Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

1. Introduction and setting of the problem. Homogenization theory has become an important tool in the investigation of processes taking place in highly heterogenous media ranging from soil to the most advanced aircraft the construction of which uses composite materials. So far, the problems solved by means of homogenization have mainly involved deterministic partial differential equations (PDEs) and the homogenization of PDEs with randomly oscillating coefficients; the great wealth of results obtained over several decades on problems of diverse classes and methodologies can be found for instance in [9,6,40,41,23,34,22,49,31,17,4,32,36,46,50,33], for the deterministic case and [13,14,18,20,24,37,19,47,48]. for the random case. Fundamental methods were subsequently developed such as the method of asymptotic expansions ( [9], [6], [40], [41]), the two scale-convergence ( [4], [32]), Tartar method of oscillating test functions and H-convergence ( [49]), the asymptotic method for non periodically perforated domains ( [23], [46]), Gconvergence ( [36]) and Γ-convergence developed by De Giorgi and his students; relevant extensions of some of these methods, including their random counterparts, have also emerged in recent times. One rapidly developping important branch of homogenization is that of numerical homogenization; see [1], [2].
However physical processes under random fluctuations are better modelled by stochastic partial differential equations (SPDEs). It was therefore natural to consider homogenization of this very important class of PDEs. Research in this direction is still at its infancy, despite the importance of such problems in both applied and fundamental sciences. Some relevant interesting work have recently been undertaken, mainly for parabolic SPDEs; see for instance [3,8,10,11,21,43,44]. We also note the closely related work [3,25,15,16] dealing with stochastic homogenization for SPDEs with small parameters. The list of references is of course not exhaustive, but a representation of the main trends in the field.
The homogenization of hyperbolic SPDEs was initiated in [27], [28,29], [30] where the authors studied the homogenization of Dirichlet problems for linear hyperbolic equation with rapidly oscillating coefficients using the method of the two-scale convergence pioneered by Nguetseng in [32] and developed by Allaire in [4] and [5]; they also dealt with the linear Neumann problem by means of Tartar's method and obtained the corresponding corrector results within these settings; [30] deals with a semilinear hyperbolic SPDE by Tartar's method.
In the present work, following the two-scale convergence method, we investigate the homogenization of a non-linear hyperbolic equation with nonlinear damping, where the intensity of the noise is also nonlinear and is assumed to satisfy Lipschitz's condition. Our investigation relies on crucial compactness results of analytic (Aubin-Lions-Simon's type) and probabilistic (Prokhorov and Skorokhod fundamental theorems) nature. It should be noted that these methods extend readily to the case when Lipschitz condition on the intensity of the noise is replaced by a mere continuity. In contrast to the linear and the semilinear cases considered in previous papers, the type of nonlinear damping and nonlinear noise in the present paper leads to new challenges in obtaining uniform a priori estimates as well as in the passage to the limit. It should be noted that the process of damping in mechanical systems is a crucial stabilizing factor when the system is subjected to very extreme tasks; mathematically this translates in some regularizing effects on the solution of the governing equations.
We are concerned with the homogenization of the initial boundary value problem with oscillating data, referred to throughout the paper as problem (P ): where u t denotes the partial derivative ∂u /∂t of u with respect to t, > 0 is a sufficiently small parameter which ultimately tends to zero, T > 0, Q is an open and bounded (at least Lipschitz) subset of R n , W = (W (t)) (t ∈ [0, T ]) an m-dimensional standard Wiener process defined on a given filtered complete probability space (Ω, F, P, (F t ) 0≤t≤T ); E denotes the corresponding mathematical expectation. For a physical motivation, we refer to [27,28], where the authors discussed real life processes of vibrational nature subjected to random fluctuations; for instance the nonlinear term B(t, u t ) stands for damping effects, the term f (t, x, x/ε, ∇u ) is the oscillating regular part of the force acting on the system and depending linearly on ∇u , while the term g(t, x, x/ε, u t )dW represents the oscillating random component of the force; it depends on u ε t . More precise assumptions on the data will be provided shortly.
Few words about the difference between the current work and previous works by the authors on homogenization of SPDEs. Compared to [27,28,29,30], the structure of problem (P ε ) is dominated by nonlinear terms such as the damping B(t, u t ), leading to L p (Q)-like norms whose combination with the predominently L 2 -like norms coming from the other terms requires special care, both in the derivation of the a priori estimates, as well as in the passage to the limit. Though, two-scale convergence method is also used in the paper [27], the model there is essentially linear. The works [43,44] deal with stochastic parabolic equations in domains with fine grained boundaries, where no conditions of periodicity hold and the methodology implemented there is a stochastic counterpart of Kruslov-Marchenko's [23] and Skrypnik's [46] homogenization theories based on potiential theory; for instance the homogenized problems in [43,44] involve an additional term of capacitary type. The investigation of a hyperbolic counterpart of these works has still not been undertaken and is somehow overdue. Finally, compared with the above mentioned works, the current paper involves a simpler proof of the convergence of the stochastic nonlinear term (its integral) thanks to a blending of two-scale convergence with a regularizing argument and a result on convergence of stochastic integrals due to Rozovskii [39,Theorem 4,P 63].
We now impose the following hypotheses on the data.
,j≤n is an n × n symmetric matrix, the components a i,j , are Y −periodic and there exists a constant α > 0 such that continuous operator for all t ∈ (0, T ) and all u, v, w ∈ W 1,p 0 (Q); (ii) There exists a constant γ > 0 such that B(t, u), u ≥ γ u p W 1,p 0 (Q) for a.e. t ∈ (0, T ) and all u ∈ W 1,p 0 (Q); (iii) There exists a positive constant β such that B(t, u) W −1,p (Q) ≤ β u p−1 W 1,p 0 (Q) for a.e. t ∈ (0, T ) and all u ∈ W 1,p 0 (Q); (iv) B(t, u) − B(t, v), u − v ≥ 0, for a.e. t ∈ (0, T ) and all u, v ∈ W 1,p 0 (Q); (v) The map t −→ B(t, u) is Lebesgue measurable in (0, T ) with values in W −1,p (Q) for all u ∈ W 1,p 0 (Q). (A3) We assume that f (t, x, y, w) is measurable with respect to (x, y) for any (t, w) ∈ (0, T ) × R n , continuous with respect to (t, w) for almost every (x, y) ∈ Q × Y , and Y -periodic with respect to y. Also there exists an R n -valued func- for any (t, w, ε) ∈ (0, T ) × L 2 (Q) × (0, ∞), with the constant C independent of ε and t. A sufficient requirement for this condition to hold is that x, y, φ) is an m-dimensional vector-function whose components g j (t, x, y, φ) satisfy the following conditions: • g j (t, x, y, φ) is Y -periodic with respect to y, measurable with respect to x and y for almost all t ∈ (0, T ) and for all φ ∈ L 2 (Q), • g j (t, x, y, φ) is continuous with respect to φ for almost all (t, x, y) ∈ (0, T ) × Q × Y , and there exists a positive constant C independent of t, x and y, such and • g j (t, x, y, ·) satisfies Lipschitz's condition with the constant L independent of t, x and y. If ||g j (t, x, y, 0)|| L 2 (Q×Y ) < ∞ for any i = 1, ..., m and any t ∈ (0, T ), the condition (1) is redundant since it follows from the Lipschitz condition (2).
From now on we use the following oscillating functions We now introduce our notion of solution; namely the strong probabilistic one. ( u ∈ L 2 Ω, F, P; C(0, T ; H 1 0 (Q)) u t ∈ L 2 Ω, F, P; C(0, T ; L 2 (Q)) ∩ L p Ω, F, P; L p (0, T ; W 1,p 0 (Q)) , . The problem of existence and uniqueness of a strong probabilistic solution of (P ) was dealt with in [38]. The relevant result is Theorem 1.2. Suppose that the assumptions (A1) − (A5) hold and let p ≥ 2. Then for fixed > 0, the problem (P ) has a unique strong probabilistic solution u in the sense of Definition 1.1.
Our goal is to show that as tends to zero the sequence of solutions (u ) converge in a suitable sense to a solution u of the following SPDE is the unique solution of the following boundary value problem: x, y, u t ) dy, a and b are suitable limits of the oscillating initial conditions a and b , respectively, W is an m-dimensional Wiener process 2. A priori estimates. Here and in the sequel, C will denote a constant independent of . In the following lemma, we obtain the energy estimates associated to problem (P ).
Lemma 2.1. Under the assumptions (A1)-(A5), the solution u of (P ) satisfies the following estimates: and Proof. The following arguments are used modulo appropriate stopping times. Itô's formula and the symmetry of A give Integrating over (0, t), t ≤ T , we get Using the assumptions (A1), (A2)(ii), (A5) and taking the supremum over t ∈ [0, T ] and the expectation on both sides of the resulting relation, we have . Using assumptions (A3), thanks to Cauchy-Schwarz's and Young's inequalities, we have where > 0. Thanks to Burkholder-Davis-Gundy's inequality, followed by Cauchy-Schwarz's inequality, the last term in 5 can be estimated as Again using Young's inequality and the assumptions (A5), we get 2E sup for > 0. Combining the estimates 6, 7, 5 and assumption (A5) and taking sufficiently small, we infer that Using Gronwall's inequality, we have The proof is complete.
The following lemma will be of great importance in proving the tightness of probability measures generated by the solution of problem (P ) and its time derivative.
Lemma 2.2. Let the conditions of Lemma 2.1 be satisfied and let p ≥ 2. Then there exists a constant C > 0 such that for any > 0 and 0 < δ < 1.
Proof. . We consider that div (A ∇φ) has been restricted to the space W −1,p (Q) and that the restriction induces a bounded mapping from W 1,p 0 (Q) to W −1,p (Q). Assume that u t is extended by zero outside the interval [0, T ] and that θ > 0. We write Then Firstly, thanks to assumption (A1), we have where we have used the fact that p ≤ 2.
Secondly, we use assumption (A2)(iii), estimate 4 and Hölder's inequality to get where we have used assumption (A3). Using 10, 11 and 12 in 9 raised to the power p , for fixed δ > 0, we get We now estimate the term involving the stochastic integral. We use the embedding Thanks to Fubini's theorem and Hölder's inequality, we have where we have used Burkholder-Davis-Gundys inequality. We now invoke assumption (A5) and estimate 3 to deduce from 14 and 15 that For the first term in the right-hand side of 13, we use Fubini's theorem, Hölder's inequality and estimate 3 to get ≤ CT δ p .
The second term on the right hand side of 13 is estimated using 4 and we get Combining 13, 16, 17 and 18, and taking into account the fact that the similar estimates hold for θ < 0, we conclude that This completes the proof.
3. Tightness property of probability measures. The following Lemmas are needed in the proof of the tightness and the study of the properties of the probability measures generated by the sequence (W, u , u t ).
We have from [45] Lemma 3.1. Let B 0 , B and B 1 be some Banach spaces such that B 0 ⊂ B ⊂ B 1 and the injection B 0 ⊂ B is compact. For any 1 ≤ p, q ≤ ∞, and 0 < s ≤ 1 let E be a set bounded in L q (0, T ; Then E is relatively compact in L p (0, T ; B) The following two lemmas are collected from [12]. Let S be a separable Banach space and consider its Borel σ-field to be B(S). We have  the symbol L (·) stands for the law of ·.
Let us introduce the space We endow Z with the norm Lemma 3.4. The above constructed space Z is a compact subset of L 2 (0, T ; L 2 (Q))× L 2 (0, T ; L 2 (Q)).
Define on (X , B(X )) the family of probability measures (Π ) by Π (A) = P(Ψ −1 (A)) for all A ∈ B(X ). Proof. We carry out the proof following a long the lines of the proof of [27, lemma 7]. For δ > 0, we look for compact subsets This is equivalent to which can be proved if we can show that Let L δ be a positive constant and n ∈ N. Then we deal with the set Using Arzela's theorem and the fact that W δ is closed in C(0, T ; R m ), we ensure the compactness of W δ in C(0, T ; R m ). From Markov's inequality where η is a nonnegative random variable and k a positive real number, we have where (2k − 1)!! = 1 · 3 · · · (2k − 1) and W i denotes the i-th component of W .
For k = 4, we have Now, let K δ , M δ be positive constants. We define Lemma 3.4 shows that D δ is compact subset of L 2 (0, T ; L 2 (Q)), for any δ > 0. It is therefore easy to see that Markov's inequality 19 gives Similarly, we let µ n , ν m be sequences of positive real numbers such that µ n , ν n → 0 as n → ∞, n µ p /p n νn < ∞ (for the series to converge we can choose ν n = 1/n 2 , µ n = 1/n α , with αp /p > 4) and define Owing to Proposition 3.1 in [7], B δ is a compact subset of L 2 (0, T ; L 2 (Q)) for any δ > 0. We have Again thanks to 19, we obtain . This completes the proof.
From Lemmas 3.2 and 3.5, there exist a subsequence {Π j } and a measure Π such that Π j Π weakly. From lemma 3.3, there exist a probability space (Ω,F,P) and X -valued random variables (W j , u j , u j t ), (W , u, u t ) such that the probability law of (W j , u j , u j t ) is Π j and that of (W , u, u t ) is Π. Furthermore, we have (W j , u j , u j t ) → (W , u, u t ) in X ,P − a.s..
Let us define the filtrationF We show thatW (t) is anF t -wiener process following [7] and [42]. Arguing as in [42], we get that (W j , u j , u j t ) satisfiesP − a.s. the problem P j in the sense of distributions.
4. Two-scale convergence. In this section, we state some key facts about the powerful two-scale convergence invented by Nguetseng [32].
The following result deals with some of the properties of the test functions which we are considering; it is a modification of Lemma 9.1 from [17, p.174].
The following theorems are of great importance in obtaining the homogenization result; for their proofs, we refer to [4], [17] and [26].
, thanks to the Lipschitz condition on g (t, x, ·). Now due to the strong convergence 20 of u ε t − u t to zero in L 2 ((0, T ) × Q),P-a.s., we get that I ε 1 → 0,P − a.s. Now we can apply 2-scale convergence for the limit of I ε 2 and indeed lim ε→0 x, y, u t ) ψ (t, x, y) dxdt,P − a.s.
Remark 1. From the assumption (A5), 28 and 23, we have the following strong convergence 5. The homogenization result. We will now study the asymptotic behaviour of the problem (P j ), when j → 0.
Theorem 5.1. Suppose that the assumptions on the data are satisfied. Let a j a, weakly in H 1 0 (Q), Then there exist a probability space Ω ,F,P, F t 0≤t≤T and random variables (u j , u j t , W j ) and (u, u t ,W ) such that the convergences 20 and 26 hold. Furthermore (u, u t ,W ) satisfies the homogenized problem (P ).
Testing problem (P j ) by the function Φ ∈ C ∞ c ((0, T ) × Q) and integrating the first term in the right-hand side by parts, we have Using estimate 3, the convergence 20 and Theorems 4.3 and 4.4, we show the twoscale convergence ). Then we can still consider Φ j as test function in 40. Thus Let us deal with these terms one by one, when j → 0. Thanks to estimate 22 and convergence 33, we have The second term can be written as follows, ] as a test function in the two-scale limit of the gradient in the first term in 42. Therefore Thanks to Hölder inequality, 22 and the fact that A j ∇u j is bounded in L ∞ (0, T ; x, x j )dxdt = 0,P − a.s..

MOGTABA MOHAMMED AND MAMADOU SANGO
Again, thanks to estimate 22 and convergence 35, we have where we have used the assumption (A3). It is easy to see that the second term in 43, converges to zero. For the first term in the right-hand side of 43, we readily have Concerning the stochastic integral, we havẽ We deal with the term involving φ (t, x). We havẽ In view of the unbounded variation of W ε t −W t , the convergence of the first term on the right-hand side of 46 needs appropriate care, in order to take advantage of theP−a.s. uniform convergence of W ε dt 1 2 Combining the above convergences, we obtain Choosing in the first stage φ = 0 and after φ 1 = 0, the problem 57 is equivalent to the following system of integral equations and By standard arguments (see [17]), equation 58 has a unique solution given by where χ(y), known as the first order corrector, is the unique solution to the following equation: div y (A(y)∇ y χ(y)) = ∇ y · A(y), in Y, χ is Y periodic.
As for the uniqueness of the solution of 59, we prove it as follows. Using 60 in 59, one obtains that 59 is the weak formulation of the equation where x, y, u t ) dy.
But the initial boundary value problem corresponding to 62 has a unique solution by [38]. It remains to show that u(x, 0) = a(x) and u t ( where we pass to the limit, to get The integration by parts, in the first term gives In view of equation 57, we deduce that for any v ∈ C ∞ c (Q). This implies that u t (x, 0) = b(x). For the other initial condition, we consider Φ j (t, x). Integration by parts in the first term of 40, gives Passing to the limit in this equation, we obtain We integrate by parts again to obtain

MOGTABA MOHAMMED AND MAMADOU SANGO
Using the same argument as before, we show that u(x, 0) = a(x). We note the triple W , u, u t is a probabilistic weak solution of (P ) which is unique. Thus by the infinite dimensional version of Yamada-Watanabe's theorem (see [35]), we get that (W, u, u t ) is the unique strong solution of (P ). Thus up to distribution (probability law) the whole sequence of solutions of (P ) converges to the solution of problem (P ). Thus the proof of Theorem 5.1 is complete.
6. Convergence of the energy. Let us introduce the energies associated with the problems (P j ) and (P ), as follows: But from Itô's formula, we have The vanishing of the expectation of the stochastic integrals is due to the fact that (g (u t ),ũ t ) and (g(u), u t ) are square integrable in time. We want to prove that the energy associated with the problem (P j ), uniformly converges to that of the corresponding homogenized problem (P ). For this purpose we need to assume some stronger assumptions on the initial data. We have the following result Now we need to show that (E j (u j )(t)), is uniformly bounded and equicontinuous on [0, T ] and hence Arzela-Ascoli's theorem concludes the proof. We have Thanks to the assumptions on the data (A3), (A4) and (A5), the a priori estimates 3 and 4, we show that For any h > 0 and t ∈ [0, T ], we get Again assumptions (A3), (A5) and Cauchy-Schwarz's inequality, give This implies the equicontinuity of the sequence {E j (u j )(t)} j , and therefore the proof is complete.
As a closing remark, we note that our results can readily be extended to the case of infinite dimensional Wiener processes taking values in appropriate Hilbert spaces; for instance cylindrical Wiener processes.