Large-domain stability of random attractors for stochastic g-Navier–Stokes equations with additive noise

This paper concerns the long term behavior of the stochastic two-dimensional g-Navier–Stokes equations with additive noise defined on a sequence of expanding domains, where the ultimate domain is unbounded and of Poincaré type. We prove that the weak continuity is uniform with respect to all expanding cocycles, which yields the equi-asymptotic compactness by using an energy equation method. Finally, we show the existence of a random attractor for the equation on each domain and the upper semi-continuity of random attractors as the bounded domain is expanded to the unbounded ultimate domain.


Introduction
The fluid dynamics of deterministic or stochastic Navier-Stokes (NS) equations has been extensively studied. For example, many properties such as existence, upper semicontinuity, regularity, and fractal dimension of an attractor were studied in the literature [4,5,8,13,19]. However, we find that most of the above-mentioned studies are given in a two-dimensional situation rather than three-dimensional one, which encourages us to do more in-depth research about the dynamic behavior of Navier-Stokes equations.
The g-NS equations in spatial dimension 2 were introduced by Roh [18] as follows: where g is a suitable smooth real-valued function. The uniqueness and existence of solutions for the g-NS equations in R n (n = 2, 3) were proved by Bae and Roh [3]. When g = 1, Eq. (1) becomes the usual 2D Navier-Stokes equations. In the last decade, the limiting behavior of solutions in terms of the existence of attractors for 2D g-NS equations has been studied in both autonomous and non-autonomous cases without the stochastic situations, see [10,11,18]. As described in [18], the 2D g-NS equations arise in a natural way when we study the standard 3D Navier-Stokes problem in a 3D thin domain O g = O × (0, εg), (O ⊂ R 2 ) which was introduced by [9,17], and we do not claim that the g-NS equations form a model of any fluid flow. They may, or may not. That they are derived from a standard 3D problem is the basis for our study. However, as we know, there are no results related to the long-time behavior of solutions for the 2D stochastic g-NS equations.
In this paper, we consider both the existence and large-domain stability of a random attractor for the stochastic 2D g-NS equations on an unbounded Poincaré-type domain O ∞ ⊂ R 2 , which is regarded as a limit of the sequence of expanding domains O k = {x ∈ O ∞ : |x| < k}.
We write the sequence of stochastic 2D g-NS equations on O k (k ∈ N := N ∪ {∞}) as a unified form: where ν, ε > 0, p is the pressure, u k is the velocity vector, W is a scalar Wiener process defined on a probability space (Ω, F, P). h(x) is a given time-independent two-dimensional vector function belonging to some Sobolev spaces which will be specified later.
The first subject is to show the existence of a random attractor A k in H g (O k ) (a special subspace of L 2 (O k )) for each k ∈ N. Due to both non-autonomy and randomness of model (2), the attractor is actually a bi-parametric set A k = {A k (τ , ω) : τ ∈ R, ω ∈ Ω} in H g (O k ) (see [21]). Even for this existence of a pullback attractor, the assumption of small noise (ε ≤ ε 0 ) seems to be necessary.
To study problem (2), the real-valued function g = g(x) ∈ W 1,∞ (O ∞ ) satisfies the following basic assumption: Using the famous energy equation method [19], we establish the existence of random attractors. More precisely, for each k ∈ N, the stochastic g-NS equations (2) have a random attractor A k in H g (O k ).
The second subject is to investigate large-domain stability of the attractor, which means that A k is stable (upper semi-continuous) at A ∞ under a suitable Hausdorff semi-distance.
Such an expanding-domain problem is contrary to the thin-domain problem, the latter was extensively investigated in the literature (see [14,15]) and time-varying domains problem [20]. However, the same difficulty arises from the fact that both A k and A ∞ lie in different phase spaces, compared with the same phase space in time-dependent stability of a pullback attractor [6,7,12].
In order to define a distance between two subsets lying in different spaces, we consider the null-expansion u k of the solution u k ∈ H g (O k ) defined by One can show easily that u k ∈ H g (O ∞ ) if u k ∈ H g (O k ). However, in general, u k = u ∞ . So, the attractor A k can be expanded to a bi-parametric set In this way, the Hausdorff semi-distance between A k and A ∞ lies in the same space H g (O ∞ ) where the Hausdorff semi-distance can be understood in the following sense: Then our aim is to prove However, the usual energy equation method is not sufficient to prove the large-domain stability from A k to A ∞ as k → ∞. We will expand each cocycle Φ k (on H g (O k )) to the null-expansion cocycle Φ k on a subspace of H g (O ∞ ) and prove that the sequence of ex- For this end, we develop the usual energy equation method from a single system to a sequence of systems, and prove weak equi-continuity of expanding cocycles { Φ k } k , which together with an energy equality can help us to establish the equi-asymptotic compactness of expanding cocycles { Φ k } k .
Finally, by proving the convergence from Φ k to Φ ∞ in H g (O ∞ ), we establish the largedomain stability (5) as desired.
This paper is organized as follows. In the next section, the functional spaces and a continuous cocycle for the stochastic g-Navier-Stokes equations are defined. In Sect. 3, we define the expanding cocycles and prove the convergence of the expanding cocycles for stochastic g-Navier-Stokes equations. We derive the uniform estimates and weak equicontinuity of the solution sequence {v k } in Sect. 4, which yields the equi-asymptotic compactness of the sequence { Φ k } k of expanding cocycles in Sect. 5. In the last section, we show the existence and large-domain stability of the attractor when the domain changes from bounded to unbounded.

Functional spaces and operators
As pointed out in Sect. 1, the unbounded domain O ∞ is of Poincaré type, and thus there exists λ ∞ > 0 such that Let O k = {x ∈ O ∞ : |x| ≤ k}, and for each k ∈ N, we use (L 2 g (O k ), · g ) to denote the space L 2 (O k ) with the following norm: By (3), one can show that m 0 ζ 2 ≤ ζ 2 g ≤ M 0 ζ 2 . So, both norms · g and · are indeed equivalent.
Also, we use (H 1 0,g (O k ), · H 1 0,g ) to denote the space H 1 0 (O k ) with the following norm: Then, by [16], there exists λ 0 > 0 (independent of k) such that which implies that the new norm is (uniformly) equivalent to the original H 1 0 (O k )-norm. To reformulate system (2), we introduce some function space: where cl X denotes the closure taken in X and are Hilbert spaces with the inner products (·, ·) g and ((·, ·)) g of and for all u = (u 1 , ). Now, we can define the g-Laplace operator as follows: Then, the first equation of (2) can be rewritten as Consider the g-orthogonal projection P g,k : L 2 g (O k ) → H g (O k ) and define the g-Stokes operator [18] by ) is a homomorphism with A g,k ≤ 1/m 0 (see [16]) and the bound is independent of k ∈ N.
Furthermore, we consider the uniform bound of operator R g,k : In this case, by [16], we have the following result.
Lemma 2.1 For each k ∈ N and u ∈ V g (O k ), we have the following uniform bounds: where λ 0 is given by (7) and · ∞ is the norm in L ∞ (O ∞ ).
In the sequel, we will define the bilinear operator B g,k : and we write B g,k (u) = B g,k (u, u) without confusion. By Roh [18], we have Therefore, we can rewrite (8) in the sense of abstract equation Conversely, for a function v : We need to estimate the norms of both expansion and restriction in H g , V g , and V * g .

Cocycles for stochastic g-NS equations
The standard probability space (Ω, F, P) will be used in this paper where F is the Borel algebra induced by the compact-open topology of Ω, and P is the Wiener is continuous in t for every ω ∈Ω, and we have the following limits: where E, Γ denote expectation and gamma function, respectively.
In this case, system (13) can be rewritten as follows: Then we can define a family of measurable mappings Φ k : where t ≥ 0. Then, for each k ∈ N, Φ k is a continuous cocycle [21] and we have for all t, s ≥ 0, τ ∈ R, and ω ∈ Ω. We now take a universe D on H g (O ∞ ), which consists of all set-valued mappings D : where D denotes the supremum of norms of all elements and λ = 1 In order to obtain the D| O k -pullback attractor A k for all k ∈ N, we make further assumptions.
Assumption G We further assume g ∈ W 1,∞ (O k ) and where λ 0 is given by (7).
By Lemma 2.3(4) and Assumption F, the restriction f | O k is still tempered: Assumption S (Small noise) The density of noise ε ∈ (0, ε 0 ] is small enough, where

Expanding cocycles
In this section, we need to expand the cocycle However, in general, Fortunately, we can show that the null-expansion Φ k is a cocycle on the closed linear subspace H k (O ∞ ) defined by A D-pullback random attractor means a bi-parametric set which is measurable, compact, invariant, and D-pullback attracting. For the concept and existence theorem, the reader can refer to [21].
Other properties of cocycle are easily verified.
Next, we prove that A k is a D k -pullback random attractor for Φ k in four steps. Step This equality implies that Since Step 3. We show the invariance.
Let w be the null-expansion of w, then w |O k = w. It follows from (27) that we have which proves the negative invariance of A k . Similarly, one can prove the positive invariance.
Step 4. We show that A k is D k -pullback attracting. Let D ∈ D k , then there is D k ∈ D k such that D = D k . By the same method as given in (28), we know that Then the attraction of A k follows from the attraction of A k .

Convergence of expanding cocycles
In this subsection, we prove the convergence of the expanding cocycles as follows.
where v ∞ = v ∞ and f (t)| O k is regarded as the null-expansion of the restriction of f (t).
. Subtracting Eq. (18) for k = ∞ from (30) and multiplying the result by gV k , we have By (10) and the trilinear property of b g,∞ , Notice that εz(θ t ω) is bounded in t ∈ [τ , τ + T], ε ≤ ε 0 and the sequence { h k : k ∈ N} is bounded in V g (O ∞ ). We infer from (12) and (32) that By Assumption G, Since A g,∞ and R g,∞ are bounded linear operators from V g (O ∞ ) to V * g , we have For the forcing term, since By the Holder inequality and Poincaré inequality, It follows from (33) to (37) that By Gronwall's lemma we get, for all t ∈ [τ , τ + T], By Lemma 2.5, v ∞ ∈ L 2 (τ , τ + T; V g (O ∞ )) and thus Since f ∈ L 2 (τ , τ + T; V * g (O ∞ )), it follows from the Holder inequality that, as k → ∞, in view of (40) and the Lebesgue controlled convergence theorem. By Assumption H and the absolute continuity of the integrals, by the convergence of the initial value, we have By the assumption that V k (τ ) 2 g → 0 as k → ∞, we infer from (39)-(42) that V k (t) 2 g → 0 as k → ∞, uniformly in t ∈ [τ , τ + T].

Lemma 4.1 There is a random variable C
Proof By (17) and the ergodic theorem, We choose large t 0 (ω) ≥ 0 and use Assumption S to obtain, for all ε ≤ ε 0 , while, for all t ≤ t 0 , Let Q be the set of rational numbers, then [-t 0 , 0]∩Q = {r 1 , r 2 , . . .} is a countable set. Hence, which is measurable in ω and so is C 0 (ω). Therefore, (43) holds true.

Weak equi-continuity
In this subsection, we show the weak equi-continuity of the solution sequence {v k }. This is different from the weak continuity for a single system as given by Rosa [19].
Proof From Lemmas 4.2 and 2.3, we can prove that, for all T > 0, We rewrite (18) as follows: where h k = P g,k h| O k . Since the norms of the operators A g,k , B g,k are bounded in k and R g,k satisfies (9), respectively, it follows from Lemma 2.3 and Assumption H that For each i ∈ N, according to Aubin's compactness theorem [2] and the compactness of From (58) and (59), by a diagonal process, we can extract an index subsequence k * of k such that We now show that v is a weak solution of Eq. (18) at k = ∞. Indeed, by the weak formulation (20), the expansion v k * satisfies that, for each and we only need to consider the convergence of the forcing term f as k * → ∞. Since → 0 as k * → ∞.
By taking the limit of (61) as k * → ∞ and noticing that A g,∞ , B g,∞ , R g,∞ are continuous operators, we see that v is a weak solution of Eq. (18) with k = ∞. By the uniqueness of the solutions, we have v(t) = v ∞ (t, τ , ω, v τ ,∞ ). Using a standard contradiction argument, we can show that the whole sequence { v k } k converges to v ∞ in the sense of (60). This proves (57).
In addition, from the strong convergence in (60), we also have On the other hand, for all where C T is positive and independent of k. Thus, it follows from (58) and (63), we see that {( v k (t), χ)} is equi-bounded and equi-continuous on [τ , τ + T] for all T > 0. This together According to the density of V(O ∞ ) in H g (O ∞ ), we show (56).
By the similar (more simple) method as given in Lemma 4.3, we have weak continuity of each v k .

Equi-asymptotic compactness
In this section, we establish an energy equation (as (75) below) for the expanding solution v k and then we use this energy equation to verify the equi-asymptotic compactness of the sequence { Φ k } k of expanding cocycles.
for problem (18) has a convergent subsequence in H g (O ∞ ).
Proof By taking σ = τ in (45), there is k 0 ∈ N such that and thus, by Lemma 2.3(1), the sequence of expanding solutions Passing to subsequence, there is By the resonance theorem, Hence, in order to prove the weak convergence in (67) is strongly convergent, it suffices to prove that, for a subsequence, By the cocycle property, we have, for each m, k ∈ N, For each m ∈ N, by taking σ = τm in (45), there is K m ∈ N such that We still use {k} to denote the index diagonal subsequence {k(k)}. Then, as k → ∞, By (70), (71), and (56), we have From (67), (72), and the uniqueness of weak limit, we have Now, we infer from (46) an energy equation O k for all k ∈ N: where g . Since the null-expansion does not change the norm, we have Hence, we can rewrite (74) on O ∞ as follows: Let s = τm and v s, Hence, we have the following estimate of the limit: For the second term, we claim that g defines a norm which is equivalent to the norm in V g (O k ). Indeed, by Lemma 2.1 and Assumption G, we see that On the other hand, by the uniform Poincaré inequality and λ = 1 3 λ 0 ν, Thus, by the Fatou lemma and weak equi-continuity (57), we obtain Similarly, from Lemma 4.

Final conclusion
In the last section, we deduce the existence and large-domain stability of the attractor when the domain changes from bounded to unbounded. Theorem 6.1 For each k ∈ N, let Φ k be the cocycle associated with the g-NS equation (18) on O k , and let D k := D| O k be the restriction of the universe D in (22). Then Φ k has a D kpullback random attractor A k in H g (O k ).
Proof By taking σ = τ in (45), we find that Φ k has an absorbing set M k given by We need to prove that ρ 1 and ρ 2 are tempered with the growth rate 3 2 λ. Indeed, by Lemma 4.1 and Assumption F, as t → +∞. Similarly, the tempered property of |z(θ s ω)| 2 + |z(θ s ω)| 4 implies that ρ 2 is tempered. Therefore, M k ∈ D k . On the other hand, by Corollary 2, Φ k is D k -pullback asymptotically compact. Therefore, it follows from the abstract result [21] that Φ k has a unique D k -pullback random attractor denoted by A k = {A k (τ , ω)}.
In addition, by Theorem 3.1, the expanded cocycle Φ k has a D k -pullback random attractor in H k (O ∞ ). This expanded attractor is just the null-expansion A k of A k .
Finally, we establish the large-domain stability (upper-semicontinuity) of random attractors as k → ∞.

Theorem 6.2
The sequence {A k } k of random attractors associated with problem (18) satisfies as k → ∞ for all τ ∈ R and ω ∈ Ω, where A k is the null-expansion of A k .
Proof The proof is similar to that of [16, Theorem VI.2] and so is omitted here.