Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains

In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (n+1)-dimensional thin region when it collapses into an n-dimensional region. We first established the existence of attractors and their properties for these equations on (n+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.


DINGSHI LI, KENING LU, BIXIANG WANG AND XIAOHU WANG
where 0 < ε ≤ 1 and g ∈ C 2 (Q, (0, +∞)) which implies that there exist positive constants γ 1 and γ 2 such that (1) Throughout this paper, we write O = Q × (0, 1) and O = Q × (0, γ 2 ). We consider the following stochastic non-autonomous reaction-diffusion equation with additive noise defined on the thin domain O ε : where τ is the initial time, ν ε is the unit outward normal vector on ∂O ε , F is a nonlinear function defined on R × O × R, G is a function defined on R × O, h is a given function defined on O, and w is a two-sided real-valued Wiener process on a probability space.
We will study the limiting behavior of system (2)-(3) when the (n+1)-dimensional thin domain O ε degenerates into an n-dimensional domain Q as ε → 0. In particular, we will construct a stochastic equation defined on the lower dimensional spatial domain Q which captures the essential dynamics of the original higher dimensional stochastic equations. To justify our limiting system, we will prove not only the convergence of solutions but also the upper semi-continuity of tempered pullback random attractors A ε (τ, ω) of (2)-(3) as ε → 0, namely, the attractor A ε (τ, ω) converges to the tempered pullback random attractor A 0 (τ, ω) of the limit equation under the usual semi-distance as ε → 0.
The need for studying random dynamical systems was pointed out by Ulam and von Neumann [33] in 1945. It has flourished since 1980s due to the discovery that stochastic ordinary differential equations generate random dynamical systems through the efforts of Arnold, Harris, Elworthy, Baxendale, Bismut, Ikeda, Kunita, Watanabe, and others. The study of global random attractors dates back to [30]. The concept of pullback attractor for autonomous stochastic systems was introduced in [16,18,31] as an extension of the global attractor for deterministic equations in [7,19,32]. Due to the unbounded fluctuations in the systems caused by the white noise, the concept of pullback global random attractor was introduced to capture the essential dynamics with possibly extremely wide fluctuations. This is significantly different from the deterministic case. There is an extensive literature on this subject, see for instant, [8,10,11,15,16,18,23,34,35,36]. It is worth mentioning that the ergodicity of stochastic 3D Navier-Stokes equations in a thin domain was recently investigated in [12,13], and the synchronization of semilinear parabolic stochastic equations in thin bounded tubular domains was studied in [9].
The outline of this paper is as follows. In the next section, we define a continuous cocycle in L 2 (O) for the stochastic equation defined on the fixed domain O. We also discuss the continuous cocycle in L 2 (Q) generated by the stochastic equation (4). Furthermore, we present the functional setting and the abstract formulation of the problem and give the main results of this paper. Section 3 is devoted to uniform estimates of the solutions for both system (2)-(3) and (4)- (5). The existence and uniqueness of tempered attractors for the stochastic equations are presented in section 4, and the upper semi-continuity of random attractors is finally proved in section 5.
2. Main results. In this section, we reformulate systems (2) and (4) and present our main results. Given τ ∈ R, consider the following stochastic equation driven by white noise and non-autonomous deterministic terms: with the initial conditionû where ν ε is the unit outward normal vector on ∂O ε , G : ) satisfies the boundary conditions, w is a twosided real-valued Wiener process on a probability space, F is a nonlinear function satisfying the following conditions: for all x ∈ O and t, s ∈ R, where p ≥ 2, λ 1 , λ 2 and γ are positive constants, We now reformulate problem (6)-(7) by following the process of [24]. Let λ ∈ (0, λ 1 ) be a fixed number and denote by for all x ∈ O and t, s ∈ R. Then it follows from (8)-(10) that there exist positive numbers α 1 , α 2 , β, c 1 and c 2 such that where ψ 1 (t, x) = ϕ 1 (t, x) + c 1 and ψ 2 (t, x) = ϕ 2 (t, x) + c 2 , for x ∈ O and t ∈ R. Substituting (12) into (6) we get for t > τ , with the initial conditionû To transform the ε-dependent domain O ε into the fixed domain O, we define The Jacobian matrix of T ε is The determinant of J is |J| = 1 εg(y * ) . Let J * be the transport of J. Then we have By [20,25] the gradient operator and the Laplace operator in x ∈ O ε and in y ∈ O are related by ∇ xû (x) = J * ∇ y u(y) and ∆ xû (x) = |J|div y |J| −1 JJ * ∇ y u(y) = 1 g div y (P ε u(y)), where we denote byû(x) = u(y), ∇ x and ∆ x are the gradient operator and the Laplace operator in x ∈ O ε respectively, div y and ∇ y are the divergence operator and the gradient operator in y ∈ O respectively, and P ε is the operator given by Given y = (y * , y n+1 ) ∈ O and t, s ∈ R, denote by F ε (t, y * , y n+1 , s) = F (t, y * , εg (y * ) y n+1 , s) , F 0 (t, y * , s) = F (t, y * , 0, s) , f ε (t, y * , y n+1 , s) = f (t, y * , εg (y * ) y n+1 , s) , f 0 (t, y * , s) = f (t, y * , 0, s) , G ε (t, y * , y n+1 ) = G (t, y * , εg (y * ) y n+1 ) , G 0 (t, y * ) = G (t, y * , 0) , and h ε (y * , y n+1 ) = h (y * , εg (y * ) y n+1 ) , h 0 (y * ) = h (y * , 0) . In terms of y = (y * , y n+1 ) ∈ O, system (16)- (17) can be written as, for t > τ , with the initial condition where ν is the unit outward normal vector on ∂O. Note that the boundary condition in (18) follows from the original boundary condition in (16). Indeed, by the transformation T ε we find that ν ε = J * ν. Then by ∇ xû Thus the boundary condition ∂û ε (t,x) ∂νε = 0 in (16) yields P ε u ε (t, y) · ν = 0 as in (18). We now define an inner product on L 2 (O) by and use H g (O) to denote L 2 (O) equipped with this inner product. Note that g is continuous on Q and satisfies (1). We find that H g (O) is a Hilbert space with norm equivalent to the standard norm of L 2 (O).

DINGSHI LI, KENING LU, BIXIANG WANG AND XIAOHU WANG
Let A ε be an unbounded operator on H g (O) given by In terms of A ε , we can reformulate system (18)- (19) as Similarly, we define an inner product (·, ·) Hg(Q) on L 2 (Q) by and use H g (Q) to denote L 2 (Q) equipped with this inner product. Define a 0 (·, ·): Let A 0 be an unbounded operator on H g (Q) given by In terms of A 0 , we can write system (4)-(5) as In this paper, we will use the standard probability space (Ω, F, P ) where Ω = {ω ∈ C (R, R) : ω (0) = 0}, F is the Borel σ-algebra induced by the compact-open topology of Ω, and P is the Wiener measure on (Ω, F). As usual, we use {θ t } t∈R to denote the measure-preserving transformations on (Ω, F, P ) given by For our purpose, we need to convert the stochastic equations with additive noise into pathwise deterministic equations with a random parameter. To that end, we consider the one-dimensional Ornstein-Uhlenbeck equation: One may easily check that a solution to (28) is given by From [2] we know that there exists a θ t -invariant set Ω ⊆ Ω of full P measure such that z(θ t ω) is continuous in t for every ω ∈ Ω, and the random variable |z(θ t ω)| is tempered. Let F 1 and P 1 be the restrictions of F and P on Ω, respectively. We will define a continuous cocycle for problem (18) For convenience, from now on, we will abuse the notation slightly and write the space where τ ∈ R. For the pathwise deterministic equation (29), we can show that if F satisfies (8)-(11), then for every ω ∈ Ω, τ ∈ R and v ε . Then given t ∈ R + , τ ∈ R, ω ∈ Ω andû ε τ ∈ L 2 (O ε ), we can define a continuous cocycleΦ ε for problem (16)-(17) by the formulâ Similarly, by introducing a new variable v 0 (t) = u 0 (t) − h 0 (y * )z (θ t ω), we can transform equation (26) into the following random partial differential equation on Q: As above, one can prove that system (4)-(5) generates a continuous cocycle in where B i Xi = sup x∈Bi x Xi . The collection of all families of tempered nonempty subsets of X i is denoted by D i , i.e., In the rest of this paper, we first prove the existence of D ε -pullback attractorÂ ε and D 0 -pullback attractor A 0 forΦ ε and Φ 0 , respectively, and then establish the upper semi-continuity ofÂ ε at ε = 0, i.e., for every τ ∈ R and ω ∈ Ω, To that end, we must show the cocycle Φ ε has a D 1 -pullback attractor A ε in L 2 (O) such that for every τ ∈ R and ω ∈ Ω, which is the main result of the last section.
For the deterministic forcing terms, we assume: For the existence of tempered absorbing sets, we further assume the following tempered condition on the deterministic forcing terms: for all σ > 0.
We remark that (33) and (34) do not require that G(s, ·) and ϕ i be bounded in L ∞ ( O) for each i = 1, 2, 3 when s → ±∞. As mentioned before, ψ 1 = ϕ 1 + c 1 and ψ 2 = ϕ 2 + c 2 for some positive constants c 1 and c 2 . Based on this fact, we find that (33) and (34) are equivalent to the following conditions, respectively: (35) and for any σ > 0 We are now in a position to present our main results of this paper. We start with the existence and uniqueness of random attractors for the continuous cocycle Φ ε and Φ 0 .
Remark 1. Since L 2 (Q) can be embedded naturally into L 2 (O) as the subspace of functions independent of y n+1 , we can consider the cocycle Φ 0 as a mapping from L 2 (Q) into L 2 (O) .
Proof. Taking the inner product of (29) with v ε in H g (O), we find that For the nonlinear term, by (13)- (14) we obtain For the third term on the right-hand side of (43), we have On the other hand, from (22), the last term on the right-hand side of (43) is bounded by Then it follows from (43)-(46) that Multiplying (47) by e λt and then integrating the resulting inequality on (τ − t, τ ) with t ≥ 0 and replacing ω by θ −τ ω, we get that for every ω ∈ Ω, This shows that there exists T = T (τ, ω, D 1 ) > 0 such that for all t ≥ T , Then the lemma follows immediately from (48).
As a consequence of Lemma 3.1, we have the following inequality which is useful for deriving the uniform estimates of solutions in H 1 ε (O).
Proof. Taking the inner product of (29) with A ε v ε in H g (O), we find that We first estimate the nonlinear term in (50) for which, by (14) and Lemma 3.3, we have On the other hand, the last two term on the right-hand side of (50) is bounded by By (50)-(52), we can get that which implied that Given t ≥ 1, τ ∈ R, ω ∈ Ω and s ∈ (τ − 1, τ ), integrating (54) on (s, τ ) we get Now integrating the above with respect to s over (τ −1, τ ) and replacing ω by θ −τ ω, we obtain that Let T = T (τ, ω, D 1 ) ≥ 1 be the positive number found in Lemma 3.2. Then it follows from the above inequality and Lemma 3.2 that, for all t ≥ T and ω ∈ Ω, For the first term on the right-hand side of (55), we have The second term on the right-hand side of (55) satisfies Thus, Lemma 3.4 follows from (55)-(57) and Lemma 3.1.
We are now in a position to establish the uniform estimates for the solution u ε of the stochastic equation (25) by using those estimates for the solution v ε of (29). Notice that for each τ ∈ R, t ≥ 0, and ω ∈ Ω, where v ε τ −t = u ε τ −t − h ε z (θ −t ω). Suppose D = {D (τ, ω) : τ ∈ R, ω ∈ Ω} is a family of nonempty subsets of L 2 (O). Based on D, given c > 0, define a family D c by If D is tempered, then we can verify that D c given by (59) is also tempered. In for some c > 0. This fact allows us to get uniform estimates on u ε immediately from (59) and those estimates on v ε as established by Lemma 3.1 and Lemma 3.4.

4.
Existence and uniqueness of pullback attractors. In this subsection, we establish the existence of D 1 -pullback attractor for Φ ε . To that end, we must show that (25) has a tempered pullback absorbing set, which is given as follows.

DINGSHI LI, KENING LU, BIXIANG WANG AND XIAOHU WANG
We are now ready to prove Theorem 2.1. Proof of Theorem 2.1. First, we know from Lemma 4.1 that Φ ε has a a closed measurable D 1 -pullback absorbing set K (τ, ω). Thanks to the compact embedding . Hence, the existence of a unique D 1 -pullback attractor for the cocycle Φ ε follows from [34] immediately. If F , G, ϕ 1 , ϕ 2 and ϕ 3 are T -periodic with respect to t, the continuous cocycle Φ ε and the absorbing set K are also T -periodic, which implies the T -periodicity of the attractor. Analogous results hold also for Φ 0 . We omit the details here.

5.
Upper semi-continuity of random attractors. In this section, we prove the upper semi-continuity of random attractors of Φ ε as ε → 0. The following estimates are needed for this purpose.
Similarly, one can prove Lemma 5.2. Assume that (8)-(11) hold. Then for every τ ∈ R, ω ∈ Ω and v 0 τ ∈ H g (Q), the solution v 0 of (31) satisfies, for all T > 0, where M is a positive constant depending on λ and T , but independent of τ and ω .