Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing

In this paper, we study the squeezing property and finite dimensionality of cocycle attractors for non-autonomous dynamical systems (NRDS). We show that the generalized random cocycle squeezing property (RCSP) is a sufficient condition to prove a determining modes result and the finite dimensionality of invariant non-autonomous random sets, where the upper bound of the dimension is uniform for all components of the invariant set. We also prove that the RCSP can imply the pullback flattening property in uniformly convex Banach space so that could also contribute to establish the asymptotic compactness of the system. The cocycle attractor for 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing is studied as an application.


1.
Introduction. Non-autonomous random dynamical systems (NRDS for short) are introduced to study evolution equations driven by time-dependent forcings and perturbed by stochastic noises, see for instance [27,50,51,28,29]. Because of the non-autonomous and stochastic features, the dynamical behavior of NRDS is relatively more complicated and deserves more efforts to go into than determininstic and autonomous random dynamical systems.
Global attractor is a useful objective to learn the dynamical behavior of a dynamical system. For NRDS there are considerably many publications on various random attractors, see for instance [50,52,31] for pullback attractors, [28,29] for cocycle attractors and most recently [29] for uniform attractors. In addition to the attracting property, the compactness and then the finite-dimensionality are interesting and significant properties, making attractors more important for studying the dynamical system, see for instance Robinson [45] and references therein. Igor Chueshov gave us nice and important results, both in the deterministic and the random case, in all the main items where the theory of infinite-dimensional dynamical systems has been developed in the last four decades: existence of attractors and its characterization (see [15,2,24,11,21]), dependence of the asymptotic behaviour on a finite number of degrees of freedom, including determining modes and squeezing property ( [14,17,23]), finite fractal dimensionality of attractors ( [16,20]), invariant manifolds ( [12,3]) or inertial manifolds ( [37,22]). Applications to important models of PDEs also focused an important part of his work (see, for instance, [13,19,18]). Science in these areas of research is really grateful on all his efforts and such a great work, which has been and it is so helpful to a big number of collaborators and researches in the field.
In this paper, we generalize the idea of squeezing property to a random cocycle squeezing property (RCSP for short) for NRDS, by which we further study the finite dimensionality of invariant sets, including cocycle attractors of NRDS. The squeezing property was firstly introduced by Foias & Temam [36] in the context of the Navier-Stokes equations, and then applied to many other classes of dissipative partial differential equation [33,47,44]. For autonomous RDS, Debussche [32] generalizes the method of [36], and the ideas of [32] motivated Flandoli and Langa [35] to define the "random squeezing property". As will be shown latter, the generalized squeezing property could imply the determining modes and finite dimensionality for cocycle attractors of NRDS, and, as the squeezing implies asymptotic compactness of the system, it could also contribute to obtain the compactness and existence of the cocycle attractor.
What is worth a note is that the finite dimensionality of cocycle attractors obtained in this paper does hold uniformly for all components of the attractor, that is, the upper bound of the dimension is uniform for all components. Applying to a 2D Navier-Stokes equation we find that the uniformity of the bound is supported by the translation-boundedness of the external force of the system ( [10,8,9]), which is known often a sufficient condition for the existence of a random uniform attractor [29].
We carry out this paper as follows. In Section 2, we introduce preliminary definitions to NRDS. In Section 3, we extend the idea of squeezing property to NRDS, and prove that it is a sufficient condition to prove a determining modes results and finite dimensionality of random non-autonomous random invariant set. In Section 4, we apply our theoretical results to a non-autonomous stochastic 2D Navier-Stokes equation.
2. Non-autonomous random dynamical systems. In this section, we first recall some general concepts related to NRDS and their cocycle attractors.
2.1. Preliminaries. In this part, we recall from [1,28,50] some basic definitions which will be used throughout this paper. Let (X, · ) be a separable Banach space, and denote the Hausdorff semi-metric between sets in X by For any set B ⊂ X, we denote by B := sup x∈B x .
Let Σ be a topological space. Most generally, we do not require compactness or boundedness (under some metric) on Σ unless otherwise stated, but we always assume a group {θ t } t∈R of operators acting on Σ satisfying • θ 0 = identity operator on Σ; • σ → θ t σ is continuous for each t fixed. In applications, space Σ refers to the symbol space of an evolution equation. A popular example is the space Σ = {g(· + s) : s ∈ R}, or its closure under some topology, for some time-dependent function g taking values in some metric space, and θ t is the translation operator θ t g = g(· + t).
Throughout this paper, for any metric space M we denote by B(M ) the Borel sigma-algebra of M .
To define non-autonomous random dynamical systems studied in this paper, we let (Ω, F, P) be a probability space, which need not be P-complete, endowed also with a flow {ϑ t } t∈R satisfying • ϑ 0 = identity operator on Ω; The two groups {θ t } t∈R and {ϑ t } t∈R acting respectively on Σ and Ω are called base flows or parametric dynamical systems. For the ease of notations, we often use θ (or ϑ), instead of θ t (or ϑ t ), when describing universal properties hold for each t ∈ R. Definition 2.1. A non-autonomous random dynamical system (NRDS for short) on X with base flows {θ t } t∈R and {ϑ t } t∈R is defined as a mapping φ(t, ω, σ, x) : (2) φ(0, ω, σ, ·) is the identity on X for each ω and σ fixed; (3) it holds the cocycle property Moreover, an NRDS φ is said to be continuous if the mapping φ(t, ω, σ, ·) is continuous for each t, ω, σ fixed.
Next we define the so-called non-autonomous random set, which plays a central role in the study of NRDS.
is called a non-autonomous random set in X if it is measurable in the sense that the mapping ω → dist(x, D(ω, σ)) is (F, B(R))-measurable for each x ∈ X and σ ∈ Σ. If each its image D σ (ω) is closed (or bounded, compact, etc.) in X, then D is called a closed (or bounded, compact, etc.) non-autonomous random set in X.
Given two non-autonomous random sets D 1 , σ (ω) for all σ ∈ Σ, ω ∈ Ω, and we then say D 1 is smaller than D 2 . Definition 2.3. A non-autonomous random set D in X is called tempered if, for any > 0, lim In the following, denote by D the class of the tempered non-autonomous random set in X. Then the universe D is inclusion-closed, i.e., if D ∈ D then each random set smaller than D belongs to D.
Before the definition of D-random cocycle attractors let us recall invariant, pullback attracting and pullback absorbing sets.
In fact, even for deterministic non-autonomous dynamical systems, pullback attraction and forward attraction have no general relationship [41,39,4]. But we have the following.

SQUEEZING AND FINITE DIMENSIONALITY OF COCYCLE ATTRACTORS 1301
Notice that ϑ t is P-preserving and Ω is invariant under ϑ t for any t > 0, so we have The proof is complete.
For each non-empty non-autonomous random set D and σ ∈ Σ, we define the random omega-limit set W(·, σ, D) of D driven by σ by Omega-limit sets are important in attractor theory. It is straightforward to have the following lemma.
For D-random cocycle attractors, Wang [50,51] studied the existence and characterization by complete trajectories. The following existence result is well known.
Theorem 2.10. [50,51] Suppose φ is a continuous NRDS with a compact Dpullback attracting set K and a closed D-pullback absorbing set B ∈ D. Then φ has a unique D-random cocycle attractor A ∈ D given by A σ (ω) = W(ω, σ, B).
Observe that in the above theorem the set K does not necessarily belong to the class D.

2.2.
Several alternative dynamical compactnesses. Theorem 2.10 shows a direct relationship between attractors and compact attracting sets. However, the existence of a compact attracting set is often the open problem. Hence, there are other dynamical compactnesses in the literature, such as asymptotic compactness, pullback omega-limit compactness, flattening and squeezing properties [42,38,30,25], which ensures the omega-limit set of a D-absorbing set is a compact D-attracting set. Now we introduce analogous concepts in the context of NRDS, and show that these dynamical compactness of an NRDS will ensure the omega-limit set of a Dabsorbing set to be a compact D-attracting set.
The above theorem implies that these dynamical compactnesses could replace the requirement of a compact D-attracting set in Theorem 2.10, since they are stronger under suitable conditions. Let us study the squeezing property in the next section.
3. The random cocycle squeezing property. In this section, we introduce the definition of squeezing property for NRDS. We prove that it is a sufficient condition to prove a determining modes results and finite dimensionality of non-autonomous random invariant set, also it can imply the pullback flattening in uniformly convex Banach space. Definition 3.1 (Random cocycle squeezing property). Suppose φ is an NRDS on X with a non-autonomous random invariant set B ∈ D. We say that φ satisfies the random cocycle squeezing property (RCSP for short) on B if there exists δ ∈ (0, 1/2), a m-dimensional orthogonal projector P : X → P X (dim P X = m) and a random variable C : Ω → R with finite expectation E(C(ω)) < ln(1/2δ), such that for each ω ∈ Ω and σ ∈ Σ and for all x, y ∈ B σ (ω), where Q = I − P .

Remark 3.2.
It is important to note that, the RCSP gives an alternative on the one-step (in time) behaviour of the solutions associated to the NRDS: either the low modes control the behavior of the high modes, that is, or the one-time evolution produces a flattening effect on the trajectories, that is, then, by (3.2) and triangle inequality, we obtain that The flattening on the one-time evolution of the trajectories is due to the condition on the expectation of C(ω), which is very important in order to prove the property of determining modes.
Suppose that φ satisfies the RCSP on A. Then, for each ω ∈ Ω and σ ∈ Σ the following result on determining modes is true: Let k ∈ R + be a number satisfying and let x, y ∈ A σ (ω) be two points for which Proof. Because of the Lipschitz condition of the NRDS, it is enough to prove that the result is true in the discrete case, that is since for any t = m + s, s ∈ [0, 1) by the cocycle property we have We can now write for 0 > 0 small enough and for e − 0 m L(ϑ m ω) we have that from (3.4) and for all m large enough and > 0 small enough such that − 0 < 0 and this implies Going back to the discrete case, we argue by contradiction (see [35]). Suppose that there exist > 0 and a sequence m j → ∞ such that Denote Q = I − P , then there exist j 0 ∈ N such that for all j j 0 we have which contradicts (3.7). Thus, for each j fixed, from (3.8) and RCSP it follows where and in the sequelC Now consider φ(m j −1, ω, σ, x) and φ(m j −1, ω, σ, y). Again, by the RCSP, either they satisfy Continue this way until we reach M j with either M j = m j or for which (M j < m j ) Then applying the RCSP M j times, we have (3.9) Notice that we now have two possibilities for the sequence {M j } j∈N .
Then, multiplying both sides in (3.9) by ek mj and taking into account that 2δ < 1 we get On the other hand, by the ergodicity of the shift ϑ t , we have that Thus, from (3.10) we have (3.12) But as, for small enoughk < k − ( + E(C(ω))), that is, k −k > ( + E(C(ω))) and by (3.5) we have exponential convergence (with exponent k) of the first m modes of the solutions, it is clear that the last expression tends to zero when j goes to infinity, which contradicts (3.7).
Case 2. Suppose that there exists a subsequence of M j , we denote again by M j , such that M j → ∞. In this situation we have again two possibilities: 13) where K is a bound for the last term in (3.13). Then, multiplying both sides in (3.13) by ek mj , we have (3.14) Taking into account that for > 0 small enough, we obtain the convergence to zero of the right expression in (3.14).
and note that this last expression tends to zero by the conditions we have for the constantsk and k.
The following theorem indicates that, under certain conditions, the RCSP is a sufficient condition for pullback omega-limit compactness, and hence for uniformly convex Banach spaces pullback flattening could be a weaker concept than RCSP.
Theorem 3.6. Suppose that a continuous NRDS φ on X has a positively invariant D-pullback absorbing set B ∈ D. Suppose further that there exists a tempered random variable R(ω) such that sup σ∈Σ B σ (ω) R(ω). If φ satisfies the RCSP, then is D-pullback omega-limit compact and hence has a D-random cocycle attractor. In addition, if X is a uniformly convex Banach space, then φ is D-pullback flattening.
Proof. Following the proof of Theorem 3.5 before (3.20) we have that, for each n ∈ N, the set can be covered by k n 0 balls of radius (2δ) n e 0 −n C(ϑsω) ds R(ϑ −n ω). so that each φ(n, ϑ −n ω, θ −n σ)B θ−nσ (ϑ −n ω) with n n 0 can be covered by a finite number of balls of radius less than . But for t = n 0 + τ , τ 0, by the positive invariance of B σ (ω) and the cocycle property for all t n 0 . Thus so the NRDS is thus D-pullback omega-limit compact. Finally, if X is a uniformly convex Banach space, then Theorem 2.13 implies that the NRDS is D-pullback flattening.
4. Applications to 2D Navier-Stokes equations. In this section we study a 2D Navier-Stokes equation as an example to apply our theoretical analysis. First, let us introduce translation compact/bounded functions which are known important in the study of uniform attractors [10,8,9,29].    From now on, we fix g(t, x) ∈ L 2 loc (R; (L 2 (O)) 2 ) to be translation bounded. From the above proposition it is clear that the family {θ t } t∈R is a well-defined base flow on Σ compact. Moreover, similar to [29, Section 6.1] one can see that Σ is in fact a compact Polish metric space, and the mapping t → θ t σ is (R, Σ)-continuous.
We consider the two-dimensional stochastic Navier-Stokes equation on O with translation bounded external forcing and scalar additive noise. This equation reads endowed with initial-boundary value condition where ν > 0 is a constant, σ ∈ Σ and Σ is the previously defined symbol space defined as the hull of a translation bounded function g. The term W (t) is a scalar Brownian motion on a probability space (Ω, F, P) specified later.
Define the operator of Stokes A : D(A) ⊂ H → H as Au = −P∆u, where P is the orthogonal projection in (L 2 (O)) 2 over H and D(A) = (H 2 (O)) 2 ∩ V . The operator A is a self-adjoint positive operator in H with compact inverse (see Temam [47]), it is known by spectral theory that there exists a sequence {λ j } ∞ j=1 satisfying 0 < λ 1 λ 2 · · · → ∞ and a sequence of vectors {e j } ∞ j=1 ⊂ D(A), which is orthonormal in H and such that Ae j = λ j e j , j = 1, 2 . . . .

Moreover, define the bilinear operator B as
By the incompressibility condition we have
By (4.14) and Lemmas 4.4 and 4.6 we have the following estimate for solutions (4.14).
for all t T , where R 1 (ω) is a tempered random variable given by (4.26), R 3 (ω) is the random variable given by (4.43) and c is a positive constant.
It is clear that from definition of R 4 (ω) and Lemma 4.6, for any p 1 satisfying (4.33) (4.48) The proof is complete.

4.5.
Random cocycle squeezing property. In this part, we show that the NRDS generated by the Navier-Stokes equation satisfies the RCSP on the D-random cocycle attractor A. Then the NRDS φ generated by the stochastic Navier-Stokes equation (4.14) with translation bounded forcing satisfies the RCSP on the D-random cocycle attractor A.
Proof. Let u and v two solutions of (4.5), then Denote by P the orthogonal projector onto the subspace of H spanned by the first m eigenfunctions associated with the Stokes operator A and Q = I − P . Taking the inner product of (4.50) with Q(u − v), we obtain and thus d dt Since By Gronwall lemma, we have (4.51) Now, we estimate the norm u − v 2 by |rot(u − v)| 2 and the latter by |u 0 − v 0 | 2 . Taking the inner product of (4.50) with u − v in H, we get Using the following inequality (see Temam [47]) and Young's inequality we have By Gronwall's lemma, we obtain Thus integrating (4.53) with respect to t in the interval (t − 1, t), with t 1 and using (4.54) , we have that Let us put ξ u = rotu and ξ v = rotv, we have Taking the inner product of (4.56) with ξ u − ξ v in H and using Agmon inequality, see Temam [47] we have that and then d dt By Gronwall's lemma we get Integrating the above relation with respect to s over (t − 1, t) with t 1 we obtain For all t 1, by (4.55) and (4.57), it follows that Using this inequality in (4.51) we have that (4.59) By (4.54), for each ω ∈ Ω, σ ∈ Σ and any u 0 , v 0 ∈ A σ (ω) we have |P (φ(t, ω, σ, u 0 ) − φ(t, ω, σ, v 0 ))| 2 e t 0 c v(s) 2 ds |u 0 − v 0 | 2 .
The proof is complete.
4.6. Determining modes and finite dimensionality of the random cocycle attractor. In this part, we show a determining modes result and finite dimensionality of the D-random cocycle attractor for the Navier-Stokes equation. The finite dimensionality of the attractor (kernel sections) for non-autonomous Navier-Stokes equations and other models goes back to the works of Chepyzhov and Vishik (see, for instance, [8,9,5,48,7]). Note that our method does not give the better upper bounds on the dimension of attractors for the non-autonomous Navier-Stokes, as the Lyapunov method (see, for instance, [6,10,40]).
Proof. It is clear that, from definition of B(ω), cf. (4.44), Lemma 4.4 and minimal property of A that there exists a tempered random variable R(ω) such that sup σ∈Σ |A σ (ω)| R(ω). Then the result follows from Theorem 3.5.