Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition

The evolution Stokes equation in a perforated domain subject to Fourier boundary condition on the boundaries of the holes is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the two-scale convergence method after extending the solution with 0 in the wholes to pass to the limit. By It\^o stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results from of Duan and Wang (Comm. Math. Phys. 275:1508--1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived.


Introduction and formulation of the problem
In this paper, we are interested in a fluid flow where the advective inertial forces are small compared with viscous forces. Starting from the evolution Stokes equations in a periodic porous medium, with a dynamical boundary condition driven by a noise source on the solid pores, the homogenized dynamic is rigorously recovered by the use of two-scale convergence method.
The homogenization of the Stokes problem in perforated domains goes back to Sánchez-Palencia in [24] where an asymptotic expansion method was used. Rigorous proofs were given later by Tartar in [7] by the energy method and by Allaire in [2] where two scale convergence method was used. The two scale convergence was first introduced by Nguetseng in 1989 in [3] and later developed by Allaire in [1]. The idea of this method was to give a rigorous justification to the asymptotic expansion method. A more general setting has been defined by Nguetseng in [4], [5] and later in [9]. The theory of the two scale convergence from the periodic to the stochastic setting has been extended by Bourgeat, A. Mikelić and Wright in [6], using techniques from ergodic theory. There is a vast literature for partial differential equations with random coefficients, where this method was used, however most of the tackled problems in this setting are not in perforated domains, (see [6], [15] and the references therein). Much less was done for homogenization of stochastic partial differential equations, in particular in perforated domains. We mention the paper [21] where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived. A comprehensive theory for solving stochastic homogenization problems has been constructed recently in [10] and [11], where a Σ-convergence method adapted to stochastic processes was developed. An application of the method to the homogenization of a stochastic Navier-Stokes type equation with oscillating coefficients in a bounded domain (without holes) has been provided.
For the deterministic Stokes or Navier Stokes equations in perforated domains we refer to: [24], [7], [2], [12], [8], [14], [13]. In [8] the Stokes problem in a perforated domain with a nonhomogeneous Fourier boundary condition on the boundaries of the holes was studied while in [14] the same problem was studied with a slip boundary condition. As far as we know, the stochastic Stokes or Navier Stokes equation in a perforated domain has not been studied.
In this paper we consider a stochastic linear Navier Stokes equation in a periodically perforated domain with a noise source. On the boundaries of the holes, we consider a dynamical Denote by O ε,k the translation of εO by εk, k ∈ Z n . We make the assumption that the holes do not intersect the boundary ∂D and we denote by K ε the set of all k ∈ Z n such that the cell εk + εY is strictly included in D. The set of all such holes will be denoted by O ε , i.e.
and set D ε := D − O ε . By this construction, D ε is a periodically perforated domain with holes of size of the same order as the period. One of the difficulties of the homogenization in perforated domains consists in the fact that the ε-problems are defined in different domains. In [7], [8], [2], [14] suitable extensions for the velocity and for the pressure were defined to overcome this difficulty.
The evolution Stokes equation in the domain D ε with a stochastic dynamical boundary condition on the boundaries of the holes is given by where u ε is the velocity of the fluid and p ε is the pressure. Using Itô's formula and stochastic calculus we are able to prove some uniform estimates in some functional spaces that are ε-dependent. Our particular boundary condition makes it difficult to extend the velocity continuously in H 1 (D) n , so we chose to use the trivial extension by 0 of the velocity as well as of the gradient of the velocity. This extension is a continuous one in L 2 (D) for which the uniform estimates still hold. Our extension is not divergence free, so we cannot expect the homogenized solution to be divergence free. Hence we cannot use test functions that are divergence free in the variational formulation, which implies that the pressure has to be included. Because of the low regularity in time of the stochastic process, we have to define a more regular in time pressure P (t) = t 0 p(s)ds in Theorem 3.3. We apply to it the same extension and we will recover in the limit the information into the coefficients of the homogenized equation.
For the convergence of the boundary integrals we use an idea from [8] to rewrite an integral on the boundary in terms of an integral on the whole domain. In this way, the integrals over the boundary ∂O ε become in the limit integrals over the whole domain D. In particular, in the case of the stochastic integral on the boundaries of the holes, we combined the idea with the use of the decomposition of the Wiener process in terms of the basis, and that it is of trace class.
Our process depends on three variables: ω, t, and x but the oscillations appear only in x, hence we will use the two scale convergence method in the space variable but with a parameter (ω, t) ∈ Ω × [0, T ]. One of the condition to be able to use two scale convergence is the uniform boundedness with respect to ε > 0 of the L 2 -norm. We were not able to show uniform bounds for ω ∈ Ω or for any t ∈ [0, T ], but only in L 2 (Ω × [0, T ] × D) n , thus we decided using this type of convergence. Most of the results concerning the two scale convergence are being extended in a straighforward way to the results of our paper. We gather the results we use in Section 4. The convergence in two scale is a stronger type of convergence than the weak convergence (see Corollary 4.3), and so the convergence to the homogenized solution will be the weak one in L 2 (Ω × [0, T ] × D) n . This convergence although weak in the deterministic sense, is strong in the probability sense.
The paper will be organized as follows: in Section 2 we formulate more precisely our problem, set the functional setting and give the assumptions needed for the rest of the paper. In Section 3 we study the problem (1.1) in the perforated domain and show the existence and uniqueness in Theorem 3.1. The estimates, needed for the two scale convergence method, are derived. We also introduce the pressure in Theorem 3.1 and setup the variational formulation to be used for the passage to the limit. In Section 4, we introduce the two scale convergence and give a number of results that we will use. In Section 5, we pass to the limit in the variational formulation (3.12). We derive the cell problem (5.30) and the equation for the homogenized solution u * in (5.38).

Preliminaries and assumptions
In this section we introduce some functional spaces and assumptions in order to study the problem (1.1).
2.1. Functional setting. Let us introduce the following Hilbert spaces equipped respectively with the inner products We introduce the bounded linear and surjective operator γ ε : H 1 (D ε ) n → H 1 2 (∂O ε ) n such that γ ε u = u| ∂O ε for all u ∈ C ∞ D ε n . γ ε is the trace operator, (see [19], pp47). We denote by H − 1 2 (∂O ε ) n the dual space of H 1 2 (∂O ε ) n . We denote by H ε the closure of V ε in L 2 ε , and by V ε the closure of V ε in H 1 ε , where 3) Let us also denote by H ε and V ε the following functional spaces Let Π ε : L 2 ε → L 2 (D ε ) n be the operator that represents the projection onto the first component, i.e. Π ε U = u, for every U = (u, u) ∈ L 2 ε , and let B ε : V ε → H − 1 2 (∂O ε ) n denote the linear operator defined by B ε u = ∂u ∂n .
H ε and V ε are separable Hilbert spaces with the inner products and norms inherited from L 2 ε and H 1 ε respectively: U)) , and H ε and V ε are also separable Hilbert spaces with the norms induced by the projection Π ε .
We denote the dual pairing between U ∈ V ε and V ∈ (V ε ) ′ by U, Assume that b is a strictely positive constant, and define the linear operator A ε : Let Q 1 and Q 2 be linear positive operators in L 2 (D) n of trace class. Let (W 1 (t)) t≥0 and (W 2 (t)) t≥0 be two mutually independent L 2 (D) n -valued Wiener processes defined on the complete probability space (Ω, F, P) endowed with the canonical filtration (F t ) t≥0 and with the covariances Q 1 and Q 2 . The expectation is denoted by E. If K and H are two separable Hilbert spaces, then we will denote by L 2 (K, H) the space of bounded linear operators that are Hilbert-Schmidt from K in H. If Q is a linear positive operator in K of trace class, then we will denote by L Q (K, H), the space of bounded linear operators that are Hilbert-Schmidt from Q 1 2 K to H, and the norm will be denoted by · Q . Let us denote by , The system (1.1) can be rewritten: Lemma 2.1. (Properties of the operator A ε ) For every ε > 0, the linear operator A ε is positive and self-adjoint in H ε .
Proof. The operator is obviously symmetric, since for every U, V ∈ H ε : and also coercive: To show that it is self-adjoint it will be enough to show that Now we use Proposition A.10, page 389 from [16] to infer that A ε is self-adjoint.
For α > 0, let us denote by (A ε ) α the α-power of the operator A ε and D((A ε ) α ) its domain. In particular, Indeed, for every ε > 0 and using the trace inequality Denote by S ε (t) the analytic semigroup generated by A ε (see [17]).
We use the change of variables x = εx + εk and add over all k ∈ K ε to obtain the result.
In the next section, we will set up the assumptions.

2.2.
Assumptions. We assume that the operator and let where {e j1 } ∞ j=1 and {e j2 } ∞ j=1 are respectively the eigenvalues for Q 1 and Q 2 , and {λ j1 } ∞ j=1 and {λ j2 } ∞ j=1 are the corresponding sequences of eigenvalues. For any element h ∈ L 2 (∂O), we define the element R ε h ∈ L 2 (∂O ε ) by Proof. We use the definition of R ε h to obtain after a change of variables: . Moreover, throughout the paper we will assume that U ε 0 = (u ε 0 , u ε 0 = εv ε 0 ) is an F 0 − measurable H ε − valued random variable and there exists a constant C independent of ε, such that for every ε > 0: 3. The microscopic model In this section, we will state the well posedness of system (1.1) and some uniform estimates wrt ε.
The mild solution U ε is also a weak solution, that is, P-a.s.
Moreover, if (2.11) holds then for every ε > 0 Proof. Since the operator A ε is the generator of a strongly continuous semigroup S ε (t), t ≥ 0 in H ε and using the assumption (2.9), then the existence and uniqueness of mild solutions in H ε is a consequence of Theorem 7.4 of [16]. The regularity in V ε is a consequence of the estimates below.
Taking the expected value yields and Now (3.3) follows from using Gronwall's lemma in (3.6).
On the other side, (3.5) implies that Moreover using the Burkholder-David-Gundy inequality and the Young inequality, we get that Now, plugging this estimate in the previous one and using Gronwall's lemma completes the proof of (3.4).
To show (3.13) we consider the extension to the whole domain of the pressure P ε (t), and compute from (3.12). We show that it is bounded uniformly for t ∈ [0, T ], and then apply the result given by Proposition 1.2 from [18] to obtain (3.13). We estimate first D P ε (t) div φdx from (3.12) and obtain by applying Hölder's inequality: We use Lemma 2.2 and the estimates (3.8) for u ε (t) and (2.11) for the initial conditions to get Ito's isometry gives We use again Lemmas 2.2 and (2.3) and then property (2.9) to obtain that E ∇ P ε (t) 2

Two scale convergence
We will summarize in this section several results about the two scale convergence that we will use throughout the paper. For the results stated without proofs, see [1] or [10]. First we establish some notations of spaces of periodic functions. We denote by C k # (Y ) the space of functions from C k (Y ), that have Y − periodic boundary values. By L 2 # (Y ) we understand the closure of C # (Y ) in L 2 (Y ) and by H 1 Definition 4.1. We say that a sequence u ε ∈ L 2 (Ω × [0, T ] × D) n two-scale converges to u ∈ L 2 (Ω × [0, T ] × D × Y ) n , and denote this convergence by for any fixed t ∈ [0, T ] and any Ψ 1 ∈ L 2 (Ω × D; C # (Y )) n . As a consequence, for Ψ 2 ∈ L 2 (Ω × [0, T ] × D; C # (Y )) n the sequence By applying Hölder's inequality, We use the dominated convergence theorem to deduce that

Homogenized equation
In order to find the homogenized equation, we first establish the two scale limits that we use in the variational formulation (3.12) to pass to the limit when ε → 0.

2) and
We use Theorem 4.5 to obtain as consequences of (5.2) and (5.3) the following two scale convergences:
We compute the limits that involve integrals over the boundaries, and we will make use of the techniques from [8] that give a way of transforming integrals over the surface in integrals over the volume.
Proof. Similar arguments discussed in (5.17) and (5.16) will be used here. We will prove: and To show (5.20) we use the functions w 1 and w ε 1 defined in (5.18) and (5.19): follows from the condition (2.9) and the definiton of w ε 1 . Now, the same computation used to show (5.16) yields: To show (5.21), let (e i2 (x)) ∞ i=1 and (λ i2 ) ∞ i=1 previously defined in (2.9). Denote by h i (s) = g 22 (s)e i2 ∈ L 2 (∂O), for each i ∈ Z + and s ∈ [0, T ]. We infer from (2.9) that We will define w i (s) similarly as w 1 to be the unique element in H 1 (Y * ) that solves: and w ε i (s) = ε 2 w i (s) · ε that will solve There exists a constant C independent of i and s such that ||w i (s)|| H 1 (Y * ) n ≤ C||h i (s)|| L 2 (∂O) n and using (5.22) we also have: Using the decomposition of φ ε and previously used computations, we have to show that: converges to 0 uniformly for t ∈ [0, T ]. We perform the following calculations: The second term of the sum will be bounded by Using (5.25) it converges to 0 uniformly for t ∈ [0, T ]. We decompose the first term similarly to (5.16) The first sum goes to 0 for any fixed N because of the weak convergence to 0 in L 2 (D) of (1 D ε (x) − |Y * |)φ 2 (x). The second sum goes to 0 when N → ∞ because of (5.22).
Using the limits obtained in the variational formulation we get (5.26)
Using these relations, elementary calculations will give us that CΛ 1 · Λ 2 = Y * ν (Λ 1 + ∇ y w Λ 1 (y)) · (Λ 2 + ∇ y w Λ 2 (y)) dy, (5.39) which implies that C is symmetric and positive definite. As a consequence, we have the following existence result: Theorem 5.5. (Well posedness of the homogenized equation) Let S * (t) t≥0 be the semigroup generated by the operator A * and assume that assumption (2.9) is satisfied. Then, the equation (5.38) admits a unique mild solution u * ∈ L 2 (Ω; C([0, T ], L 2 (D) n ) ∩ L 2 (0, T ; H 1 0 (D) n )), given by u * (t) =|Y * |S * (t)u * 0 + |∂O|  (1) The solution of (5.38) is not divergence free, so the homogenized equation does not contain a pressure term. However, the effect of the pressure P ε (t) appears implicitely through the matrix C, hence the cell problem. (3) The convergence of the sequence u ε to the limit u * is strong in the probabilistic sense, but weak in the deterministic sense. In particular, the sequence u ε of the extensions by 0 of u ε inside O ε converges to u * weakly in L 2 (Ω × [0, T ] × D) n .