ_{2}^{ϵ} u_{2}^{ϵ}, χ_{2}^{ϵ} u_{3}^{ϵ}]$ and $S = C (0, T; R^{n}) \times L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; L^{2} (Q))$ be equipped with the Borel σ-algebra $B (S)$ . Let $ϕ_{ϵ}$ be the $(S, B (S))$ -valued measurable map defined on $(Ω, F, P)$ by $\begin{matrix} ϕ_{ϵ} : ω \mapsto (B (ω), {\vec{u}}^{ϵ}), with B = (B_{1}, B_{2}, B_{3}) . \end{matrix}$ We introduce the probability measures $π^{ϵ}$ on $(S, B (S))$ defined by $\begin{matrix} π^{ϵ} (S) = P (ϕ_{ϵ}^{- 1} (S)), for all S \in B (S) . \end{matrix}$
Lemma 5.4.
The family of probability measures ${π^{ϵ} : ϵ > 0}$ is tight in $(S, B (S))$ .

The proof is carried out following [4], see also [30] and [21,31].

By Prokhorov’s result, (Lemma 5.1), there exists a subsequence ${π_{ϵ_{j}}}$ of ${π_{ϵ}}$ and a probability measure π such that $\begin{matrix} π_{ϵ_{j}} ⇀ π weakly in S . \end{matrix}$ Using Lemma 5.2 due to Skorokhod, there exists a probability space $(\tilde{Ω}, \tilde{F}, \tilde{P})$ and $S$ -valued random variables $(B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}})$ and $(\tilde{B}, \vec{u})$ defined on $(\tilde{Ω}, \tilde{F}, \tilde{P})$ such that the probability law of $(B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}})$ is $π_{ϵ_{j}}$ and that of $(\tilde{B}, \vec{u})$ is π. Furthermore, $\begin{matrix} (5.1) & (B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}}) ⟶ (\tilde{B}, \vec{u}) in S \tilde{P} -a.s., \end{matrix}$ where $B^{ϵ_{j}} = (B_{1}^{ϵ_{j}}, B_{2}^{ϵ_{j}}, B_{3}^{ϵ_{j}})$ , ${\vec{u}}^{ϵ_{j}} = (u_{1}^{ϵ_{j}}, u_{2}^{ϵ_{j}}, u_{3}^{ϵ_{j}})$ , $\vec{u} = [u_{1}, u_{2}, u_{3}]$ , and $\tilde{B} = ({\tilde{B}}_{1}, {\tilde{B}}_{2}, {\tilde{B}}_{3})$ .

$B^{ϵ_{j}}$ and $\tilde{B}$ are Wiener processes on $(\tilde{Ω}, \tilde{F}, \tilde{P})$ and the pair $(B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}})$ satisfies problem $(P^{ϵ_{j}})$ as stipulated in the following;
Theorem 5.5.
For any $\vec{ψ} = [ψ_{1}, ψ_{2}, ψ_{3}] \in C^{\infty} (Q) \times C^{\infty} (Q) \times C^{\infty} (Q)$ and $t \in [0, T]$ . The sequence $(B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}})$ satisfies $\tilde{P} -a.s.$ , the relations $\begin{array}{l} {({\vec{u}}^{ϵ_{j}} (t, \cdot), \vec{ψ})}_{H} & = {({\vec{u}}^{ϵ_{j}} (0, \cdot), \vec{ψ})}_{H} + \sum_{i = 1}^{2} \int_{0}^{t} {(χ_{i}^{ϵ} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}), \nabla ψ_{i})}_{L^{2} (Q)} d s \\ + \int_{0}^{t} {(χ_{2}^{ϵ} μ_{3}^{ϵ_{j}} (x, ϵ_{j} \nabla u_{3}^{ϵ_{j}}), ϵ_{j} \nabla ψ_{3})}_{L^{2} (Q)} d s + \sum_{i = 1}^{2} \int_{0}^{t} {(χ_{i}^{ϵ} f_{i}^{ϵ_{j}}, ψ_{i})}_{L^{2} (Q)} d B^{ϵ_{j}} (s) \\ (5.2) & + \int_{0}^{t} {(χ_{2}^{ϵ} f_{3}^{ϵ_{j}}, ψ_{3})}_{L^{2} (Q)} d B_{3}^{ϵ_{j}} (s), \end{array}$ with $({\vec{u}}^{ϵ_{j}} (0, \cdot), \vec{ψ}) = ({\vec{u}}^{0} (\cdot), \vec{ψ})$ , in the sense of distributions.

Since $({\vec{u}}^{ϵ_{j}})$ satisfies the same type of problem as $(P^{ϵ})$ , we have the following corresponding estimates for $({\vec{u}}^{ϵ_{j}})$ ; $\begin{matrix} (5.3) & \begin{matrix} \tilde{E} sup_{0 ⩽ t ⩽ T} {‖ χ_{1}^{ϵ} u_{1}^{ϵ_{j}} ‖}_{L^{2} (Q)}^{2} + \tilde{E} sup_{0 ⩽ t ⩽ T} {‖ χ_{2}^{ϵ} u_{2}^{ϵ_{j}} ‖}_{L^{2} (Q)}^{2} \\ + \tilde{E} sup_{0 ⩽ t ⩽ T} {‖ χ_{2}^{ϵ} u_{3}^{ϵ_{j}} ‖}_{L^{2} (Q)}^{2} + \tilde{E} \int_{0}^{T} {‖ χ_{1}^{ϵ} \nabla u_{1}^{ϵ_{j}} ‖}_{L^{p} (Q)}^{p} d t \\ + \tilde{E} \int_{0}^{T} {‖ χ_{2}^{ϵ} \nabla u_{2}^{ϵ_{j}} ‖}_{L^{p} (Q)}^{p} d t + \tilde{E} \int_{0}^{T} {‖ ϵ χ_{2}^{ϵ} \nabla u_{3}^{ϵ_{j}} ‖}_{L^{p} (Q)}^{p} d t ⩽ C, \end{matrix} \end{matrix}$ and on the dual $V^{'}$ we have, $\begin{matrix} \tilde{E} sup_{| h | ⩽ δ} \int_{0}^{T - h} {‖ {\vec{u}}^{ϵ_{j}} (t + h) - {\vec{u}}^{ϵ_{j}} (t) ‖}_{V^{'}}^{p^{'}} d t ⩽ C max {δ^{\frac{1}{p - 1}}, δ^{\frac{p^{'}}{4}}}, δ ⩽ 1, \end{matrix}$ where ${\vec{u}}^{ϵ_{j}} = (u_{1}^{ϵ_{j}}, u_{2}^{ϵ_{j}}, u_{3}^{ϵ_{j}})$ .
6. Homogenization process

In this section, we derive the homogenized problem using the two-scale convergence. We start by introducing some theorems that will be useful in the convergence process. For their proofs, we refer to [1,8].

6.1. Some results on two-scale convergence

The following results are the-time dependent version of some fundamental results on [8].

Theorem 6.1.
Let ${φ^{ϵ}}$ be a bounded sequence of functions in $L^{p} (0, T; L^{p} (Q))$ with $1 < p ⩽ \infty$ . Then there exists subsequence ${φ^{ϵ^{'}}}$ and a function $φ \in L^{p} (0, T; L^{p} (Q \times Y))$ such that ${φ^{ϵ^{'}}}$ is two-scale convergent to φ.
Theorem 6.2.
Let ${φ^{ϵ}}$ be a sequence satisfying the assumptions of Theorem 6.1 . Furthermore, let ${φ^{ϵ}}$ be bounded in $L^{p} (0, T; W_{0}^{1, p} (Q))$ . Then
there exists a subsequence ${φ^{ϵ^{'}}}$ and a couple of functions $(φ, φ_{1})$ with $φ \in L^{p} (0, T; W_{0}^{1, p} (Q))$ and $φ_{1} \in L^{p} ((0, T) \times Q; W_{per}^{1, p} (Y))$ such that up to a subsequence, $\nabla φ^{ϵ} \overset{2 - s}{\to} \nabla_{x} φ (x) + \nabla_{y} φ_{1} (x, y)$ .

there exists a function $φ_{0} \in L^{2} ((0, T) \times Q; W_{per}^{1, p} (Y))$ such that up to a subsequence, $φ^{ϵ} \overset{2 - s}{\to} φ_{0} (x, y)$ and $ϵ \nabla φ^{ϵ} \overset{2 - s}{\to} \nabla_{y} φ_{0} (x, y)$ .

Proposition 6.3.
Let $φ^{ϵ}$ be a sequence of functions in $L^{p} (Q)$ such that $φ^{ϵ}$ two-scale converges to $φ_{0} (x, y)$ in $L^{p} (Q \times Y)$ . Then $φ^{ϵ}$ converges weakly to $φ (x)$ in $L^{p} (Q)$ , where $\begin{matrix} φ (x) = \int_{Y} φ_{0} (x, y) d y, in L^{p} (Q) . \end{matrix}$ Furthermore, we have $\begin{matrix} lim_{ϵ \to 0} {‖ φ^{ϵ} ‖}_{L^{p} (Q)} ⩽ ‖ φ_{0} ‖_{L^{p} (Q \times Y)} ⩽ ‖ φ ‖_{L^{p} (Q)} . \end{matrix}$

6.2. Passage to the limit

Now we study the asymptotic behaviour of $(P^{ϵ_{j}})$ as $ϵ_{j} \to 0$ using the two-scale convergence method.

Lemma 6.4.

For the sequences $(u_{i}^{ϵ})$ , there exists subsequences $(u_{i}^{ϵ_{j}})$ such that, we have the following two-scale convergences, $\tilde{P}$ -almost surely: $\begin{matrix} χ_{i}^{ϵ} u_{i}^{ϵ_{j}} \overset{2 - s}{\to} χ_{i} (y) u_{i} (t, x), (i = 1, 2), χ_{2}^{ϵ} u_{3}^{ϵ_{j}} \overset{2 - s}{\to} χ_{2} (y) U_{3} (t, x, y), \\ χ_{i}^{ϵ} \nabla u_{i}^{ϵ_{j}} \overset{2 - s}{\to} χ_{i} (y) [\nabla u_{i} (t, x) + \nabla_{y} U_{i} (t, x, y)], (i = 1, 2), \\ χ_{2}^{ϵ} ϵ \nabla u_{3}^{ϵ_{j}} \overset{2 - s}{\to} χ_{2} (y) \nabla_{y} U_{3} (t, x, y), \\ where U_{i} \in L^{p} ((0, T) \times Q; W_{per}^{1, p} (Y)), i = 1, 2, 3 . \end{matrix}$ Furthermore, there exist some functions ${\vec{g}}_{i} \in L^{p^{'}} ((0, T) \times Q \times Y^{n})$ , $u^{*} \in L^{2} (Q \times Y)$ , for $i = 1, 2, 3$ such that the following convergences hold $\tilde{P}$ -almost surely; $\begin{matrix} χ_{i}^{ϵ} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}) \overset{2 - s}{\to} χ_{i} (y) {\vec{g}}_{i} (t, x, y), (i = 1, 2), χ_{2}^{ϵ} μ_{3}^{ϵ_{j}} (x, ϵ_{j} \nabla u_{3}^{ϵ_{j}}) \overset{2 - s}{\to} χ_{2} (y) {\vec{g}}_{3} (t, x, y), \\ χ_{i}^{ϵ} u_{i}^{ϵ_{j}} (T, \cdot) \overset{2 - s}{\to} χ_{i} (y) u_{i}^{*} (x), (i = 1, 2), χ_{2}^{ϵ} u_{3}^{ϵ_{j}} (T, \cdot) \overset{2 - s}{\to} χ_{2} (y) u_{3}^{*} (x), \end{matrix}$ and $\begin{matrix} χ_{i}^{ϵ} f_{i}^{ϵ} (t, x) \overset{2 - s}{\to} χ_{i} f_{i} (t, x, y) \in L^{2} ((0, T) \times Q \times Y), i = 1, 2, 3 . \end{matrix}$

Proof.

According to Lemma 4.3, the sequences ${\vec{u}}^{ϵ} = [u_{1}^{ϵ}, u_{2}^{ϵ}, u_{3}^{ϵ}]$ and $\nabla {\vec{u}}^{ϵ} = [\nabla u_{1}^{ϵ}, \nabla u_{2}^{ϵ}, \nabla u_{3}^{ϵ}]$ are bounded in $L^{2} (Q_{i}^{ϵ})$ and $L^{p} (Q_{i}^{ϵ})$ respectively. Since by definition, $χ_{1}^{ϵ} u_{1}^{ϵ}$ is zero in $Q ∖ Q_{1}^{ϵ}$ , and $χ_{2}^{ϵ} u_{2}^{ϵ}$ , $χ_{2} u_{3}^{ϵ}$ is zero in $Q ∖ Q_{2}^{ϵ}$ , we have the estimates (4.8) and (5.3) for the subsequences $χ_{1}^{ϵ} u_{1}^{ϵ_{j}}$ , $χ_{2}^{ϵ} u_{2}^{ϵ_{j}}$ , $χ_{2}^{ϵ} u_{3}^{ϵ_{j}}$ .

By Theorem 6.2(1) and Theorem 2.9 in [1], we have the following two-scale convergences, $\tilde{P}$ -a.s., $\begin{array}{l} χ_{i}^{ϵ} u_{i}^{ϵ_{j}} \overset{2 - s}{\to} χ_{i} (y) u_{i} (t, x), i = 1, 2, \\ χ_{i}^{ϵ} \nabla u_{i}^{ϵ_{j}} \overset{2 - s}{\to} χ_{i} (y) [\nabla u_{i} (t, x) + \nabla_{y} U_{i} (t, x, y)], i = 1, 2, \end{array}$ where $u_{i} \in L^{p} (0, T; W_{0}^{1, p} (Q))$ and $U_{i} \in L^{p} ((0, T) \times Q; W_{per}^{1, p} (Y) ∖ R)$ , for $i = 1, 2$ .

Using the same argument and Theorem 6.1, $χ_{i}^{ϵ} f_{i}^{ϵ_{j}}$ ( $i = 1, 2$ ) and $χ_{2}^{ϵ} f_{3}^{ϵ_{j}}$ are bounded in $L^{2} ((0, T) \times Q)$ . Hence, the subsequences two-scale converge to $χ_{i} (y) f_{i} (t, x, y)$ ( $i = 1, 2$ ) and $χ_{2} (y) f_{3} (t, x, y)$ respectively in $L^{2} ((0, T) \times Q \times Y)$ , $\tilde{P}$ -a.s.

Theorem 6.2(2) gives $\begin{matrix} χ_{2} u_{3}^{ϵ_{j}} \overset{2 - s}{\to} χ_{2} (y) U_{3} (t, x, y), \tilde{P} -a.s. and ϵ χ_{2}^{ϵ} \nabla u_{3}^{ϵ_{j}} \overset{2 - s}{\to} \nabla_{y} U_{3} (t, x, y) \tilde{P} -a.s. . \end{matrix}$ Lastly, from $A (3.1)$ , we have $\begin{matrix} \int_{0}^{T} \int_{Q} {| χ_{i}^{ϵ} μ_{i} (\frac{x}{ϵ_{j}}, \nabla u_{i}^{ϵ}) |}^{p^{'}} d x d t & ⩽ \int_{0}^{T} \int_{Q} χ_{i}^{ϵ} {| μ_{i} (\frac{x}{ϵ_{j}}, \nabla u_{i}^{ϵ}) |}^{p^{'}} d x d t \\ ⩽ C \int_{0}^{T} \int_{Q} χ_{i}^{ϵ} {| \nabla u_{i}^{ϵ} |}^{p - 1 (p^{'})} d x d t + \int_{0}^{T} \int_{Q} χ_{i}^{ϵ} k^{p^{'}} d x d t \\ ⩽ C \int_{0}^{T} \int_{Q} χ_{i}^{ϵ} {| \nabla u_{i}^{ϵ} |}^{p} d x d t (i = 1, 2) . \end{matrix}$ Hence, by Theorem 6.1, $χ_{i}^{ϵ} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ})$ ( $i = 1, 2$ ) is bounded in $L^{p^{'}} ((0, T); L^{p^{'}} (Q))$ , and similarly, $χ_{2}^{ϵ} μ_{3}^{ϵ_{j}} (x, \nabla u_{3}^{ϵ})$ is bounded in $L^{p^{'}} ((0, T); L^{p^{'}} (Q))$ . Consequently, the sequences two-scale converge to ${\vec{g}}_{i} (t, x, y)$ ( $i = 1, 2$ ) and ${\vec{g}}_{3} (t, x, y)$ $\tilde{P}$ -a.s., respectively with ${\vec{g}}_{i} (t, x, y) \in L^{p} ((0, T) \times Q \times Y^{n})$ . □

Before we proceed with the homogenization process, we establish the conditions on the interface $Γ_{1, 2}$ .

Let $u^{ϵ_{j}} = χ_{1}^{ϵ} u_{1}^{ϵ_{j}} + χ_{2}^{ϵ} (α u_{2}^{ϵ_{j}} + β u_{3}^{ϵ_{j}}) \in L^{p} (0, T; W^{1, p} (Q))$ . Owing to the transmission conditions on $Γ_{1, 2}^{ϵ}$ in $(P^{ϵ})$ , we have $\begin{matrix} γ_{1}^{ϵ_{j}} u^{ϵ_{j}} = γ_{1}^{ϵ_{j}} u_{1}^{ϵ_{j}} = α γ_{2}^{ϵ_{j}} u_{2}^{ϵ_{j}} + β γ_{2}^{ϵ_{j}} u_{3}^{ϵ_{j}} = γ_{2}^{ϵ_{j}} u^{ϵ_{j}}, on Γ_{1, 2}^{ϵ} . \end{matrix}$ Thus $\begin{matrix} ϵ_{j} \nabla u^{ϵ_{j}} = ϵ_{j} χ_{1}^{ϵ} \nabla u_{1}^{ϵ_{j}} + χ_{2}^{ϵ} (α ϵ_{j} \nabla u_{2}^{ϵ_{j}} + β ϵ_{j} \nabla u_{3}^{ϵ_{j}}) \in L^{p} ((0, T) \times Q), \tilde{P} -a.s. . \end{matrix}$ Hence, according to Lemma 6.4, $\begin{matrix} u^{ϵ_{j}} \overset{2 - s}{\to} χ_{1} (y) u_{1} (t, x) + χ_{2} (y) (α u_{2} (t, x) + β U_{3} (t, x, y)), \end{matrix}$ and $\begin{matrix} ϵ_{j} \nabla u^{ϵ_{j}} \overset{2 - s}{\to} χ_{2} (y) β \nabla_{y} U_{3} (t, x, y) . \end{matrix}$ Let $\vec{ϕ} \in D (Q, C_{per}^{\infty} (Y^{3}))$ , we have $\begin{matrix} \int_{Q} ϵ_{j} \nabla u^{ϵ_{j}} (t, x) \vec{ϕ} (x, \frac{x}{ϵ_{j}}) d x & = - \int_{Q} u^{ϵ_{j}} (t, x) ϵ_{j} \nabla (\vec{ϕ} (x, \frac{x}{ϵ_{j}})) d x \\ = - \int_{Q} u^{ϵ_{j}} (t, x) [ϵ_{j} \nabla \vec{ϕ} (x, \frac{x}{ϵ_{j}}) + \nabla_{y} \vec{ϕ} (x, \frac{x}{ϵ_{j}})] d x . \end{matrix}$ Taking the two-scale limits on both sides give $\begin{matrix} (6.1) & \begin{matrix} \int_{Q} \int_{Y} β χ_{2} (y) \nabla_{y} U_{3} (t, x, y) \vec{ϕ} (x, y) d y d x \\ = - \int_{Q} \int_{Y} [χ_{1} (y) u_{1} (t, x) + χ_{2} (y) (α u_{2} (t, x) + β U_{3} (t, x, y))] \nabla_{y} \vec{ϕ} (x, y) d y d x . \end{matrix} \end{matrix}$ The left hand side of (6.1) can be written as $\begin{matrix} \int_{Q} \int_{Y_{2}} β \nabla_{y} U_{3} (t, x, y) ϕ (x, y) d y d x & = - \int_{Q} \int_{Y_{2}} β U_{3} \nabla_{y} ϕ (x, y) d y d x \\ + \int_{Q} \int_{\partial Y_{2}} β U_{3} (t, x, y) ϕ (x, y) \cdot {\vec{ν}}_{2} d S_{y} d x \end{matrix}$ while the right hand side of (6.1) can be written as $\begin{matrix} - \int_{Q} \int_{Y_{1}} u_{1} (t, x) \nabla_{y} \vec{ϕ} (x, y) d y d x - \int_{Q} \int_{Y_{2}} (α u_{2} (t, x) + β U_{3} (t, x, y)) \nabla_{y} \vec{ϕ} d y d x . \end{matrix}$ From (6.1) we see that $\begin{matrix} \int_{Q} \int_{\partial Y_{2}} β U_{3} (t, x, y) ϕ (x, y) \cdot {\vec{ν}}_{2} d S_{y} d x \\ = - \int_{Q} \int_{Y_{1}} u_{1} (t, x) \nabla_{y} \vec{ϕ} (x, y) d y d x - \int_{Q} \int_{Y_{2}} α u_{2} (t, x) \nabla_{y} \vec{ϕ} d y d x \\ = - \int_{Q} \int_{\partial Y_{1}} u_{1} (t, x) \vec{ϕ} (x, s) \cdot ν_{1} d S_{y} d x - \int_{Q} \int_{\partial Y_{2}} α u_{2} (t, x) \vec{ϕ} (x, s) \cdot ν_{2} d S_{y} d x . \end{matrix}$ Since $U_{3}$ and $\vec{ϕ}$ are periodic on $Γ_{2, 2}$ , this implies that $\begin{matrix} β U_{3} + α u_{2} = u_{1} on \partial Y_{1} \cap \partial Y_{2} \equiv Γ_{1, 2} . \end{matrix}$ Now we state our main result.

Theorem 6.5.

Suppose the assumptions $A (1) - A (4)$ are satisfied. Then there exists a probability space $(\tilde{Ω}, \tilde{F}, \tilde{P}, {(\tilde{F})}_{0 ⩽ t ⩽ T})$ and random variables $(B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}})$ and $(\tilde{B}, \vec{u})$ such that $\begin{matrix} (B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}}) \to (\tilde{B}, \vec{u}) in S \tilde{P} -a.s. \end{matrix}$ where ${\vec{u}}^{ϵ_{j}} = [u_{1}^{ϵ_{j}}, u_{2}^{ϵ_{j}}, u_{3}^{ϵ_{j}}]$ and $\vec{u} = [u_{1}, u_{2}, u_{3}]$ such that $(B^{ϵ_{j}}, {\vec{u}}^{ϵ_{j}})$ is the solution of problem $(P^{ϵ_{j}})$ and $(\tilde{B}, \vec{u})$ satisfies the homogenized problems ( 3.4 ), ( 3.5 ) and ( 3.6 ); recall that $S = C (0, T; R^{n}) \times L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; L^{2} (Q)) \times L^{2} (0, T; L^{2} (Q))$ .

Proof.

Let $ψ_{i} \in D (0, T; C_{0}^{\infty} (Q))$ , $i = 1, 2$ , and $ϕ_{i} \in D ((0, T) \times Q; C_{per}^{\infty} (Y))$ , $i = 1, 2, 3$ , with $β ϕ_{3}^{ϵ_{j}} (t, x, y) = ψ_{1} (t, x) + α ψ_{2} (t, x)$ for $y \in Γ_{1, 2}$ .

We take the triple $\begin{matrix} [ψ_{1} (t, x) + ϵ_{j} ϕ_{1} (t, x, \frac{x}{ϵ_{j}}), ψ_{2} (t, x) + ϵ_{j} ϕ_{2} (t, x, \frac{x}{ϵ_{j}}), ϕ_{3}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}})] \end{matrix}$ in $L^{2} (0, T; H^{ϵ})$ as a test function, where we define $\begin{matrix} ϕ_{3}^{ϵ_{j}} (t, x, y) = ϕ_{3} (t, x, y) + \frac{ϵ_{j}}{β} ϕ_{1} (t, x, y) - \frac{ϵ_{j} α}{β} ϕ_{2} (t, x, y) . \end{matrix}$ Substituting these test functions in the weak formulation (5.2) of problem $(P^{ϵ})$ , we get $\begin{array}{l} - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q_{i}^{ϵ}} c_{i}^{ϵ_{j}} u_{i}^{ϵ_{j}} (t) [ψ_{i t} (t, x) + ϵ_{j} ϕ_{i t} (t, x, \frac{x}{ϵ_{j}})] d x d t \\ - \int_{0}^{T} \int_{Q_{2}^{ϵ}} c_{3}^{ϵ_{j}} u_{3}^{ϵ_{j}} (t) ϕ_{3 t}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}}) d x d t \\ + \int_{Q_{i}^{ϵ}} c_{i}^{ϵ_{j}} u_{i}^{ϵ_{j}} (T, x) [ψ_{i} (T, x) + ϵ_{j} ϕ_{i} (T, x, \frac{x}{ϵ_{j}})] d x \\ + \int_{Q_{2}^{ϵ}} c_{3}^{ϵ_{j}} u_{3}^{ϵ_{j}} (T, x) ϕ_{3}^{ϵ_{j}} (T, x, \frac{x}{ϵ_{j}}) d x \\ - \sum_{i = 1}^{2} \int_{Q_{i}^{ϵ}} c_{i}^{ϵ_{j}} u_{i}^{ϵ_{j}} (0, x) [ψ_{i} (0, x) + ϵ_{j} ϕ_{i} (0, x, \frac{x}{ϵ_{j}})] d x - \int_{Q_{2}^{ϵ}} c_{3}^{ϵ_{j}} u_{3}^{ϵ_{j}} (0, x) ϕ_{3}^{ϵ_{j}} (0, x, \frac{x}{ϵ_{j}}) d x \\ = - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q_{i}^{ϵ}} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}) \nabla [ψ_{i} (t, x) + ϵ_{j} ϕ_{i} (t, x, \frac{x}{ϵ_{j}})] d x d t \\ - \int_{0}^{T} \int_{Q_{2}^{ϵ}} μ_{3}^{ϵ_{j}} (x, ϵ_{j} \nabla u_{3}^{ϵ_{j}}) ϵ_{j} [\nabla ϕ_{3}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}}) + \frac{1}{ϵ_{j}} \nabla_{y} ϕ_{3}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}})] d x d t \\ + \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q_{i}^{ϵ}} f_{i}^{ϵ_{j}} [ψ_{i} (t, x) + ϵ_{j} ϕ_{i} (t, x, \frac{x}{ϵ_{j}})] d x d B^{ϵ_{j}} (t) \\ (6.2) & + \int_{0}^{T} \int_{Q_{2}^{ϵ}} f_{3}^{ϵ_{j}} ϕ_{3}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}}) d x d B_{3}^{ϵ_{j}} (t) . \end{array}$ Let us determine the limit of each term in this relation using the two-scale result in Lemma 6.4.

For the first term on the left hand side we have, $\begin{matrix} lim_{ϵ_{j} \to 0} \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q_{i}^{ϵ}} c_{i}^{ϵ_{j}} u_{i}^{ϵ_{j}} (t) [ψ_{i t} (t, x) + ϵ_{j} ϕ_{i t} (t, x, \frac{x}{ϵ_{j}})] d x d t \\ = lim_{ϵ_{j} \to 0} \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} c_{i}^{ϵ_{j}} χ_{i}^{ϵ} u_{i}^{ϵ_{j}} (t, x) ψ_{i t} (t, x) d x d t \\ + lim_{ϵ_{j} \to 0} ϵ_{j} \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} c_{i}^{ϵ_{j}} χ_{i}^{ϵ} u_{i}^{ϵ_{j}} (t, x) ϕ_{i t} (t, x, \frac{x}{ϵ_{j}}) d x d t \\ = \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} c_{i} (y) u_{i} (t, x) ψ_{i t} (t, x) d x d t \tilde{P} -a.s. . \end{matrix}$ The second term yields $\begin{matrix} lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q_{2}^{ϵ}} c_{3}^{ϵ_{j}} u_{3}^{ϵ_{j}} (t, x) ϕ_{3 t}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}}) d x d t \\ = lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q} c_{3}^{ϵ_{j}} χ_{2}^{ϵ} u_{3}^{ϵ_{j}} (t, x) ϕ_{3 t}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}}) d x d t \\ = \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) ϕ_{3 t} (t, x, y) d x d t \tilde{P} -a.s. . \end{matrix}$ Similarly, using the definitions of $χ_{i}^{ϵ}$ and taking the limit at $ϵ_{j} \to 0$ on the remaining terms on the left hand side of (6.2) give $\tilde{P} -a.s.$ , $\begin{matrix} \sum_{i = 1}^{2} \int_{Q} \int_{Y_{i}} c_{i} (y) u_{i}^{*} (x) ψ_{i} (T, x) d y d x + \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{*} (x) ϕ_{3} (T, x, y) d y d x \\ - \sum_{i = 1}^{2} \int_{Q} \int_{Y_{i}} c_{i} (y) u_{i}^{0} (x) ψ_{i} (0, x) d y d x - \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{0} (x) ϕ_{3} (0, x, y) d y d x . \end{matrix}$ Taking the limit as $ϵ_{j} \to 0$ in the first and second terms on the right hand side of (6.2), we obtain $\begin{matrix} - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} {\vec{g}}_{i} (t, x, y) [\nabla ψ_{i} (t, x) + \nabla_{y} ϕ_{i} (t, x, y)] d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} (t, x, y) \nabla_{y} ϕ_{3} (t, x, y) d y d x d t \tilde{P} -a.s. . \end{matrix}$ Lastly, we deal with the limits of the last two terms on the right hand side of (6.2); which are stochastic integrals.

For the integral involving $f_{1}^{ϵ}$ , we have $\begin{matrix} (6.3) & \begin{matrix} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) [ψ_{1} (t, x) + ϵ_{j} ϕ_{1} (t, x, \frac{x}{ϵ_{j}})] d x d B_{1}^{ϵ_{j}} (t) \\ = \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d B_{1}^{ϵ_{j}} (t) \\ + \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ϵ_{j} ϕ_{1} (t, x, \frac{x}{ϵ_{j}}) d x d B_{1}^{ϵ_{j}} (t) . \end{matrix} \end{matrix}$ We start with the first term on the right hand side of (6.3). Since $B_{1}^{ϵ_{j}} (t)$ has unbounded variations, some care is needed. We first split the integral as $\begin{matrix} (6.4) & \begin{matrix} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d B_{1}^{ϵ_{j}} (t) \\ = \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) \\ + \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d {\tilde{B}}_{1} (t) . \end{matrix} \end{matrix}$ For the first term on the right hand side of (6.4), we adopt the process of regularization for $χ_{1}^{ϵ_{j}} f^{ϵ_{j}} (t, x)$ with respect to t in the following form $\begin{matrix} χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}} (t) = \frac{1}{λ} \int_{0}^{T} ρ (- \frac{t - s}{λ}) χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (s, x) d s, for λ > 0, \end{matrix}$ where ρ is a standard mollifier.

Now we have that $χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}}$ is differentiable with respect to t and satisfies the following relation $\begin{matrix} \int_{0}^{T} {‖ χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}} (t) ‖}_{L^{2} (Q)}^{2} d t ⩽ \int_{0}^{T} {‖ χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} ‖}_{L^{2} (Q)}^{2} d t, \forall λ > 0 and \forall ϵ_{j} > 0, \end{matrix}$ and $\begin{matrix} χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}} (t, x) \to χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) strongly in L^{2} ((0, T) \times Q) as λ \to 0 . \end{matrix}$ We write the first term on the right hand side of (6.4) as $\begin{matrix} (6.5) & \begin{matrix} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) \\ = \int_{0}^{T} \int_{Q} χ_{1}^{ϵ} ϵ_{j} f_{1, λ}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) \\ + \int_{0}^{T} \int_{Q} [χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) - χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}}] ψ_{1} (t, x) d x d (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) . \end{matrix} \end{matrix}$ Since $χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}}$ is differentiable, we integrate by parts on the first term in the right hand side of (6.5) to get $\begin{matrix} (6.6) & \begin{matrix} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) \\ = \int_{Q} (B^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}} ψ_{1} (t, x) d s |_{0}^{T} \\ - \int_{0}^{T} \int_{Q} (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) \partial_{t} (χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}} (t, x) ψ_{1} (t, x)) d x d t . \end{matrix} \end{matrix}$ The condition on $χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}}$ and $ψ_{1}$ together with the convergence of $B_{1}^{ϵ_{j}}$ to ${\tilde{B}}_{1} (t)$ in $C ([0, T])$ $\tilde{P}$ -a.s., give that the right hand side of (6.6) is bounded by a positive number $κ_{1} (λ) η_{1} (ϵ_{j})$ , where $η_{1} (ϵ_{j})$ vanishes as ϵ tends to zero, while $κ_{1} (λ)$ is finite.

Thanks to Burhölder–Davis Gundy inequality and the convergence of $χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}}$ to $χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}}$ , the second term on the right hand side of (6.5) is estimated as $\begin{matrix} \tilde{E} | \int_{0}^{T} \int_{Q} [χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) - χ_{1}^{ϵ_{j}} f_{1, λ}^{ϵ_{j}} (t)] ψ_{1} (t, x) d x d (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) | ⩽ κ_{2} (λ), \end{matrix}$ where $κ_{2} (λ)$ converge to zero as $λ \to 0$ .

Hence from (6.5), we have $\begin{matrix} \tilde{E} | \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d (B_{1}^{ϵ_{j}} (t) - {\tilde{B}}_{1} (t)) | ⩽ κ_{1} (λ) η_{1} (ϵ_{j}) + κ_{2} (λ) . \end{matrix}$ From (6.4), we conclude that $\begin{matrix} | \tilde{E} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d B_{1}^{ϵ_{j}} (t) - \tilde{E} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d {\tilde{B}}_{1} (t) | \\ ⩽ κ_{1} (λ) η_{1} (ϵ_{j}) + κ_{2} (λ) . \end{matrix}$ Passing to the limit as $ϵ_{j} \to 0$ , we get $\begin{matrix} lim_{ϵ_{j} \to 0} | \tilde{E} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d B_{1}^{ϵ_{j}} (t) - \tilde{E} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d {\tilde{B}}_{1} (t) | \\ ⩽ κ_{2} (λ) . \end{matrix}$ But since the left hand side of this relation is independent of λ, and $κ_{2} (λ) \to 0$ as $λ \to 0$ , we can pass to the limit on both sides as $λ \to 0$ to get $\begin{matrix} lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d B_{1}^{ϵ_{j}} (t) = lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d {\tilde{B}}_{1} (t) . \end{matrix}$ By Lemma 6.4, we have the two-scale convergence of $χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}}$ to $χ_{1} (y) f_{1} (t, x, y)$ $\tilde{P}$ -a.s., which implies weak convergence

Hence by using the convergence result for stochastic integrals in [28] (Theorem 4, pg 63), we get $\begin{matrix} lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ψ_{1} (t, x) d x d B_{1}^{ϵ_{j}} (t) = \int_{0}^{T} \int_{Q} χ_{1} (y) f_{1} (t, x, y) ψ_{1} (t, x) d x d {\tilde{B}}_{1} (t), \\ \tilde{P} -a.s. . \end{matrix}$ Now we show that $\begin{matrix} lim_{ϵ_{j} \to 0} ϵ_{j} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ϕ_{1} (t, x, \frac{x}{ϵ_{j}}) d x d B_{1}^{ϵ_{j}} (t) = 0, \tilde{P} -a.s. . \end{matrix}$ With the assumptions of $f_{1}^{ϵ_{j}}$ and Burkhölder–Davis–Gundy’s inequality, we have $\begin{array}{l} lim_{ϵ_{j} \to 0} ϵ_{j} \tilde{E} sup_{t \in [0, T]} | \int_{0}^{T} \int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ϕ_{1} (t, x, \frac{x}{ϵ_{j}}) d x d B_{1}^{ϵ_{j}} (t) | \\ ⩽ C lim_{ϵ_{j} \to 0} ϵ_{j} \tilde{E} {(\int_{0}^{T} {(\int_{Q} χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} (t, x) ϕ_{1} (t, x, \frac{x}{ϵ_{j}}) d x)}^{2} d t)}^{\frac{1}{2}} \\ ⩽ C lim_{ϵ_{j} \to 0} ϵ_{j} \tilde{E} {(\int_{0}^{T} {‖ χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} ‖}_{L^{2} (Q)} {‖ ϕ_{1} (t, x, \frac{x}{ϵ_{j}}) ‖}_{L^{2} (Q)} d t)}^{\frac{1}{2}} \\ ⩽ C lim_{ϵ_{j} \to 0} ϵ_{j} {(\int_{0}^{T} {‖ χ_{1}^{ϵ_{j}} f_{1}^{ϵ_{j}} ‖}_{L^{2} (Q)} d t)}^{\frac{1}{2}} \to 0 \tilde{P} -a.s. . \end{array}$ Combining the above convergences, we assert that $\begin{matrix} lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q} χ_{1}^{ϵ} f_{1}^{ϵ_{j}} [ψ_{1} (t, x) + ϵ_{j} ϕ_{1} (t, x, \frac{x}{ϵ_{j}})] d x d B_{1}^{ϵ_{j}} (t) \\ = \int_{0}^{T} \int_{Q} \int_{Y_{1}} f_{1} (t, x, y) ψ_{1} (t, x) d x d {\tilde{B}}_{1} (t) \tilde{P} -a.s. . \end{matrix}$ Similarly, $\begin{matrix} lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q} χ_{2}^{ϵ} f_{2}^{ϵ_{j}} [ψ_{2} (t, x) + ϵ_{j} ϕ_{2} (t, x, \frac{x}{ϵ_{j}})] d x d B_{2}^{ϵ_{j}} (t) \\ = \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{2} (t, x, y) ψ_{2} (t, x) d x d {\tilde{B}}_{2} (t), \tilde{P} -a.s. \end{matrix}$ and $\begin{matrix} lim_{ϵ_{j} \to 0} \int_{0}^{T} \int_{Q} χ_{2}^{ϵ} f_{3}^{ϵ_{j}} ϕ_{3}^{ϵ_{j}} (t, x, \frac{x}{ϵ_{j}}) d x d B_{3}^{ϵ_{j}} (t) \\ = \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ϕ_{3} (t, x, y) d x d {\tilde{B}}_{3} (t), \tilde{P} -a.s. . \end{matrix}$ Combining all the above convergences yield $\begin{array}{l} - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} c_{i} (y) u_{i} (t, x) ψ_{i t} (t, x) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) ϕ_{3 t} (t, x, y) d y d x d t \\ + \sum_{i = 1}^{2} \int_{Q} \int_{Y_{i}} c_{i} (y) u_{i}^{*} (x) ψ_{i} (T, x) d y d x + \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{*} (x) ϕ_{3} (T, x, y) d y d x \\ - \sum_{i = 1}^{2} \int_{Q} \int_{Y_{i}} c_{i} (y) u_{i}^{0} (x) ψ_{i} (0, x) d y d x - \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{0} (x) ϕ_{3} (0, x, y) d y d x \\ + \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} {\vec{g}}_{i} (t, x, y) [\nabla ψ_{i} (t, x) + \nabla_{y} ϕ_{i} (t, x, y)] d y d x d t \\ + \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} (t, x, y) \nabla_{y} ϕ_{3} (t, x, y) d y d x d t \\ - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} f_{i} (t, x, y) ψ_{i} (t, x) d y d x d {\tilde{B}}_{i} (t) \\ (6.7) & - \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ϕ_{3} (t, x, y) d y d x d {\tilde{B}}_{3} (t) = 0, \tilde{P} -a.s. \end{array}$ let us decouple equation (6.7) by making specific choices of test functions $ψ_{1}$ , $ψ_{2}$ , $ϕ_{1}$ , $ϕ_{2}$ , $ϕ_{3}$ .

Let $ψ_{1}$ be such that $ψ_{1} (t) = 0$ at $t = 0$ and $t = T$ and choose $ϕ_{3}$ such that $β ϕ_{3} (t, x, y) = ψ_{1} (t, x)$ for $y \in Y_{2}$ and $ψ_{2}, ϕ_{1}, ϕ_{2} = 0$ . Then we get the following equation; $\begin{array}{l} - \int_{0}^{T} \int_{Q} \int_{Y_{1}} c_{1} (y) u_{1} (t, x) ψ_{1 t} (t, x) d y d x d t - \frac{1}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) ψ_{1 t} (t, x) d y d x d t \\ + \int_{Q} \int_{Y_{1}} c_{1} (y) u_{1}^{*} (x) ψ_{1} (T, x) d y d x + \frac{1}{β} \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{*} (x) ψ_{1} (T, x) d y d x \\ - \int_{Q} \int_{Y_{1}} c_{1} (y) u_{1}^{0} (x) ψ_{1} (0, x) d y d x - \frac{1}{β} \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{0} (x) ψ_{1} (0, x) d y d x \\ + \int_{0}^{T} \int_{Q} \int_{Y_{1}} {\vec{g}}_{1} (t, x, y) \nabla ψ_{1} (t, x) d y d x d t \\ + \frac{1}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} (t, x, y) \nabla_{y} ψ_{1} (t, x) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{1}} f_{1} (t, x, y) ψ_{1} (t, x) d y d x d {\tilde{B}}_{1} (t) \\ - \frac{1}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ψ_{1} (t, x) d y d x d {\tilde{B}}_{3} (t) = 0 . \end{array}$ Integrating by parts with respect to t in the first and second terms on the left hand side and with respect to x on the seventh term gives $\begin{array}{l} \int_{0}^{T} \int_{Q} \int_{Y_{1}} c_{1} (y) d u_{1} (t, x) ψ_{1} (t, x) d y d x + \frac{1}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) d U_{3} (t, x, y) ψ_{1} (t, x) d y d x \\ - \int_{0}^{T} \int_{Q} \int_{Y_{1}} \nabla \cdot {\vec{g}}_{1} (t, x, y) ψ_{1} (t, x) d y d x d t + \int_{0}^{T} \int_{\partial Q} \int_{Y_{1}} {\vec{g}}_{1} (t, x, y) \cdot \vec{ν_{1}} ψ_{1} (t, x) d y d S_{x} d t \\ - \frac{1}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} \nabla_{y} \cdot {\vec{g}}_{3} (t, x, y) ψ_{1} (t, x) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{1}} f_{1} (t, x, y) ψ_{1} (t, x) d y d x d {\tilde{B}}_{1} (t) \\ - \frac{1}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ψ_{1} (t, x) d y d x d {\tilde{B}}_{3} (t) \\ (6.8) & = 0 . \end{array}$ This is the weak formulation of the following macro-fissure equation; $\begin{matrix} (6.9) & \begin{matrix} (\int_{Y_{1}} c_{1} (y) d y) d u_{1} (t, x) + \frac{1}{β} (\int_{Y_{2}} c_{3} (y) d U_{3} (t, x, y) d y) \\ = \nabla \cdot (\int_{Y_{1}} {\vec{g}}_{1} (t, x, y) d y) d t + (\int_{Y_{1}} f_{1} (t, x, y) d y) d {\tilde{B}}_{1} (t) \\ + \frac{1}{β} (\int_{Y_{2}} f_{3} (t, x, y) d y) d {\tilde{B}}_{3} (t) . \end{matrix} \end{matrix}$ Similarly, let $ψ_{1}, ϕ_{1}, ϕ_{2} = 0$ and $ψ_{2}$ is such that $ψ_{2} (t, x) = 0$ at $t = 0$ and $t = T$ and $ϕ_{3}$ be such that $β ϕ_{3} = - α ψ_{2}$ , for $y \in Y_{1}$ Then we obtain the following $\begin{array}{l} - \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{2} (y) u_{2} (t, x) ψ_{2 t} (t, x) d y d x d t + \frac{α}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) ψ_{2 t} (t, x) d y d x d t \\ + \int_{Q} \int_{Y_{2}} c_{2} (y) u_{2}^{*} (x) ψ_{2} (T, x) d y d x - \frac{α}{β} \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{*} (x) ψ_{2} (T, x) d y d x \\ - \int_{Q} \int_{Y_{2}} c_{2} (y) u_{2}^{0} (x) ψ_{2} (0, x) d y d x + \frac{α}{β} \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{0} (x) ψ_{2} (0, x) d y d x \\ + \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{2} (t, x, y) \nabla ψ_{2} (t, x) d y d x d t \\ - \frac{α}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} (t, x, y) \nabla_{y} ψ_{2} (t, x) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{2} (t, x, y) ψ_{2} (t, x) d y d x d {\tilde{B}}_{2} (t) \\ + \frac{α}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ψ_{2} (t, x) d y d x d {\tilde{B}}_{3} (t) = 0 . \end{array}$ Integrating by parts with respect to t in the first and second terms on the left hand side and with respect to x on the seventh term gives $\begin{array}{l} \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{2} (y) d u_{2} (t, x) ψ_{2} (t, x) d y d x - \frac{α}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) d U_{3} (t, x, y) ψ_{2} (t, x) d y d x \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} \nabla \cdot {\vec{g}}_{2} (t, x, y) ψ_{2} (t, x) d y d x d t \\ + \int_{0}^{T} \int_{\partial Q} \int_{Y_{2}} {\vec{g}}_{2} (t, x, y) \cdot \vec{ν_{2}} ψ_{2} (t, x) d y d S_{x} d t \\ + \frac{α}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} \nabla_{y} \cdot {\vec{g}}_{3} (t, x, y) ψ_{2} (t, x) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{2} (t, x, y) ψ_{2} (t, x) d y d x d {\tilde{B}}_{2} (t) \\ + \frac{α}{β} \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ψ_{2} (t, x) d y d x d {\tilde{B}}_{3} (t) \\ (6.10) & = 0 . \end{array}$ Since $ψ_{2} \in D ((0, T); C_{0}^{\infty} (Q))$ is arbitrary, we have that (6.10) is the weak formulation of the following macro-matrix equation; $\begin{matrix} (6.11) & \begin{matrix} (\int_{Y_{2}} c_{2} (y) d y) d u_{2} (t, x) - \frac{α}{β} (\int_{Y_{2}} c_{3} (y) d U_{3} (t, x, y) d y) \\ = \nabla \cdot (\int_{Y_{2}} {\vec{g}}_{2} (t, x, y) d y) d t + (\int_{Y_{2}} f_{2} (t, x, y) d y) d {\tilde{B}}_{2} (t) \\ - \frac{α}{β} (\int_{Y_{2}} f_{3} (t, x, y) d y) d {\tilde{B}}_{3} (t) . \end{matrix} \end{matrix}$ Next, let $ψ_{1}, ψ_{2}, ϕ_{1}, ϕ_{2} = 0$ and $ϕ_{3}$ is such that $ϕ_{3} (t) = 0$ at $t = 0$ and $t = T$ on $Γ_{1, 2}$ ., together with $β U_{3} + α u_{2} = u_{1}$ , on $\partial Y_{1} \cap \partial Y_{2} = Γ_{1, 2}$ , we get the cell system $\begin{matrix} - \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) U_{3} (t, x, y) ϕ_{3 t} (t, x, y) d y d x d t + \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{*} (x) ϕ_{3} (T, x, y) d y d x \\ - \int_{Q} \int_{Y_{2}} c_{3} (y) u_{3}^{0} (x) ϕ_{3} (0, x, y) d y d x + \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} (t, x, y) \nabla_{y} ϕ_{3} (t, x, y) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ϕ_{3} (t, x, y) d y d x d {\tilde{B}}_{3} (t) = 0 . \end{matrix}$ Integrating by parts with respect to t on the first and third terms on the left hand side gives $\begin{array}{rcl} (6.12) & \begin{matrix} - \int_{0}^{T} \int_{Q} \int_{Y_{2}} c_{3} (y) d U_{3} (t, x, y) ϕ_{3} (t, x, y) d y d x + \int_{0}^{T} \int_{Q} \int_{Y_{2}} \nabla_{y} \cdot {\vec{g}}_{3} (t, x, y) ϕ_{3} (t, x, y) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{\partial Y_{2}} {\vec{g}}_{3} (t, x, y) \cdot \vec{ν_{2}} ϕ_{3} (t, x, y) d S_{y} d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) ϕ_{3} (t, x, y) d y d x d {\tilde{B}}_{3} (t) = 0 . \end{matrix} \end{array}$ This is a weak formulation of the following the cell system; $\begin{matrix} (6.13) & \begin{matrix} c_{3} (y) d U_{3} (t, x, y) = \nabla_{y} \cdot {\vec{g}}_{3} (t, x, y) + f_{3} (t, x, y) d {\tilde{B}}_{3} (t), y \in Y_{2} \\ U_{3} and {\vec{g}}_{3} \cdot ν are Y -periodic on Γ_{2, 2}, \\ β U_{3} = u_{1} - α u_{2} on Γ_{1, 2} . \end{matrix} \end{matrix}$ Lastly, letting $ψ_{1}, ψ_{2}, ϕ_{3} = 0$ and $ϕ_{1}, ϕ_{2} \in D ((0, T) \times Q; C_{per}^{\infty} (Y))$ , we obtain $\begin{matrix} \int_{0}^{T} \int_{Q} \int_{Y_{1}} {\vec{g}}_{1} (t, x, y) \nabla_{y} ϕ_{1} (t, x, y) d y d x d t + \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{2} (t, x, y) \nabla_{y} ϕ_{2} (t, x, y) d y d x d t, \end{matrix}$ which is weak formulation of the following system of equations; $\begin{matrix} (6.14) & \begin{matrix} \nabla_{y} \cdot {\vec{g}}_{i} (t, x, y) = 0 y \in Y_{i}, \\ {\vec{g}}_{i} \cdot \vec{ν} = 0 on Γ_{1, 2} and \vec{g} \cdot \vec{ν} is Y -periodic on \partial Y_{i} \cap \partial Y for i = 1, 2 . \end{matrix} \end{matrix}$ We will have our homogenized problem when we have identified the terms ${\vec{g}}_{1}$ , ${\vec{g}}_{2}$ and ${\vec{g}}_{3}$ .

Next we split (6.7) using special choices of test functions $ψ_{1}$ , $ψ_{2}$ , $ψ_{3}$ , $ϕ_{1}$ , $ϕ_{2}$ , $ϕ_{3}$ in order to be able to use Ito’s formula.

In the first stage, we choose $ψ_{2}, ϕ_{1}, ϕ_{2}, ϕ_{3} = 0$ and $ψ_{1} \in D (0, T; W_{0}^{1, p} (Q))$ , we have $\begin{matrix} (6.15) & (\int_{Y_{1}} c_{1} (y) d y) d u_{1} (t, x) = \nabla \cdot (\int_{Y_{1}} {\vec{g}}_{1} (t, x, y) d y) + (\int_{Y_{1}} f_{1} (t, x, y) d y) d {\tilde{B}}_{1} (t) . \end{matrix}$ Next we choose $ψ_{1}, ϕ_{1}, ϕ_{2}, ϕ_{3} = 0$ and $ψ_{2} \in D (0, T; W_{0}^{1, p} (Q))$ , we have $\begin{matrix} (6.16) & (\int_{Y_{2}} c_{2} (y) d y) d u_{2} (t, x) = \nabla \cdot (\int_{Y_{2}} {\vec{g}}_{2} (t, x, y) d y) + (\int_{Y_{2}} f_{2} (t, x, y) d y) d {\tilde{B}}_{2} (t) . \end{matrix}$ Now we choose $ψ_{1}, ψ_{2}, ϕ_{1}, ϕ_{2} = 0$ and $ϕ_{3} \in D ([0, T] \times Q; W_{per}^{1, p} (Y))$ , we have $\begin{matrix} (6.17) & c_{3} (y) d U_{3} (t, x, y) = \nabla_{y} \cdot {\vec{g}}_{3} (t, x, y) + (\int_{Y_{2}} f_{3} (t, x, y) d y) d {\tilde{B}}_{3} (t) . \end{matrix}$ Next we choose $ψ_{1}, ψ_{2}, ϕ_{2}, ϕ_{3} = 0$ and $ϕ_{1} \in D ([0, T] \times Q; W_{per}^{1, p} (Y))$ , we have $\begin{matrix} (6.18) & \int_{0}^{T} \int_{Q} \int_{Y_{1}} {\vec{g}}_{1} (t, x, y) \nabla_{y} ϕ_{1} (t, x, y) d y d x d t = 0 . \end{matrix}$ Lastly, we choose $ψ_{1}, ψ_{2}, ϕ_{1}, ϕ_{3} = 0$ and $ϕ_{2} \in D ([0, T] \times Q; W_{per}^{1, p} (Y))$ , we have $\begin{matrix} (6.19) & \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{2} (t, x, y) \nabla_{y} ϕ_{2} (t, x, y) d y d x d t = 0 . \end{matrix}$

Ito’s formula on (6.15)–(6.17) at $t = T$ and adding (6.18) and (6.19) we get $\begin{array}{rcl} (6.20) & \begin{matrix} \frac{1}{2} \sum_{i = 1}^{2} \int_{Q} \int_{Y_{i}} c_{1} (y) {| u_{i} (T) |}^{2} d y d x + \frac{1}{2} \int_{Q} \int_{Y_{2}} c_{3} (y) {| U_{3} (T) |}^{2} d y d x \\ - \frac{1}{2} \int_{Q} \int_{Y_{2}} c_{3} (y) {| U_{3} (0) |}^{2} d y d x - \frac{1}{2} \sum_{i = 1}^{2} \int_{Q} \int_{Y_{i}} c_{1} (y) {| u_{i} (0) |}^{2} d y d x \\ + \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} {\vec{g}}_{i} (t, x, y) \nabla u_{i} (t, x) d y d x d t \\ + \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} (t, x, y) \nabla_{y} U_{3} (t, x, y) d y d x d t \\ - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} f_{i} (t, x, y) u_{i} (t, x) d y d x d B_{i} (t) \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} (t, x, y) U_{3} (t, x, y) d y d x d B_{3} (t) \\ - \frac{1}{2} \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} {| f_{i} (t, x, y) |}^{2} d y d x d t - \frac{1}{2} \int_{0}^{T} \int_{Q} \int_{Y_{2}} {| f_{3} (t, x, y) |}^{2} d y d x d t \\ + \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} {\vec{g}}_{i} (t, x, y) \nabla_{y} U_{i} (t, x, y) d y d x d t \\ = 0, \tilde{P} -a.s. . \end{matrix} \end{array}$ Now we identify ${\vec{g}}_{1}$ , ${\vec{g}}_{2}$ , ${\vec{g}}_{3}$ . For this we use Minty’s trick (Lemma 2.2) [18], see also [4] and [7]. Let $\vec{φ}, \vec{ξ} \in C_{0}^{\infty} {([0, T] \times Q; C_{per}^{\infty} (Y))}^{3}$ and $η_{1}, η_{2}, η_{3} \in C_{0}^{\infty} ([0, T] \times Q; C_{per}^{\infty} (Y))$ and for $ϵ > 0$ , we define the functions $\begin{array}{l} λ_{i}^{ϵ} (t, x) = χ_{i} (\frac{x}{ϵ}) \nabla u_{i} (t, x) + ϵ χ_{i} (\frac{x}{ϵ}) \nabla η_{i} (t, x, \frac{x}{ϵ}) + σ \vec{φ} (t, x, \frac{x}{ϵ}), i = 1, 2, \\ λ_{3}^{ϵ} (t, x) = χ_{2} (\frac{x}{ϵ}) (ϵ \nabla η_{3} (t, x, \frac{x}{ϵ})) + σ \vec{ξ} (t, x, \frac{x}{ϵ}) . \end{array}$ Since $μ_{i} (\frac{x}{ϵ}, λ_{i}^{ϵ} (x, t))$ and $λ_{i}^{ϵ} (t, x)$ ( $i = 1, 2, 3$ ) arises from an admissible test function, we have the following two-scale convergence $\begin{array}{l} λ_{i}^{ϵ} (t, x) \overset{2 - s}{\to} λ_{i} (t, x, y) = χ_{i} (y) \nabla u_{i} (t, x) + χ_{i} (y) \nabla_{y} η_{i} (t, x, y) + σ φ (t, x, y) (i = 1, 2), \\ λ_{3}^{ϵ} (t, x) \overset{2 - s}{\to} λ_{3} (t, x, y) = χ_{2} (y) \nabla_{y} η_{3} (t, x, y) + σ \vec{ξ} (t, x, y), \end{array}$ by $A (3.2)$ , we get $\begin{matrix} \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} (χ_{i}^{ϵ_{j}} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}) - χ_{i}^{ϵ_{j}} μ_{i}^{ϵ_{j}} (x, λ_{i}^{ϵ_{j}})) (\nabla u_{i}^{ϵ_{j}} - λ_{i}^{ϵ_{j}}) d x d t \\ + \int_{0}^{T} \int_{Q} (χ_{2}^{ϵ_{j}} μ_{3}^{ϵ_{j}} (x, ϵ_{j} \nabla u_{3}^{ϵ_{j}}) - χ_{2}^{ϵ_{j}} μ_{3}^{ϵ_{j}} (x, λ_{3}^{ϵ_{j}})) (ϵ_{j} \nabla u_{3}^{ϵ_{j}} - λ_{3}^{ϵ_{j}}) d x d t ⩾ 0 . \end{matrix}$ Expanding the above inequality yields $\begin{matrix} (6.21) & \begin{matrix} \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} χ_{i}^{ϵ_{j}} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}) \nabla u_{i}^{ϵ_{j}} d x d t - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} χ_{i}^{ϵ_{j}} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}) λ_{i}^{ϵ_{j}} d x d t \\ - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} χ_{i}^{ϵ_{j}} μ_{i}^{ϵ_{j}} (x, λ_{i}^{ϵ_{j}}) (\nabla u_{i}^{ϵ_{j}} - λ_{i}^{ϵ_{j}}) d x d t \\ + \int_{0}^{T} \int_{Q} χ_{i}^{ϵ_{j}} μ_{3}^{ϵ_{j}} (x, ϵ \nabla u_{3}^{ϵ_{j}}) ϵ_{j} \nabla u_{3}^{ϵ_{j}} d x d t \\ - \int_{0}^{T} \int_{Q} χ_{2}^{ϵ_{j}} μ_{3}^{ϵ_{j}} (x, ϵ \nabla u_{3}^{ϵ_{j}}) λ_{3}^{ϵ_{j}} d x d t \\ - \int_{0}^{T} \int_{Q} χ_{2}^{ϵ_{j}} μ_{3}^{ϵ_{j}} (x, λ_{3}^{ϵ_{j}}) (ϵ_{j} \nabla u_{3}^{ϵ_{j}} - λ_{3}^{ϵ_{j}}) d x d t ⩾ 0 . \end{matrix} \end{matrix}$ Recall that Ito’s formula on $(P^{ϵ_{j}})$ yields $\begin{matrix} (6.22) & \begin{matrix} \sum_{i = 1}^{2} c_{i}^{ϵ} (x) {| u_{i}^{ϵ_{j}} (t) |}^{2} + c_{3}^{ϵ} (x) {| u_{3}^{ϵ_{j}} (t) |}^{2} + 2 \sum_{i = 1}^{2} \int_{0}^{t} (μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}), \nabla u_{i}^{ϵ_{j}} (s)) d s \\ + 2 \int_{0}^{t} (μ_{3}^{ϵ_{j}} (x, ϵ \nabla u_{3}^{ϵ_{j}}), ϵ \nabla u_{3}^{ϵ_{j}} (s)) d s = \sum_{i = 1}^{2} c_{i}^{ϵ} (x) {| u_{i}^{ϵ_{j}} (0) |}^{2} + c_{3}^{ϵ} (x) {| u_{3}^{ϵ_{j}} (0) |}^{2} \\ + 2 \sum_{i = 1}^{2} \int_{0}^{t} (f_{i}^{ϵ_{j}} (s), u_{i}^{ϵ_{j}} (s)) d B^{ϵ_{j}} (s) + 2 \int_{0}^{t} (f_{3}^{ϵ_{j}} (s), u_{3}^{ϵ_{j}} (s)) d B_{3}^{ϵ_{j}} (s) \\ + \sum_{i = 1}^{2} \int_{0}^{t} {‖ f_{i}^{ϵ_{j}} ‖}_{L^{2} (Q_{i}^{ϵ})}^{2} d s + \int_{0}^{t} {‖ f_{3}^{ϵ_{j}} ‖}_{L^{2} (Q_{2}^{ϵ})}^{2} d s, \tilde{P} -a.s. . \end{matrix} \end{matrix}$ Adding a suitable zero to (6.21) and using (6.22) gives $\begin{array}{l} \frac{1}{2} \sum_{i = 1}^{2} \int_{Q} c_{i}^{ϵ_{j}} (x) χ_{i}^{ϵ_{j}} {| u_{i}^{ϵ_{j}} (0) |}^{2} d x - \frac{1}{2} \sum_{i = 1}^{2} \int_{Q} c_{i}^{ϵ_{j}} (x) χ_{i}^{ϵ_{j}} {| u_{i}^{ϵ_{j}} (T) |}^{2} d x \\ + \frac{1}{2} \int_{Q} c_{3}^{ϵ_{j}} (x) χ_{2}^{ϵ_{j}} {| u_{3}^{ϵ_{j}} (0) |}^{2} d x - \frac{1}{2} \int_{Q} c_{3}^{ϵ_{j}} (x) χ_{2}^{ϵ_{j}} {| u_{3}^{ϵ_{j}} (T) |}^{2} d x \\ + \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} χ_{i}^{ϵ_{j}} f_{i}^{ϵ_{j}} u_{i}^{ϵ_{j}} d x d B_{i} (t) + \sum_{i = 1}^{2} \frac{1}{2} \int_{0}^{T} \int_{Q} {| χ_{i}^{ϵ_{j}} f_{i}^{ϵ_{j}} |}^{2} d x d t \\ + \int_{0}^{T} \int_{Q} χ_{2}^{ϵ_{j}} f_{3}^{ϵ_{j}} u_{3}^{ϵ_{j}} d x d B_{3} (t) + \frac{1}{2} \int_{0}^{T} \int_{Q} {| χ_{2}^{ϵ_{j}} f_{3}^{ϵ_{j}} |}^{2} d x d t \\ - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} χ_{i}^{ϵ_{j}} μ_{i}^{ϵ_{j}} (x, \nabla u_{i}^{ϵ_{j}}) λ_{i}^{ϵ_{j}} d x d t \\ - \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} χ_{i}^{ϵ_{j}} μ_{i}^{ϵ_{j}} (x, λ_{i}^{ϵ_{j}}) (\nabla u_{i}^{ϵ_{j}} - λ_{i}^{ϵ_{j}}) d x d t \\ - \int_{0}^{T} \int_{Q} χ_{2}^{ϵ_{j}} μ_{3}^{ϵ_{j}} (x, ϵ \nabla u_{3}^{ϵ_{j}}) λ_{3}^{ϵ_{j}} d x d t \\ (6.23) & - \int_{0}^{T} \int_{Q} χ_{2}^{ϵ_{j}} μ_{3}^{ϵ_{j}} (x, λ_{3}^{ϵ_{j}}) (ϵ \nabla u_{3}^{ϵ_{j}} - λ_{3}^{ϵ_{j}}) d x d t ⩾ 0 . \end{array}$ Recall that $\nabla u_{i}^{ϵ_{j}} \overset{2 - s}{\to} \nabla u_{i} (t, x) + \nabla_{y} U_{i} (t, x, y)$ , $i = 1, 2$ and $ϵ \nabla u_{3}^{ϵ_{j}} \overset{2 - s}{\to} \nabla_{y} U_{3} (t, x, y)$ . We now take the limit as $ϵ_{j} \to 0$ to get $\begin{array}{l} \frac{1}{2} \sum_{i = 1}^{2} \int_{Q} \int_{Y_{i}} c_{i} (y) {| u_{i} (0) |}^{2} d y d x + \frac{1}{2} \int_{Q} \int_{Y_{2}} c_{3} (y) {| U_{3} (0) |}^{2} d y d x \\ + \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} f_{i} u_{i} d y d x d {\tilde{B}}_{i} (t) + \frac{1}{2} \sum_{i = 1}^{2} \int_{0}^{T} \int_{Q} \int_{Y_{i}} | f_{i} |^{2} d y d x d t \\ + \int_{0}^{T} \int_{Q} \int_{Y_{2}} f_{3} U_{3} d y d x d {\tilde{B}}_{3} (t) + \frac{1}{2} \int_{0}^{T} \int_{Q} \int_{Y_{2}} | f_{3} |^{2} d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{1}} {\vec{g}}_{1} (\nabla u_{1} + \nabla_{y} η_{1} + σ φ) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} μ_{1} (y, λ_{1}) (\nabla_{y} U_{1} - \nabla_{y} η_{1} - σ φ) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{1}} {\vec{g}}_{2} (\nabla u_{2} + \nabla_{y} η_{2} + σ φ) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} μ_{2} (y, λ_{2}) (\nabla_{y} U_{2} - \nabla_{y} η_{2} - σ φ) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} (\nabla_{y} η_{3} + σ ξ) d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} μ_{3} (y, λ_{3}) (\nabla_{y} U_{3} - \nabla_{y} η_{3} - σ ξ) d y d x d t \\ ⩾ \frac{1}{2} lim_{ϵ_{j} \to 0} \int_{Q_{1}^{ϵ}} c_{1}^{ϵ_{j}} (x) {| u_{1}^{ϵ_{j}} (T) |}^{2} d x + \frac{1}{2} lim_{ϵ_{j} \to 0} \int_{Q_{2}^{ϵ}} c_{2}^{ϵ_{j}} (x) {| u_{2}^{ϵ_{j}} (T) |}^{2} d x \\ (6.24) & + \frac{1}{2} lim_{ϵ_{j} \to 0} \int_{Q_{2}^{ϵ}} c_{3}^{ϵ_{j}} (x) {| u_{3}^{ϵ_{j}} (T) |}^{2} d x, \end{array}$ where we omit the variable $(t, x, y)$ in order to avoid cumbersome writing.

We use (6.20) in (6.24) and replace $η_{i} (t, x, y)$ by $U_{i} (t, x, y)$ ( $i = 1, 2, 3$ ) to get $\begin{matrix} (6.25) & \begin{matrix} - \int_{0}^{T} \int_{Q} \int_{Y_{1}} {\vec{g}}_{1} σ \vec{φ} d y d x d t + \int_{0}^{T} \int_{Q} \int_{Y_{1}} μ_{1} (y, \nabla u_{1} + \nabla_{y} U_{1} + σ \vec{φ}) σ \vec{φ} d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{2} σ \vec{φ} d y d x d t + \int_{0}^{T} \int_{Q} \int_{Y_{2}} μ_{2} (y, \nabla u_{2} + \nabla_{y} U_{2} + σ \vec{φ}) σ \vec{φ} d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} σ \vec{ξ} d y d x d t + \int_{0}^{T} \int_{Q} \int_{Y_{2}} μ_{3} (y, \nabla_{y} U_{3} + σ \vec{ξ}) σ \vec{ξ} d y d x d t \\ ⩾ \frac{1}{2} lim_{ϵ_{j} \to 0} \int_{Q_{1}^{ϵ}} c_{1}^{ϵ} (x) {| u_{1}^{ϵ} (T) |}^{2} d x + \frac{1}{2} lim_{ϵ_{j} \to 0} \int_{Q_{2}^{ϵ}} c_{2}^{ϵ} (x) {| u_{2}^{ϵ} (T) |}^{2} d x \\ + \frac{1}{2} lim_{ϵ_{j} \to 0} \int_{Q_{2}^{ϵ}} c_{3}^{ϵ} (x) {| u_{3}^{ϵ} (T) |}^{2} d x - \frac{1}{2} \int_{Q} \int_{Y_{1}} c_{1} (y) {| u_{1} (T) |}^{2} d y d x \\ - \frac{1}{2} \int_{Q} \int_{Y_{2}} c_{2} (y) {| u_{2} (T) |}^{2} d y d x - \frac{1}{2} \int_{Q} \int_{Y_{2}} c_{3} (y) {| U_{3} (T) |}^{2} d y d x . \end{matrix} \end{matrix}$ Now let us set $\vec{φ} = χ_{1} {\vec{θ}}_{1} + χ_{2} {\vec{θ}}_{2}$ where we take $θ_{i} \in C_{0}^{\infty} ([0, T] \times Q; C^{\infty} (Y_{i}))$ ( $i = 1, 2$ ). Using Proposition 6.3, the right hand side of (6.25) is nonnegative. Thus (6.25) becomes $\begin{matrix} - \int_{0}^{T} \int_{Q} \int_{Y_{1}} {\vec{g}}_{1} σ {\vec{θ}}_{1} d y d x d t + \int_{0}^{T} \int_{Q} \int_{Y_{1}} μ_{1} (y, \nabla u_{1} + \nabla_{y} U_{1} + σ {\vec{θ}}_{1}) σ {\vec{θ}}_{1} d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{2} σ {\vec{θ}}_{2} d y d x d t + \int_{0}^{T} \int_{Q} \int_{Y_{2}} μ_{2} (y, \nabla u_{2} + \nabla_{y} U_{2} + σ {\vec{θ}}_{2}) σ {\vec{θ}}_{2} d y d x d t \\ - \int_{0}^{T} \int_{Q} \int_{Y_{2}} {\vec{g}}_{3} σ \vec{ξ} d y d x d t + \int_{0}^{T} \int_{Q} \int_{Y_{2}} μ_{3} (y, \nabla_{y} U_{3} + σ \vec{ξ}) σ \vec{ξ} d y d x d t ⩾ 0 . \end{matrix}$ Following Bensoussan’s argument in [4], first we divide the above equation by σ and then let $σ \to 0$ to obtain $\begin{matrix} \int_{0}^{T} \int_{Q} \int_{Y_{1}} [μ (y, \nabla u_{1} + \nabla_{y} U_{1}) - {\vec{g}}_{1}] {\vec{θ}}_{1} d y d x d t \\ + \int_{0}^{T} \int_{Q} \int_{Y_{2}} [μ (y, \nabla u_{2} + \nabla_{y} U_{2}) - {\vec{g}}_{2}] {\vec{θ}}_{2} d y d x d t \\ + \int_{0}^{T} \int_{Q} \int_{Y_{1}} [μ (y, \nabla_{y} U_{3}) - {\vec{g}}_{3}] \vec{ξ} d y d x d t ⩾ 0 \forall {\vec{θ}}_{1}, {\vec{θ}}_{2}, \vec{ξ} . \end{matrix}$ Hence owing to Minty’s trick (Lemma 2.2) [18] as implemented by Bensoussan in [4], we conclude that $\begin{matrix} {\vec{g}}_{1} (t, x, y) = μ_{1} (y, \nabla u_{1} + \nabla_{y} U_{1}) in (0, T) \times Q \times Y_{1} \tilde{P} -a.s., \\ {\vec{g}}_{2} (t, x, y) = μ_{2} (y, \nabla u_{2} + \nabla_{y} U_{2}) in (0, T) \times Q \times Y_{2} \tilde{P} -a.s., \\ {\vec{g}}_{3} (t, x, y) = μ_{3} (y, \nabla_{y} U_{3}) in (0, T) \times Q \times Y_{2} \tilde{P} -a.s. . \end{matrix}$ We note that $\vec{u} = [u_{1}, u_{2}, u_{3}]$ and $\tilde{B} = ({\tilde{B}}_{1}, {\tilde{B}}_{2}, {\tilde{B}}_{3})$ are unique probabilistic weak solutions of the homogenized problems (3.4), (3.5) and (3.6). Thus by the infinite dimensional version of Yamada–Watanabe’s theorem (see [26]), we get that $\vec{u} = [u_{1}, u_{2}, u_{3}]$ and $\tilde{B} = ({\tilde{B}}_{1}, {\tilde{B}}_{2}, {\tilde{B}}_{3})$ are the unique strong solutions of (3.4), (3.5) and (3.6). Thus up to distribution (probability law) the whole sequence of solutions of $(P^{ϵ})$ converges to the solutions of (3.4), (3.5) and (3.6). This completes the proof of Theorem 6.5. □

References

Allaire, Homogenization and two-scale convergence, SIAM Journal of Mathematical Analysis 23(6) (1992), 1482–1518. doi:10.1137/0523084.

Arbogast,

DouglasJr. and

Hornung, Modeling of naturally fractured reservoirs by formal homogenization techniques, in: Frontiers in Pure and Applied Mathematics,

Dautray , ed., Elsevier, 1991, pp. 1–19.

G.I.

Barenblatt,

I.P.

Zheltov and

I.N.

Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks[strata], Journal of Applied Mathematics and Mechanics 24(5) (1960), 1286–1303. doi:10.1016/0021-8928(60)90107-6.

Bensoussan, Some existence results for stochastic partial differential equations, in: Stochastic Partial Differential Equations and Applications,

Da Prato and

Tubaro, eds, Pitman Research Notes in Mathematics, Vol. 268, 1992, pp. 37–53.

Billingsley, Convergence of Probability Measures, 2nd edn, Wiley Series in Probability and Statistics, Wiley-Interscience, 1999.

Bourgeat,

Mikelić and

Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math. 456 (1994), 19–51.

Chiado Piat and

Defranceschi, Homogenization of monotone operators, Nonlinear Analysis: Theory, Methods and Applications 14(9) (1990), 717–732. doi:10.1016/0362-546X(90)90102-M.

Cioranescu and

Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Application, Oxford University Press, 1999.

G.W.

Clark, Derivation of microstructure models of fluid flow by homogenization, Journal of Mathematical Analysis 226(2) (1998), 364–376. doi:10.1006/jmaa.1998.6085.

10.

G.W.

Clark and

R.E.

Showalter, Two-scale convergence of a model for flow in a partially fissured medium, Electronic Journal of Differential Equations 1999(2) (1999), 1.

11.

D.J.

Coffeirld and

A.M.

Spagnuolo, A model for single phase flow in layered porous media, Electronic Journal of Differential Equations 2010(94) (2010), 1.

12.

Douglas,

Peszyńska and

R.E.

Showalter, Single phase flow in partially fissured media, Transport in Porous Media 28 (1997), 285–306. doi:10.1023/A:1006562120265.

13.

L.C.

Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, Vol. 74, American Mathematical Society, 1990.

14.

L.C.

Evans, Partial Differential Equations, AMS, Providence, 1998.

15.

Hornung, Homogenization and Porous Media, Springer-Verlag, Berlin, New York, 1997.

16.

N.V.

Krylov and

B.L.

Rozovskii, Stochastic evolution equations, Journal of Soviet Mathematics 16(4) (1981), 1233–1277. doi:10.1007/BF01084893.

17.

G.J.

Minty, Monotone (nonlinear) operators in Hilbert space, Duke Mathematical Journal 29(3) (1962), 341–346.

18.

G.J.

Minty, On a “monotonicity” method for the solution of nonlinear equations in Banach spaces, Proceedings of the National Academy of the United States of America 50(6) (1963), 1038–1041. doi:10.1073/pnas.50.6.1038.

19.

Mohammed, Homogenization of nonlinear hyperbolic stochastic equation via Tartar’s method, J. Hyper. Differential Equations 14(02) (2017), 323–340. doi:10.1142/S0219891617500096.

20.

Mohammed and

Ahmed, Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains, Asymptotic Analysis 120 (2020), 123–149.

21.

Mohammed and

Sango, Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence, Asymptotic Analysis 91(3–4) (2015), 341–371. doi:10.3233/ASY-141269.

22.

Mohammed and

Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptotic Analysis 97(3–4) (2016), 301–327. doi:10.3233/ASY-151355.

23.

Mohammed and

Sango, Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing, Networks and Heterogeneous Media 14(2) (2019), 341–369. doi:10.3934/nhm.2019014.

24.

Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal of Mathematical Analysis 20(3) (1989), 608–623. doi:10.1137/0520043.

25.

Nguetseng,

R.E.

Showalter and

J.-L.

Woukeng, Diffusion of a single phase fluid through a general deterministic partially fissured medium, Electronic Journal of Differential Equations 2014(164) (2014), 1.

26.

Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Mathematicae 426 (2004), 1–63. doi:10.4064/dm426-0-1.

27.

Pardoux, Equations aux derivees Partielles Stochastiques Non Lineaires Monotones, These, Universite Paris VI, 1975.

28.

Rozovskii, Stochastic Evolution Systems: Linear Theory and Applications to Nonlinear Filtering, Springer, Netherlands, 1990.

29.

Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer-Verlag, Berlin, New York, 1980.

30.

Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations, Forum Mathematicum 25(5) (2013), 931–965.

31.

Sango, Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries, Communications in Mathematical Sciences 12(2) (2014), 345–382. doi:10.4310/CMS.2014.v12.n2.a7.

32.

R.E.

Showalter and

N.J.

Walkington, Micro-structure models of diffusion in fissured media, Journal of Mathematical Analysis and Applications 155(1) (1991), 1–20. doi:10.1016/0022-247X(91)90023-S.

33.

J.E.

Warren and

P.J.

Root, The behavior of naturally fractured reservoirs, Society of Petroleum Engineers Journal 3(3) (1963), 245–255. doi:10.2118/426-PA.

34.

Wright, On a diffusion of a single-phase slightly compressible fluid through a randomly fissured medium, Mathematical Methods in the Applied Sciences 24 (2001), 805–825. doi:10.1002/mma.243.

35.

Zhang, Monotonicity methods in PDE, Charleton University Mathematics Journal 1(1) (2010), 1–10.