Random Dynamics and Limiting Behaviors for 3D Globally Modified Navier-Stokes Equations Driven by Colored Noise

Random Dynamics and Limiting Behaviors for 3D Globally Modified Navier-Stokes Equations Driven by Colored Noise Tom´as Caraballoa,b∗ , Zhang Chenc† , Dandan Yangc‡ aDepartamento de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain bDepartment of Mathematics, Wenzhou University, Wenzhou 325035, China cSchool of Mathematics, Shandong University, Jinan 250100, China Abstract. This paper is mainly concerned with the long-term random dynamics for the non-autonomous 3D globally modified Navier-Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero. Keywords. Globally modified Navier-Stokes equations, random attractor, invariant measure, random Liouville type theorem, limit measure. AMS 2020 Mathematics Subject Classification. 37L40, 37L55, 35B41, 76F20. Contents 1 Introduction 2 ∗E-mail address: [email protected] †E-mail address: zc[email protected] ‡E-mail address: dany[email protected] 1 2 Existence of random attractors 6 2.1 Cocycle for GMNSE with colored noise .......................... 9 2.2 Construction of pullback random absorbing sets ..................... 9 2.3 Existence of random attractors .............................. 12 3 Existence of invariant measures 14 4 Random Liouville type theorem 19 5 Limiting behaviors of invariant measures 25 5.1 Limiting behaviors of invariant measures as ε→0................... 25 5.2 Limiting behaviors of invariant measures as δ→0................... 28 5.2.1 Existence of invariant measures for GMNSE with additive white noise . . . . 28 5.2.2 Convergence of invariant measures from colored noise to white noise ..... 33 1 Introduction In this paper, we investigate the pullback random attractors and invariant measures of the following non-autonomous 3D globally modified Navier-Stokes equations (GMNSE) with nonlinear colored noise:                  ∂u ∂t −ν∆u+FN(kuk) [(u(t)·∇)u] + ∇p=f(t) + εG(u)Gδ(θtω) in O×(τ, ∞), div u= 0 in O×(τ, ∞), u= 0 on Γ ×(τ, ∞), u(x, τ) = uτ(x), x ∈ O, (1.1) where ε∈(0,1], O ⊂ R3is an open bounded set with regular boundary Γ, ν > 0 is the viscosity coefficient, udenotes the velocity field, FN(·) : (0,∞)→(0,1] is defined as FN(r) = min 1,N r where r, N > 0, pdenotes the pressure, f∈L2 loc(R;H) is an external force field, Gis a nonlinear diffusion term, Gδis a colored noise with correlation time δ∈(0,1], which will be specified later. Navier-Stokes equations can be used to describe the time evolution of an incompressible fluid, which are important in fluid and mechanics and turbulence. The well-posedness, regularity and asymptotic behaviors of solutions of 2D Navier-Stokes equations have been studied in the literature, see, e.g., [30,32] and references therein. However, as we all know, the uniqueness of weak solutions and the existence of strong solutions of 3D Navier-Stokes equations are still open problems due to the nonlinear convection term. It is worth mentioning that in order to overcome the difficulties caused by the nonlinear convection term, T. Caraballo and his collaborators processed a class of 3D globally modified Navier-Stokes equations in [3], and proved the existence and uniqueness 2 of solutions as well as the existence of V-attractors. After that, the existence and asymptotic behaviors of solutions of 3D GMNSE have been investigated in [16,22,28,45]. And the invariant measures as well as the Liouville type theorem of 3D GMNSE have been examined in [45,49] and the references therein. Since noise is ubiquitous in many physical systems due to the lack of knowledge of certain physical parameters as well as measurement errors arising in experiments or modeling, it is of great significance to consider noise influences on Navier-Stokes equations. For stochastic 2D Navier- Stokes equations, the existence of solutions, random attractors and invariant measures have been reported in [1,13,14,31]. But there are a few results on the 3D GMNSE with stochastic perturbations. Recently, reference [15] investigated the well-posedness and the convergence of solutions for 3D GMNSE with white noise. In addition, it is also significant to study the pathwise dynamics of such system, for which we need to define a pathwise random dynamical system via the solution operators. However, the existence of such a random dynamical system is unknown for nonlinear white noise until now, which results in the pathwise random attractors theory cannot be applied to this case. Therefore, systems driven by colored noise have drawn considerable attention in the literature, see, e.g. [10,19,38,39,42] and the references therein. In fact, many physical systems can be simulated by colored noise rather than white noise [29]. The colored noise is a kind of autocorrelated noise, and can be used to approximate the Wiener process [33]. From the perspective of theoretical and practical value, we aim to study the non-autonomous 3D GMNSE with nonlinear colored noise (1.1), which can be considered as an approximation of the corresponding system with white noise in some sense. We also would like to mention some research on the approximation of stochastic differential equations driven by white noise [18,20,43]. In this paper, we will first investigate the existence and uniqueness of pullback random attractors of (1.1). Recently, the pullback random attractors for 2D Navier-Stokes equations with colored noise have been reported in [19]. However, the argument therein cannot be directly applied to 3D Navier- Stokes equations with colored noise due to differences regarding Sobolev embedding inequalities for nonlinear convection term. Thanks to the modifying factor FN(·) in (1.1), we are able to establish the existence of pullback random attractors for 3D GMNS with colored noise. More precisely, by the Galerkin method and the property of the modifying factor FN(·), the well-posedness of (1.1) can be shown, which leads to the existence of a random dynamical system. Then, based on the uniform estimates of solutions in spaces Hand Vas well as the compactness of injection of Vinto H, we examine the pullback asymptotic compactness of solutions of (1.1), which together with [36, Theorem 2.23] induces the existence of random attractors. For the long-term behaviors of stochastic partial differential equation, the reader is referred to [2,39,42,44] and the reference therein. Another major goal of this paper is to study the invariant measures of non-autonomous 3D 3 GMNSE with nonlinear colored noise (1.1), which can provide important information about longterm dynamics and be used to identify statistical equilibrium. Based on the existence of random attractors of (1.1), we will first prove the existence of invariant measures supported on such attractors. In the deterministic case, the existence of invariant measures has been studied by many experts, see, e.g., [4,23] for autonomous systems and [17,27] for non-autonomous systems. It is worth mentioning that [13,48] investigated the existence of invariant measures for autonomous random dynamical systems. However, these results in above papers cannot be applied to nonautonomous system with colored noise (1.1). Recently, by virtue of the theory of non-autonomous random dynamical system in [36] together with the generalized Banach limit in [17], reference [9] constructed a family of invariant measures for general non-autonomous random dynamical systems. In order to apply such abstract theory to system (1.1), we need to verify the system (1.1) is jointly continuous in initial time and initial value (see Lemma 3.1), by which we can obtain the existence of invariant measures supported on the random attractors (see Theorem 3.2). In addition, if the non-autonomous term f(t) is periodic in t, we further show such invariant measure is also periodic. We also refer the readers to [1,7,11,14,21,24,26,37,41] for the existence of invariant measures and periodic measures of Markov semigroups generated by infinite-dimensional stochastic differential equations. Then we also derives that the invariant measures constructed in above satisfy a random Liouville type equation. For deterministic differential systems, it has been shown that the time average invariant measures satisfy a Liouville type equation in [12,17,27,46,47]. Hereafter, reference [48] established that invariant measures of autonomous 2D Navier-Stokes equations with additive white noise also satisfy a random Liouville type equation. Despite some advances in this direction, there is no relevant result for non-autonomous GMNSE with nonlinear colored noise. Indeed, it is difficult to verify the continuity of the mapping u7→ F(t, θtω, u(t)),Ψ0(u)in Hfor given test function Ψ in order to use the invariant property of invariant measures, where Fis given in (4.2). To overcome this issue, what we first do is to approximate Ψ(u) by using the Galerkin projection technique. Then together with the form of the construction of invariant measures and the cocycle property of the non-autonomous random dynamical system, and taking the limit, we will show that the invariant measures supported on the pullback random attractors satisfy a random Liouville type equation (see Theorem 4.1). In order to show the limiting relationship of invariant measures between GMNSE with colored noise and the corresponding deterministic equations, we further study the limiting behaviors of invariant measures of (1.1) as the noise intensity parameter εgoes to zero, which also means the zero-noise limit is observable as noise is non-negligible in real world. Note that the limiting behaviors of invariant measures have been investigated in [34,35] for deterministic equations, in [5,6,8,25] for autonomous stochastic systems, and in [40] for non-autonomous stochastic systems. 4 Different from [40] where discussed the convergence of invariant measures of time inhomogeneous transition operators, we will study the convergence of invariant measures of the non-autonomous cocycle associated with (1.1). For that purpose, we will first prove the set of all random attractors of (1.1) for ε∈(0,1] is precompact (see Lemma 5.1). And then, together with [9, Theorem 4.1] and the convergence of solutions with respect to noise intensity (see Lemma 5.2), we can obtain that any limit of invariant measures of (1.1) must be an invariant measure of the corresponding deterministic equations as ε→0. On the other hand, we also interested in the approximation of invariant measures of GMNSE with additive white noise by those of GMNSE with additive colored noise. Therefore, we consider the limiting behaviors of invariant measures of GMNSE with additive colored noise as the correlation time δof the colored noise tends to zero. More precisely, we will consider the following GMNSE with additive white noise:                  ∂u ∂t −ν∆u+FN(kuk) [(u(t)·∇)u] + ∇p=f(t) + βdW dt in O×(τ, ∞), div u= 0 in O×(τ, ∞), u= 0 on Γ ×(τ, ∞), u(x, τ) = uτ(x), x ∈ O, (1.2) where β∈D(A) which will be specified later in Section 2. We prove the equation (1.2) has a family of invariant measures supported on the pullback random attractors (see Theorem 5.2), which can be considered as the limit of the invariant measures of the following random equations with colored noise as δ→0:                  ∂uδ ∂t −ν∆uδ+FN(kuδk) [(uδ(t)·∇)uδ] + ∇p=f(t) + βGδ(θtω) in O ×(τ, ∞), div uδ= 0 in O×(τ, ∞), uδ= 0 on Γ ×(τ, ∞), uδ(x, τ) = uτ(x), x ∈ O, (1.3) which is a particular case of (1.1). Indeed, we show that any limit of a sequence of invariant measures of the random equations (1.3) must be an invariant measure of the stochastic equations (1.2) as δ→0. This paper is organized as follows. In the next section, we show the existence of pullback random attractors for GMNSE with nonlinear colored noise (1.1). In Section 3, we prove the existence of invariant measures of (1.1). In Section 4, we derive that such invariant measures satisfy a random Liouville type equation. Section 5is devoted to the limiting behaviors of invariant measures with respect to noise intensity εand correlation time δ. 5 2 Existence of random attractors In this section, we will investigate the existence of pullback random attractors of GMNSE with nonlinear colored noise (1.1). For this purpose, let us first recall some results from the theory of random dynamical system, see [36] for more details. Let (Ω,F,P) be a standard probability space, where Ω = {ω∈C(R,R) : ω(0) = 0},Fis its Borel σ-algebra, and Pis the Wiener measure on (Ω,F). There exists a classical group {θt}t∈R acting on (Ω,F,P) is given by θtω(·) = ω(t+·)−ω(t) for any t∈Rand ω∈Ω. Let (X, d) be a complete separable metric space with Borel σ-algebra B(X). Let D={D(τ, ω) : τ∈R, ω ∈Ω}be a family of nonempty subsets of Xsatisfying lim t→−∞ ect sup u∈D(τ+t,θtω) d(x, 0) = 0 for every c > 0. Such Dis said to be tempered in X. And denote by Dthe class of all tempered families of nonempty subsets of X. Definition 2.1 ([36]).A mapping Φ : R+×R×Ω×X→Xis called a continuous cocycle on X over (Ω,F,P,{θt}t∈R)if for all τ∈R,ω∈Ωand t, s ∈R+, (i) Φ(·, τ, ·,·) : R+×Ω×X→Xis (B(R+)×F ×B(X),B(X))-measurable; (ii) Φ(0, τ, ω, ·)is the identity on X; (iii) Φ(t+s, τ, ω, ·) = Φ(t, τ +s, θsω, ·)◦Φ(s, τ, ω, ·); (iv) Φ(t, τ, ω, ·) : X→Xis continuous. If, in addition, there exists T > 0such that for every t∈R+,τ∈Rand ω∈Ω, Φ(t, τ +T, ω, ·) = Φ(t, τ, ω, ·), then Φis called a periodic cocycle on Xwith period T. Definition 2.2 ([36]).A family K={K(τ, ω) : τ∈R, ω ∈Ω}∈Dis called a D-pullback absorbing set for Φif for all τ∈R,ω∈Ωand for every D∈ D, there exists T0=T0(D, τ, ω)>0such that Φ(t, τ −t, θ−tω, D(τ−t, θ−tω)) ⊆K(τ, ω),∀t⩾T0. If, in addition, for all τ∈Rand ω∈Ω,K(τ, ω)is a closed nonempty subset of Xand Kis measurable in ωwith respect to F, then we say Kis a closed measurable D-pullback absorbing set for Φ. Definition 2.3 ([36]).A family A={A(τ, ω) : τ∈R, ω ∈Ω}∈Dis called a D-pullback random attractor for Φif for every τ∈Rand ω∈Ω, (i) Ais measurable in ωwith respect to F, and A(τ, ω)is compact in X; (ii) Ais invariant, that is, for all t⩾0,Φ(t, τ, ω, A(τ, ω)) = A(τ+t, θtω); (iii) Aattracts all sets in D, that is, for all D∈ D, lim t→+∞dX(Φ(t, τ −t, θ−tω, D(τ−t, θ−tω)), A(τ, ω)) = 0, 6 where dXis the Hausdorff semi-metric given by dX(A, B) = sup a∈A inf b∈Bd(a, b)for any A⊆Xand B⊆X. If, in addition, there exists T > 0such that for any τ∈Rand ω∈Ω,A(τ+T, ω) = A(τ, ω), then we say Ais periodic with period T. Next, the colored noise will be introduced. Given δ > 0, consider the one-dimensional stochastic equation dGδ+1 δGδdt =1 δdW, (2.1) where Wis a two-sided real-valued Wiener process defined on (Ω,F,P). Let Gδ: Ω →Rbe a random variable given by Gδ(ω) = 1 δZ0 −∞ es δdW, ∀ω∈Ω. Then, one can check that Gδ(θtω) is the unique stationary solution of (2.1). The process Gδ(θtω) is called a real-valued colored noise which is a stationary Gaussian process with mean zero and variance 1 2δ. And there exists a {θt}t∈R-invariant subset (still denoted by) Ω of full measure, such that for any ω∈Ω, lim t→±∞ |Gδ(θtω)| t= 0,for any 0 < δ ⩽1,(2.2) lim t→±∞ 1 tZt 0Gδ(θsω)ds =E[Gδ] = 0,uniformly for 0 < δ ⩽1.(2.3) Now, let us recall some abstract spaces which will be frequently used in the sequel. V=φ∈(C∞ 0(O))3:∇·φ= 0, H= closure of Vin (L2(O))3with inner product (·,·) and associate norm |·|, V= closure of Vin (H1 0(O))3with inner product ((·,·)) and associate norm k·k, H0= dual space of H,V0= dual space of Vwith norm k·kV0, h·,·i denotes the dual pairing between Vand V0. It is clear that V⊂H≡H0⊂V0, where the injections are dense and compact. Next, consider the operator A:V→V0defined by hAu, vi= ((u, v)). And denote D(A) = (H2(O))3∩V. Then for any u∈D(A), Au =−P4uis the Stokes operator, where Pis the ortho-projector from (L2(O))3onto H. We denote by 0 < λ1⩽λ2⩽··· the eigenvalues of A. In this context, the Poincar´e inequality reads λ1|u|2⩽kuk2,∀u∈V. (2.4) Then define a trilinear form b(·,·,·) as follows, b(u, v, w) = 3 X i,j=1 ZO ui ∂vj ∂xi wjdx, ∀u, v, w ∈V, 7 and set hBN(u, v), wi=bN(u, v, w) = FN(kvk)b(u, v, w). Recall that there exists a constant C > 0 depending only on Osuch that |b(u, v, w)|⩽Ckukkvk|w|1 2kwk1 2,∀u, v, w ∈V, (2.5) |b(u, v, w)|⩽Ckukkvk1 2|Av|1 2|w|,∀u∈V, v ∈D(A), w ∈H. (2.6) For any p, r, N > 0, one can obtain that |FN(p)−FN(r)|⩽1 NFN(p)FN(r)|p−r|,(2.7) |FN(p)−FN(r)|⩽|p−r| r,(2.8) |FM(p)−FN(p)|⩽|M−N| p.(2.9) Moreover, we assume the nonlinear diffusion term G(u) = αu +h(u) + βwith α⩾0, β∈Hand h:H→Hsatisfying that (G1) there exists Lh>0 such that for all u, v ∈H, |h(u)−h(v)|⩽Lh|u−v|, (G2) there exist β1, β2>0 and κ∈[0,1) such that for all u∈H, |(h(u), u)|⩽β1|u|1+κ+β2. In order to obtain some energy estimates, we need an additional assumption. (F1) Let γ∈(0, νλ1) be a fixed number and Zt −∞ eγs|f(s)|2ds < +∞for all t∈R. Sometimes, the following tempered condition is also needed. (F2) For every c > 0, lim t→−∞ ect Z0 −∞ eγs|f(s+t)|2ds = 0. A solution of equation (1.1) will be considered in the following sense. Definition 2.4. Suppose uτ∈H. A function u(·)is called a weak solution of (1.1)if u∈ L2(τ, T;V)∩C([τ, T], H)for any T > τ, such that for all v∈Vand t⩾τ, (u(t), v) + νZt τhAu(s), vids +Zt τ FN(ku(s)k)b(u(s), u(s), v)ds = (uτ, v) + Zt τ (f(s), v)ds +εZt τGδ(θsω) (G(u(s)), v)ds, in the sense of distribution on [τ, +∞). 8 2.1 Cocycle for GMNSE with colored noise In this subsection, we will investigate the well-posedness of (1.1) and define a non-autonomous cocycle based on the solution operators. To that end, we first show the existence and uniqueness of solutions to (1.1). In fact, given ω∈Ω, (1.1) can be viewed as a deterministic problem. Thus, by the Galerkin method as in [3], one can verify that for any ω∈Ω, τ∈Rand uτ∈H, if (G1) and (G2) hold, problem (1.1) has a unique solution u(·, τ, ω, uτ) in the sense of of Definition 2.4. And this solution is continuous with respect to initial data in Hand is (F,B(H))-measurable in ω∈Ω. Moreover, the solution usatisfies the energy equality |u(t)|2+ 2νZt sku(r)k2dr =|u(s)|2+ 2 Zt s (f(r), u(r))dr + 2εZt sGδ(θrω)(G(u(r)), u(r))dr, ∀τ⩽s⩽t. (2.10) Now, given t∈R+,τ∈R,ω∈Ω and uτ∈H, we define a mapping Uby U(t, τ, ω, uτ) = u(t+τ, τ, θ−τω, uτ),(2.11) where we recall that u(·, τ, θ−τω, uτ) is the solution of (1.1) corresponding to τ,θ−τωand uτ. This mapping Uis a continuous non-autonomous cocycle in H. Denote by Dthe class of all families of nonempty subsets D={D(t, ω) : t∈R, ω ∈Ω}⊂P(H) such that for any c > 0, τ∈Rand ω∈Ω, lim t→−∞ ect sup ξ∈D(τ+t,θtω)|ξ|2!= 0, where P(H) denotes the family of all nonempty subsets of H. 2.2 Construction of pullback random absorbing sets In this subsection, we will find a D-pullback random absorbing set for the cocycle Uassociated with (1.1). We start with the uniform estimates on the solutions of (1.1) in H. Lemma 2.1. Suppose (G1), (G2) and (F1) hold. Then for every ε, δ ∈(0,1],τ∈R,ω∈Ωand D={D(t, ω) : t∈R, ω ∈Ω}∈D, there exists T1=T1(τ, ω, D)>0such that, for all t⩾T1and k∈[−1,0], |u(τ+k, τ −t, θ−τω, uτ−t)|2 ⩽1 + MZk −∞ eRs k(νλ1−2εαGδ(θrω))dr|f(s+τ)|2+|Gδ(θsω)|+|Gδ(θsω)|2 1−κ+|Gδ(θsω)|2ds := RH(τ, ω, ε, δ, k),(2.12) where uτ−t∈D(τ−t, θ−tω)and M > 1is independent of τ,ω,Dand ε, δ ∈(0,1]. 9 We now estimate the first term on the right-hand side of (3.4). For short, u(s, τ −t, θ−τω, u0) and u(s, τ −t, θ−τω, u∗ 0) are denoted by u(s) and ˆu(s), respectively. By (1.1), we have 2|u(τ)−ˆu(τ)|2+ 4νZτ τ−tku(s)−ˆu(s)k2ds = 2|u0−u∗ 0|2+ 4εZτ τ−tGδ(θsω)G(u(s)) −G(ˆu(s)), u(s)−ˆu(s)ds (3.5) + 4 Zτ τ−tFN(kˆu(s)k)b(ˆu(s),ˆu(s), u(s)−ˆu(s)) −FN(ku(s)k)b(u(s), u(s), u(s)−ˆu(s))ds. For the second term on the right-hand side of (3.5), by (G1) and H¨older’s inequality, we obtain 4εZτ τ−tGδ(θsω)G(u(s)) −G(ˆu(s)), u(s)−ˆu(s)ds ⩽4αZτ τ−tGδ(θsω)|u(s)−ˆu(s)|2ds + 4LhZτ τ−t|Gδ(θsω)||u(s)−ˆu(s)|2ds. (3.6) For the third term on the right-hand side of (3.5), it follows from (2.5), (2.7) and Young’s inequality that 4Zτ τ−tFN(kˆu(s)k)b(ˆu(s),ˆu(s), u(s)−ˆu(s)) −FN(ku(s)k)b(u(s), u(s), u(s)−ˆu(s))ds = 4 Zτ τ−tFN(kˆu(s)k)b(ˆu(s)−u(s),ˆu(s), u(s)−ˆu(s)) + [FN(kˆu(s)k)−FN(ku(s)k)] b(u(s),ˆu(s), u(s)−ˆu(s))ds ⩽2νZτ τ−tku(s)−ˆu(s)k2ds + 2 e CZτ τ−t|u(s)−ˆu(s)|2ds, which, together with (3.5) and (3.6), shows that 2|u(τ)−ˆu(τ)|2+ 2νZτ τ−tku(s)−ˆu(s)k2ds ⩽2|u0−u∗ 0|2+ 2 Zτ τ−th2 (α+Lh)|Gδ(θsω)|+e Ci|u(s)−ˆu(s)|2ds. Then by Gronwall’s inequality, we have, 2|u(τ)−ˆu(τ)|2⩽2|u0−u∗ 0|2eRτ τ−t[2(α+Lh)|Gδ(θsω)|+e C]ds, which implies that for any t⩾0 with |t−t∗|<1, 2|u(τ)−ˆu(τ)|2⩽2|u0−u∗ 0|2eRτ τ−t∗−1[2(α+Lh)|Gδ(θsω)|+e C]ds. Therefore, there exists 0 < ζ1<1 such that for any (t, u0)∈[0,+∞)×Hwith |t−t∗|< ζ1and |u0−u∗ 0|< ζ1, 2|u(τ)−ˆu(τ)|2< 2.(3.7) 16 Next, the last term of (3.4) will be estimated. Without loss of generality, we assume t∗< t. Since uis the solution of (1.1), we have 2|u(τ, τ −t, θ−τω, u∗ 0)−u(τ, τ −t∗, θ−τω, u∗ 0)|2 = 2|u(τ, τ −t∗, θ−τω, u(τ−t∗, τ −t, θ−τω, u∗ 0)) −u(τ, τ −t∗, θ−τω, u∗ 0)|2.(3.8) Similarly to (3.5), by (3.8), we find that 2|u(τ, τ −t, θ−τω, u∗ 0)−u(τ, τ −t∗, θ−τω, u∗ 0)|2 ⩽2|u(τ−t∗, τ −t, θ−τω, u∗ 0)−u∗ 0|2eRτ τ−t∗[2(α+Lh)|Gδ(θsω)|+e C]ds. Note that |u(τ−t∗, τ −t, θ−τω, u∗ 0)−u∗ 0|2=|u(τ−t∗, τ −t, θ−τω, u∗ 0)|2−|u∗ 0|2 −2u(τ−t∗, τ −t, θ−τω, u∗ 0)−u∗ 0, u∗ 0.(3.9) For the first and second terms on the right-hand side of (3.9), by (2.14), we have |u(τ−t∗, τ −t, θ−τω, u∗ 0)|2−|u∗ 0|2 ⩽Zτ−t∗ τ−t (2εαGδ(θs−τω)−νλ1)|u(s, τ −t, θ−τω, u∗ 0)|2ds +4 νλ1Zτ−t∗ τ−t|f(s)|2ds +Z−t∗ −t (2β2|Gδ(θsω)|+Cν|Gδ(θsω)|2 1−κ+8 νλ1|Gδ(θsω)|2|β|2)ds, (3.10) which, together with Gronwall’s inequality and the continuity of Gδ(θtω) in t, implies that there exists M1(τ, t∗, ω, u∗ 0)>0 such that for any t⩾0 with |t−t∗|<1, sup τ−t−1⩽s⩽τ−t∗|u(s, τ −t, θ−τω)|+|u∗ 0|⩽M1(τ, t∗, ω, u∗ 0).(3.11) It then follows from (3.10) and (3.11) that there exists 0 < ζ2< ζ1such that for any t⩾0 with |t−t∗|< ζ2, |u(τ−t∗, τ −t, θ−τω, u∗ 0)|2−|u∗ 0|2< 8M2(τ, t∗, ω),(3.12) where M2(τ, t∗, ω) = eRτ τ−t∗[2(α+Lh)|Gδ(θsω)|+e C]ds. For the last term of (3.9), by the density of Vin H, we deduce that there exists an element f u∗ 0∈Vsuch that |f u∗ 0−u∗ 0|< 64M1(τ, t∗, ω, u∗ 0)M2(τ, t∗, ω). Therefore, we have for any t⩾0 with |t−t∗|< ζ2, u(τ−t∗, τ −t, θ−τω, u∗ 0)−u∗ 0, u∗ 0 ⩽u(τ−t∗, τ −t, θ−τω, u∗ 0)−u∗ 0,f u∗ 0+u(τ−t∗, τ −t, θ−τω, u∗ 0)−u∗ 0, u∗ 0−f u∗ 0 17 ⩽kf u∗ 0k Zτ−t∗ τ−tkd dsu(s, τ −t, θ−τω, u∗ 0)k2 V∗ds!1 2 (t−t∗)1 2+ 32M2(τ, t∗, ω).(3.13) By (2.5), (3.11), (G1) and the fact that Gδ(θtω) is continuous in t, we can deduce that Zτ−t∗ τ−tkd dsu(s, τ −t, θ−τω, u∗ 0)k2 V∗ds < +∞, which, along with (3.13), implies that there exists 0 < ζ3< ζ2such that, for any t⩾0 with |t−t∗|< ζ3, u(τ−t∗, τ −t, θ−τω, u∗ 0)−u∗ 0, u∗ 0⩽ 16M2(τ, t∗, ω).(3.14) Therefore, from (3.4), (3.7), (3.9), (3.12) and (3.14), it follows that there exists ζ=ζ3∈(0,1) such that, for any (t, u0)∈[0,+∞)×Hwith |t−t∗|< ζ and |u0−u∗ 0|< ζ, |U(t, τ −t, θ−tω, u0)−U(t∗, τ −t∗, θ−t∗ω, u∗ 0)|2< , as desired. Now, we are going to present the main result of this section. Theorem 3.2. Suppose (G1), (G2), (F1) and (F2) hold. Fix a generalized Banach limit LIMt→−∞ and let ξ(·) : R→Hbe a continuous mapping such that {ξ(t)}t∈R∈ D. Then, for any N > 0 and ε, δ ∈(0,1], there exists a family of probability measures {µt,ω}t∈R,ω∈Ωof process Usuch that µt,ω is an invariant measure for Uand is supported on A(t, ω), and for any real-valued continuous functional ϕon H,τ∈Rand P-a.s. ω∈Ω, LIM t→−∞ 1 −tZ0 t ϕ(U(−s, τ +s, θsω, ξ(τ+s))) ds =ZH ϕ(u)µτ,ω(du) = ZA(τ,ω) ϕ(u)µτ,ω(du) = LIM t→−∞ 1 −tZ0 t"ZA(τ+s,θsω) ϕ(U(−s, τ +s, θsω, u)) µτ+s,θsω(du)#ds. If, in addition, f(t)is a T-periodic function in tand ξ(·) : R→His a T-periodic continuous map in t, then the invariant measure µt,ω is T-periodic. Proof. It follows from Theorem 2.1 that the non-autonomous random dynamical system Uassociated with (1.1) has a D-pullback random attractor Ain H. Then, by Lemma 3.1 and Theorem 3.1, the existence of invariant measures can be obtained. In addition, if f(t, ·) is a T-periodic function in t, then we can deduce that the system Uand the random attractor Aare also periodic with T. By using Theorem 3.1 again, the periodicity of such invariant measures can be derived successfully. 18 Proposition 3.1. Let µτ,ω be an invariant probability measure for Φ(−t, τ +t, θtω, v)on X, and let K(τ, ω)be a closed set that is absorbing for Φ(−t, τ +t, θtω, v). Then, µτ,ω (X\K(τ, ω)) = 0. Proof. For every r > 0, let Br(t, ω) = {u(t, ω)∈X:ku(t, ω)kX⩽r}. Thanks to the fact that K(τ, ω) is a closed absorbing set, there is a time tr⩾0 such that Φ(−t, τ +t, θtω, ·)Br(τ+t, θtω)⊂ K(τ, ω) for all t⩾tr, which implies that Br(τ+t, θtω)⊂Φ(−t, τ +t, θtω, ·)−1K(τ, ω). Therefore, µτ+t,θtωΦ(−t, τ +t, θtω, ·)−1K(τ, ω)⩾µτ+t,θtω(Br(τ+t, θtω)). Noting that µτ+t,θtω(Φ(−t, τ + t, θtω, ·)−1K(τ, ω)) = µτ,ω(K(τ, ω)), hence 1⩾µτ,ω(K(τ, ω)) = µτ+t,θtωΦ(−t, τ +t, θtω, ·)−1K(τ, ω)⩾µτ+t,θtω(Br(τ+t, θtω)). In addition, X=[ r>0 Br(τ, ω) with Br1(τ, ω)⊂Br2(τ, ω) if r1⩽r2, then µτ,ω(Br(τ, ω)) →µτ,ω(X) as r→+∞. Therefore, µτ,ω(K(τ, ω)) = 1 which implies the desired result. Remark 3.1. From Lemma 2.3, it follows that Uhas a closed pullback absorbing set e K={e K(τ, ω) : τ∈R, ω ∈Ω}in V, where e K(τ, ω) = {u∈V:kuk2⩽RV(τ, ω, ε, δ)}, and RV(τ, ω, ε, δ)is a constant given in Lemma 2.3. Then together with Proposition 3.1, we have that µt,ω(H\e K(t, ω)) = 0 for any t∈Rand ω∈Ω. Let us denote kvk= +∞if v∈H\V. Then note that e K(τ, ω)⊂V, therefore, one can deduce that for any t∈Rand ω∈Ω, ZHkuk2µt,ω(du) = Ze K(t,ω)kuk2µt,ω(du)<+∞. 4 Random Liouville type theorem In this section, we will investigate that the family of measures {µt,ω}t∈R,ω∈Ωconstructed in Section 3satisfies a random Liouville equation. To that end, let Tdenote the class of real-valued functionals Ψ on H, which are bounded on bounded subsets of Hand satisfy: (1) for any u∈V, the first-order Fr´echet derivative Ψ0(u) exists, (2) Ψ0(u)∈Vfor all u∈V, and u7−→ Ψ0(u) is continuous and bounded as a functional from Vto V, (3) for every global solution u(t) := u(t, t0, ω, u0) of (1.1), there holds for any ω∈Ω, d dtΨ(u(t)) = F(t, θtω, u(t)),Ψ0(u(t)),(4.1) where F(t, θtω, u) = −νAu −BN(u, u) + f(t) + εG(u)Gδ(θtω).(4.2) Remark 4.1. The above set Tis not empty by referring [17] to construct some cylindrical test functionals. Indeed, for given ψ∈C1 0(O),gi∈V, i = 1,2,··· , m, let Ψ(u) := ψ((u, g1),(u, g2),··· ,(u, gm)),∀u∈H. 19 Then by the chain rule of Fr´echet derivative, we have Ψ0(u) = m X i=1 ∂iψ((u, g1),(u, g2),··· ,(u, gm)) gi,(4.3) where ∂iψis the derivative of ψwith respect to the i-th variable. Obviously, it is continuous and bounded from Vto V. In addition, if u(t)is the solution of (1.1), it follows from (4.3)and the differential chain rule that (4.1)holds. In order to use the invariance property of µt,ω, we need to verify that for any test function Ψ∈ T, the mapping u7→ F(t, θtω, u),Ψ0(u)is continuous in H. But such mapping may not be continuous in H. Thus, what we first do is to approximate Ψ(u) by Ψm(u) = Ψ(Pmu) for any test function Ψ ∈ T. And then letting m→+∞, the desired result can be obtained. We first state an auxiliary lemma. Lemma 4.1. Let Ψm(u) = Ψ(Pmu), then for any t∈Rand ω∈Ω, ZHΨ0(u), F(t, θtω, u)µt,ω(du) = lim m→+∞ZHΨ0 m(u), F(t, θtω, u)µt,ω(du).(4.4) Also, we can obtain that the mapping u7→ Ψ0 m(u), F(t, θtω, u)(4.5) is continuous on H. Proof. Step 1. We will first claim that for any u∈V, lim m→+∞F(t, θtω, u),Ψ0 m(u)=F(t, θtω, u),Ψ0(u).(4.6) It is sufficient to prove that given u∈Vand  > 0, there is N0>0 such that for any m>N0, F(t, θtω, u),Ψ0(u)−F(t, θtω, u),Ψ0 m(u)< . Indeed, for any u∈V, we have F(t, θtω, u),Ψ0(u)−F(t, θtω, u),Ψ0 m(u) =νΨ0(u)−Ψ0(Pmu),−Au+νΨ0(Pmu), APmu−Au +−BN(u, u),Ψ0(u)+BN(Pmu, Pmu),Ψ0(Pmu) +Ψ0(u)−Ψ0(Pmu), f(t) + εG(u)Gδ(θtω) +Ψ0(Pmu), f(t)−Pmf(t) + ε(G(u)−G(Pmu)) Gδ(θtω).(4.7) For the first and second terms on the right-hand side of (4.7), we obtain that νΨ0(u)−Ψ0(Pmu),−Au+νΨ0(Pmu), APmu−Au 20 ⩽νΨ0(u)−Ψ0(Pmu)kuk+νsup u∈VΨ0(u)kPmu−uk, which, together with the fact that lim m→+∞ku−Pmuk2= 0, and the continuity of Ψ0in V, implies that there exists N1>0 such that, for any m>N1, νΨ0(u)−Ψ0(Pmu),−Au+νΨ0(Pmu), APmu−Au< 4.(4.8) For the third and fourth terms on the right-hand side of (4.7), by (2.5) and (2.7), we obtain that −BN(u, u),Ψ0(u)+BN(Pmu, Pmu),Ψ0(Pmu) =−FN(kuk)bu−Pmu, u, Ψ0(u)+ (FN(kPmuk)−FN(kuk)) bPmu, u, Ψ0(u) +FN(kPmuk)bPmu, Pmu, Ψ0(u)−bPmu, u, Ψ0(u) +FN(kPmuk)bPmu, Pmu, Ψ0(Pmu)−bPmu, Pmu, Ψ0(u) ⩽3CN ku−Pmuksup u∈VΨ0(u)+CN kukΨ0(Pmu)−Ψ0(u),(4.9) which, along with the fact that lim m→+∞ku−Pmuk2= 0, and the continuity of Ψ0in V, shows that there exists N2> N1such that, for any m>N2, −BN(u, u),Ψ0(u)+BN(Pmu, Pmu),Ψ0(Pmu)< 4.(4.10) For the fifth term on the right-hand side of (4.7), we obtain that Ψ0(u)−Ψ0(Pmu), f(t) + εG(u)Gδ(θtω) ⩽1 √λ1Ψ0(u)−Ψ0(Pmu)|f(t)|+α+Lh √λ1kuk+|h(0)|+|β||Gδ(θtω)|, which, together with the fact that lim m→+∞ku−Pmuk2= 0, and the continuity of Ψ0in V, implies that there exists N3> N2such that, for any m>N3, Ψ0(u)−Ψ0(Pmu), f(t) + εG(u)Gδ(θtω)< 4.(4.11) For the last term of (4.7), by the fact that lim m→+∞ku−Pmuk2= 0, and the boundedness of Ψ0in V, we deduce that there exists N4> N3such that, for all m > N4, Ψ0(Pmu), f(t)−Pmf(t) + ε(G(u)−G(Pmu)) Gδ(θtω)< 4.(4.12) Therefore, by (4.7)-(4.12), the desired result can be obtained. Step 2. We will first claim that for any u∈V, lim m→+∞ZHF(t, θtω, u),Ψ0 m(u)µt,ω(dv) = ZHF(t, θtω, u),Ψ0(u)µt,ω(dv).(4.13) 21 For any m∈Nand u∈V, we have F(t, θtω, u),Ψ0 m(u) ⩽(ν+CN) sup u∈VkΨ0(u)kkuk+α+Lh √λ1kuk+|β||Gδ(θtω)|+|f(t)|sup u∈VkΨ0(u)k := g(t, θtω, u). It then follows from Remark 3.1 that ZH g(t, θtω, u)µt,ω(du) = Ze K(t,ω) g(t, θtω, u)µt,ω(du)<+∞, which, together with (4.6) and the Lebesgue dominated convergence theorem, shows (4.13) holds. Step 3. The mapping u7→ F(t, θtω, u),Ψ0 m(u)is continuous in H. In order to do that, we need to prove that given u0∈Hand  > 0, then there exists ζ∈(0,1] such that for any u∈Hwith |u−u0|< ζ, F(t, θtω, u),Ψ0 m(u)−F(t, θtω, u0),Ψ0 m(u0)< . Noting that Ψ0 m(u)=Ψ0(Pmu)Ψm, we have −νAu −BN(u, u) + f(t) + εG(u)Gδ(θtω),Ψ0 m(u) −−νAu0−BN(u0, u0) + f(t) + εG(u0)Gδ(θtω),Ψ0 m(u0) =−νAPmu, Ψ0(Pmu)−−νAPmu0,Ψ0(Pmu0)+Pmf(t),Ψ0(Pmu)−Ψ0(Pmu0) +−BN(Pmu, Pmu),Ψ0(Pmu)−−BN(Pmu0, Pmu0),Ψ0(Pmu0) +εG(Pmu)Gδ(θtω),Ψ0(Pmu)−εG(Pmu0)Gδ(θtω),Ψ0(Pmu0).(4.14) For the first and second terms on the right-hand side of (4.14), we obtain that −νAPmu, Ψ0(Pmu)−−νAPmu0,Ψ0(Pmu0) =νΨ0(Pmu0)−Ψ0(Pmu), APmu+νΨ0(Pmu0), APmu0−APmu ⩽νλm(|u0|+ 1) √λ1Ψ0(Pmu)−Ψ0(Pmu0)+ sup u∈VkΨ0(u)kνλm √λ1|u−u0|, which, together with the fact that kPmuk2⩽λm|u|2, and the continuity of Ψ0in V, implies that there exists ζ1∈(0,1] such that for any u∈Hwith |u−u0|< ζ1, −νAPmu, Ψ0(Pmu)−−νAPmu0,Ψ0(Pmu0)< 4.(4.15) For the third term on the right-hand side of (4.14), it follows from the fact that kPmuk2⩽λm|u|2, and the continuity of Ψ0in V, that there exists ζ2∈(0, ζ1] such that, for all u∈Hwith |u−u0|< ζ2, Pmf(t),Ψ0(Pmu)−Ψ0(Pmu0)< 4.(4.16) 22 For the fourth and fifth terms on the right-hand side of (4.14), similar to (4.9), by (2.5) and (2.7), we obtain that −BN(Pmu, Pmu),Ψ0(Pmu)−−BN(Pmu0, Pmu0),Ψ0(Pmu0) ⩽3CNpλmu−u0sup u∈VΨ0(u)+CNpλmu0Ψ0(Pmu0)−Ψ0(Pmu), which, together with the fact that kPmuk2⩽λm|um|2, and the continuity of Ψ0in V, implies that there exists ζ3∈(0, ζ2] such that, for any u∈Hwith |u−u0|< ζ3, −BN(Pmu, Pmu),Ψ0(Pmu)−−BN(Pmu0, Pmu0),Ψ0(Pmu0)< 4.(4.17) For the last two terms of (4.14), we have εG(Pmu)Gδ(θtω),Ψ0(Pmu)−εG(Pmu0)Gδ(θtω),Ψ0(Pmu0) ⩽G(Pmu)−G(Pmu0)|Gδ(θtω)|Ψ0(Pmu)+G(Pmu0)|Gδ(θtω)|Ψ0(Pmu0)−Ψ0(Pmu) ⩽|Gδ(θtω)| √λ1 sup u∈VΨ0(u)G(Pmu)−G(Pmu0) +|Gδ(θtω)| √λ1(α+Lh)|u0|+|β|+|h(0)|Ψ0(Pmu0)−Ψ0(Pmu), which, together with the fact that kPmuk2⩽λm|um|2, the continuity of Ψ0in V, and the continuity of Gin H, shows that there exists ζ4∈(0, ζ3] such that for any u∈Hwith |u−u0|< ζ4, εG(Pmu)Gδ(θtω),Ψ0(Pmu)−εG(Pmu0)Gδ(θtω),Ψ0(Pmu0)< 4.(4.18) Therefore, by (4.14)-(4.18), the continuity of the mapping u7→ F(t, θtω, u),Ψ0 m(u)in Hcan be obtained. Next, we will show the random Liouville type theorem for the non-autonomous random dynamical system Uassociated with (1.1). Theorem 4.1. Suppose (G1), (G2), (F1) and (F2) hold. Then for any Ψ∈ T , the family of measures {µt,ω}t∈R,ω∈Ωconstructed in Section 3satisfies the random Liouville equation: ZA(τ,ω) Ψ(v)µτ,ω(dv)−ZA(σ,θ−(τ−σ)ω)Ψ(v)µσ,θ−(τ−σ)ω(dv) =Zτ σZA(η,θ−(τ−η)ω)F(η, θη−τω, v),Ψ0(v)µη,θ−(τ−η)ω(dv)dη, for all τ⩾σand ω∈Ω. Proof. For any τ⩾σ,s⩽0 and Ψ ∈ T , by (4.1), we have Ψ (u(τ;σ+s, θ−τω, v)) −Ψ (u(σ;σ+s, θ−τω, v)) 23 =Zτ σF(η, θη−τω, u(η;σ+s, θ−τω, v)),Ψ0(u(η;σ+s, θ−τω, v))dη. (4.19) Since τ⩾σ, it follows from (2.11), (4.4), (4.19) and the definition of invariant measure that ZA(τ,ω) Ψ(v)µτ,ω(dv)−ZA(σ,θ−(τ−σ)ω)Ψ(v)µσ,θ−(τ−σ)ω(dv) = LIM t→−∞ 1 −tZ0 tZH ΨU(τ−σ−s, σ +s, θ−(τ−σ−s)ω, v)µσ+s,θ−(τ−σ−s)ω(dv)ds −LIM t→−∞ 1 −tZ0 tZH ΨU(−s, σ +s, θ−(τ−σ−s)ω, v)µσ+s,θ−(τ−σ−s)ω(dv)ds = LIM t→−∞ 1 −tZ0 tZHZτ σF(η, θη−τω, u(η;σ+s, θ−τω, v)), Ψ0(u(η;σ+s, θ−τω, v))dη µσ+s,θ−(τ−σ−s)ω(dv)ds = LIM t→−∞ 1 −tZ0 tZτ σhlim m→+∞ZHF(η, θη−τω, u(η;σ+s, θ−τω, v)), Ψ0 m(u(η;σ+s, θ−τω, v))µσ+s,θ−(τ−σ−s)ω(dv)idη ds. (4.20) From (2.11) and the cocycle property of U, it follows that u(η;σ+s, θ−τω, v) =U(η−σ−s, σ +s, θ−(τ−σ−s)ω, v) =U(η−σ, σ, θ−(τ−σ)ω, ·)◦U(−s, σ +s, θ−(τ−σ−s)ω, v).(4.21) In addition, by (4.5) and Lemma 3.1, we can deduce that the mapping v7→ Fη, θη−τω, U(η−σ, σ, θ−(τ−σ)ω, v),Ψ0 mU(η−σ, σ, θ−(τ−σ)ω, v) (4.22) is continuous. Then, by (4.20)-(4.22), and Definition 3.2 about invariant measures, we obtain ZA(τ,ω) Ψ(v)µτ,ω(dv)−ZA(σ,θ−(τ−σ)ω)Ψ(v)µσ,θ−(τ−σ)ω(dv) =Zτ σhlim m→+∞ZHF(η, θη−τω, U(η−σ, σ, θ−(τ−σ)ω, v)), Ψ0 m(U(η−σ, σ, θ−(τ−σ)ω, v))µσ,θ−(τ−σ)ω(dv)idη =Zτ σhlim m→+∞ZHF(η, θη−τω, v),Ψ0 m(v)µη,θ−(τ−η)ω(dv)idη. (4.23) Then, by using (4.5) again, it follows from (4.23) that ZA(τ,ω) Ψ(v)µτ,ω(dv)−ZA(σ,θ−(τ−σ)ω)Ψ(v)µσ,θ−(τ−σ)ω(dv) =Zτ σZA(η,θ−(τ−η)ω)F(η, θη−τω, v),Ψ0(v)µη,θ−(τ−η)ω(dv)dη. The proof is complete. 24 5 Limiting behaviors of invariant measures In this section, we are devoted to the limiting behaviors of invariant measures with respect to noise intensity εand correlation time δ, respectively. 5.1 Limiting behaviors of invariant measures as ε→0 In this subsection, we will discuss the limiting relationship of invariant measures between the GMNSE with colored noise and the corresponding deterministic GMNSE as parameter ε→0. To indicate the dependence of solutions on ε, we will write the solution of (1.1) as uε, and the corresponding cocycle as Uε. As proved in the previous section, Uεhas a D-pullback attractor Aε∈ D in Hand a closed measurable D-pullback absorbing set Kεgiven by, for any τ∈Rand ω∈Ω, Kε(τ, ω) = u∈H:|u|2⩽RH(τ, ω, ε, δ, 0),(5.1) where RH(τ, ω, ε, δ, 0) is as in Lemma 2.2. In addition, by (2.16), we can deduce that for any ε∈(0,1], RH(τ, ω, ε, δ, 0) ⩽1 + MZs0 −∞ eγs|f(s+τ)|2+|Gδ(θsω)|+|Gδ(θsω)|2 1−κ+|Gδ(θsω)|2ds +MZ0 s0 eRs 0(νλ1+2α|Gδ(θrω)|)dr|f(s+τ)|2+|Gδ(θsω)|+|Gδ(θsω)|2 1−κ+|Gδ(θsω)|2ds := RH(τ, ω, δ, 0),(5.2) where s0is as in (2.16), and 0 < RH(τ, ω, δ, 0) <+∞is independent of ε∈(0,1]. It then follows from (5.1) and (5.2) that [ 0<ε⩽1 Aε(τ, ω)⊆[ 0<ε⩽1 Kε(τ, ω)⊆B(τ, ω),(5.3) where B(τ, ω), for every τ∈Rand ω∈Ω, is given by B(τ, ω) = u∈H:|u|2⩽RH(τ, ω, δ, 0). If ε= 0, then system (1.1) reduces to the following deterministic system:                  ∂u ∂t −ν∆u+FN(kuk) [(u(t)·∇)u] + ∇p=f(t) in O ×(τ, ∞), div u= 0 in O×(τ, ∞), u= 0 on Γ ×(τ, ∞), u(x, τ) = uτ(x), x ∈ O. (5.4) 25 By (5.13), we see that |U0(t, τ −t, θ−tω, u0)−U0(t∗, τ −t∗, θ−t∗ω, u∗ 0)|2 =|u(τ, τ −t, θ−τω, u0)−u(τ, τ −t∗, θ−τω, u∗ 0)|2 ⩽2|v(τ, τ −t, θ−τω, v0)−v(τ, τ −t, θ−τω, v∗ 0)|2 + 2|v(τ, τ −t, θ−τω, v∗ 0)−v(τ, τ −t∗, θ−τω, v∗ 0)|2,(5.22) where v0=u0−βz(θ−tω) and v∗ 0=u∗ 0−βz(θ−t∗ω). We now estimate the first term on the right-hand side of (5.22). By (5.12), we have d dt|v(τ)−ˆv(τ)|2+ 2νkv(t)−ˆv(t)k2 = 2FN(kˆv+βzk)b(ˆv+βz, ˆv+βz, v −ˆv)−FN(kv+βzk)b(v+βz), v +βz, v −ˆv).(5.23) For the last term of (5.23), similar to (4.9), by (2.5), (2.7) and the Young inequality, we see that FN(kˆv+βzk)b(ˆv+βz, ˆv+βz, v −ˆv)−FN(kv+βzk)b(v+βz, v +βz, v −ˆv) ⩽e C|v−ˆv|2+νkv−ˆvk2, which, along with (5.23) and Gronwall’s inequality, shows that for any t > 0 with |t−t∗|<1, 2|v(τ, τ −t, θ−τω, v0)−v(τ, τ −t, θ−τω, v∗ 0)|2 ⩽4|u0−u∗ 0|2+|βz(θ−tω)−βz(θ−t∗ω)|2ee C(t∗+1), which, together with the continuity of z(θtω) in t, shows that there exists 0 < ζ1<1 such that for any |t−t∗|< ζ1and |u0−u∗ 0|< ζ1, 2|v(τ, τ −t, θ−τω, v0)−v(τ, τ −t, θ−τω, v∗ 0)|2< 2.(5.24) Next, the last term of (5.22) will be estimated. Without loss of generality, we assume t∗< t. Since vis the solution of (5.12), we have 2|v(τ, τ −t, θ−τω, v∗ 0)−v(τ, τ −t∗, θ−τω, v∗ 0)|2 = 2|v(τ, τ −t∗, θ−τω, v(τ−t∗, τ −t, θ−τω, v∗ 0)) −v(τ, τ −t∗, θ−τω, v∗ 0)|2, which, together with (5.23) and Gronwall’s inequality, shows that 2|v(τ, τ −t, θ−τω, v∗ 0)−v(τ, τ −t∗, θ−τω, v∗ 0)|2 ⩽2|v(τ−t∗, τ −t, θ−τω, v∗ 0)−v∗ 0|2ee Ct∗ = 2 |v(τ−t∗, τ −t, θ−τω, v∗ 0)|2−|v∗ 0|2−2v(τ−t∗, τ −t, θ−τω, v∗ 0), v∗ 0ee Ct∗. 32 Then, by a similar technique to the one used in Lemma 3.1, we can derive that there exists 0< ζ2< ζ1such that, for any |t−t∗|< ζ2, 2|v(τ, τ −t, θ−τω, v∗ 0)−v(τ, τ −t∗, θ−τω, v∗ 0)|2< 2, which, along with (5.22) and (5.24), shows the desired result. Now, we present the existence of invariant measures and periodic invariant measures of (5.9). Theorem 5.2. Suppose (F1) and (F2) hold. Fix a generalized Banach limit LIMt→−∞ and let ξ:R→Hbe a continuous mapping such that {ξt}t∈R∈ D. Then for any N > 0, there exists a family of probability measures {µ0 t,ω}t∈R,ω∈Ωof process U0such that µ0 t,ω is an invariant measure for U0and is supported on A0(t, ω). If, in addition, f(t)is a T-periodic function in tand ξ:R→His a T-periodic continuous map in t, then the invariant measure µ0 t,ω is T-periodic. 5.2.2 Convergence of invariant measures from colored noise to white noise Next, the limiting behaviors of invariant measures of (5.8) will be investigated. Noting that system (5.8) is a particular case of (1.1), it follows from Theorem 3.2 that there exists an invariant measure µδ t,ω supported on the attractor Aδ(t, ω) for any t∈Rand ω∈Ω. In order to investigate the limiting behaviors of invariant measures, we need to introduce a new transformation, which is useful to show the convergence of solutions of (5.8) as δ→0. Let vδ(t, τ, ω) = uδ(t, τ, ω)−βyδ(θtω),(5.25) where yδis the solution of the following random equation driven by colored noise dyδ=−yδdt + Gδ(θtω)dt. By [19, Lemma 3.2], we have lim δ→0yδ(θtω) = z(θtω),uniformly on [τ, τ +T] with τ∈Rand T > 0; (5.26) lim t→±∞ |yδ(θtω)| |t|= 0,uniformly for 0 < δ ⩽1 2; (5.27) lim t→±∞ 1 tZt 0 yδ(θsω)ds = 0,uniformly for 0 < δ ⩽1 2; (5.28) lim δ→0 E[|yδ(ω)|] = E[|z(ω)|].(5.29) By (5.8) and (5.25), we have dvδ dt +νAvδ+FN(kvδ+βyδk)(vδ+βyδ)·∇(vδ+βyδ)=f+βyδ−νyδAβ, (5.30) with initial value vδ,τ =uτ−βyδ(θτω). 33 Given ω∈Ω, (5.30) is a deterministic equation. Therefore, under the same conditions as before, (5.30) has a unique solution vδin the sense of Definition 2.4. Moreover, the solution vδis (F,B(H))- measurable in ω∈Ω, and depends continuously on vδ,τ in H. Based on this, a continuous cocycle Uδ:R+×R×Ω×H→Hfor (5.8) can be defined by Uδ(t, τ, ω, uτ) = vδ(t+τ, τ, θ−τω, vδ,τ ) + βyδ(θtω) = uδ(t+τ, τ, θ−τω, uτ),(5.31) where vδ,τ =uτ−βyδ(ω). Next, the uniform estimates on the solutions of (5.8) will be established. Lemma 5.6. Suppose (F1) and (F2) hold. Then, for every δ∈(0,1 2],Uδhas a closed measurable D-pullback absorbing set Kδ={Kδ(τ, ω) : τ∈R, ω ∈Ω} ∈ D as given by Kδ(τ, ω) = u∈H:|u|2⩽Rδ,H(τ, ω, 0), where Rδ,H(τ, ω, 0) = 1 + 2|βyδ(ω)|2+ 2 e CZ0 −∞ e−νλ1s|f(s+τ)|2+|βyδ(θsω)|2+kβyδ(θsω)k2ds, and e Cis a positive constant independent of τ,ωand δ. Proof. Similar to the proof of Lemma 5.3, we can derive that |vδ(τ, τ −t, θ−τω, vδ,τ−t)|2 ⩽e−νλ1t|vδ,τ−t|2+e CZ0 −∞ e−νλ1s|f(s+τ)|2+|βyδ(θsω)|2+kβyδ(θsω)k2ds. (5.32) By (F1) and (5.27), the last term of (5.32) is well-defined. It follows from (5.25) and (5.32) that |uδ(τ, τ −t, θ−τω, uδ,τ−t)|2 ⩽4e−νλ1t|uδ,τ−t|2+ 4e−νλ1t|βyδ(θ−tω)|2+ 2|βyδ(ω)|2 + 2 e CZ0 −∞ e−νλ1s|f(s+τ)|2+|βyδ(θsω)|2+kβyδ(θsω)k2ds, which, together with (5.11) and the fact that uδ,τ−t∈D(τ−t, θ−tω) and D∈ D, implies that there exists T2=T2(τ, ω, D)>0 such that, for all t⩾T2, |uδ(τ, τ −t, θ−τω, uδ,τ−t)|2 ⩽1+2|βyδ(ω)|2+ 2 e CZ0 −∞ e−νλ1s|f(s+τ)|2+|βyδ(θsω)|2+kβyδ(θsω)k2ds, which implies that, for any t⩾T2, uδ(τ, τ −t, θ−τω, D(τ−t, θ−τω)) ⊆Kδ(τ, ω).(5.33) In addition, by (F2) and (5.27), we can obtain that Kδis tempered in H, which, along with (5.33), completes the proof. 34 Lemma 5.7. Suppose (F1) and (F2) hold. Then for every δ∈(0,1 2],τ∈R,ω∈Ωand D= {D(t, ω) : t∈R, ω ∈Ω}∈D, we have that, for all t⩾T2, kuδ(τ, τ −t, θ−τω, uδ,τ−t)k2⩽21 + e CLδ,V (τ, ω)+2kβyδ(ω)k2+ 2 e CZτ τ−1|f(s)|2ds + 2 e CZ0 −1|βyδ(θsω)|2+kβyδ(θsω)k2+ν|yδ(θsω)Aβ|2ds :=Rδ,V (τ, ω),(5.34) where Lδ,V (τ, ω) = Rδ,H(τ, ω, −1)+ e CZτ τ−1|f(s)|2ds+e CZ0 −1|βyδ(θsω)|2+kβyδ(θsω)k2ds,uδ,τ−t∈ D(τ−t, θ−tω), and e C > 0independent of τ,ω,δand D. Proof. The proof is similar to Lemma 5.4 and the details are omitted. The following lemma is concerned with the convergence of solutions of (5.8) as δ→0. Lemma 5.8. For every T0>0,ω∈Ω, and bounded subset K⊆H, we have for any 0⩽t⩽T0, lim δ→0sup u0∈K|Uδ(t, τ −t, θ−tω, u0)−U0(t, τ −t, θ−tω, u0)|= 0. Proof. It follows from (5.13) and (5.31) that |Uδ(t, τ −t, θ−tω, u0)−U0(t, τ −t, θ−tω, u0)| =|uδ(τ, τ −t, θ−τω, u0)−u0(τ, τ −t, θ−τω, u0)| ⩽|vδ(τ, τ −t, θ−τω, vδ,0)−v0(τ, τ −t, θ−τω, v0)|+|βz(ω)−βyδ(ω)|,(5.35) where vδ,0=u0−βyδ(θ−tω) and v0=u0−βz(θ−tω). By (5.12) and (5.30), we have d dt|vδ−v|2+ 2νkvδ−vk2 =−2FN(kvδ+βyδk)b(vδ+βyδ, vδ+βyδ,Λ) + 2FN(kv0+βzk)b(v0+βz, v0+βz, Λ) + 2βyδ−βz, Λ−2ν(yδ−z)Aβ, Λ,(5.36) where Λ = vδ−v0. For the first and second terms on the right-hand side of (5.36), similar to (4.9), by (2.5), (2.7) and Young’s inequality, we find −2FN(kvδ+βyδk)b(vδ+βyδ, vδ+βyδ,Λ) + 2FN(kv0+βzk)b(v0+βz, v0+βz, Λ) ⩽ν 2kΛk2+e C|Λ|2+e Ckβk2|yδ−z|2.(5.37) For the last two terms of (5.36), we have 2βyδ−βz, Λ−2ν(yδ−z)Aβ, Λ⩽2|β||yδ−z||Λ|+ 2νkβkkΛk|yδ−z| 35 ⩽ν 2kΛk2+e C|Λ|2+e C|β|2+kβk2|yδ−z|2. which, together with (5.36), (5.37) and Gronwall’s inequality, implies that, for any t∈[0, T0], |vδ(τ, τ −t, θ−τω, vδ,0)−v0(τ, τ −t, θ−τω, v0)|2 ⩽|vδ,0−v0|2ee CT0+e C(|β|2+kβk2)Z0 −T0|yδ(θsω)−z(θsω)|2e−e Csds. (5.38) Therefore, by (5.35) and (5.38), we obtain that for any t∈[0, T0], |Uδ(t, τ −t, θ−tω, u0)−U0(t, τ −t, θ−tω, u0)|2 ⩽2|βz(θ−tω)−βyδ(θ−tω)|2ee CT0+ 2|βz(ω)−βyδ(ω)|2 + 2 e C(|β|2+kβk2)ee CT0Z0 −T0|yδ(θsω)−z(θsω)|2ds, which, together with (5.26), shows the desired result. We also need the following compactness result. Lemma 5.9. Suppose (F1) and (F2) hold. Then, for every τ∈Rand ω∈Ω, there exists 0< δ0⩽1 2such that the union [ 0<δ⩽δ0 Aδ(τ, ω)is precompact in H. Proof. Note that Z0 −∞ e−νλ1s|βyδ(θsω)|2+kβyδ(θsω)k2ds = (|β|+kβk2)Zs1 −∞ e−νλ1s|yδ(θsω)|2ds + (|β|+kβk2)Z0 s1 e−νλ1s|yδ(θsω)|2ds, (5.39) where s1<0 will be determined later. By (5.27), we see that there exists s1<0 such that, for any s⩽s1and δ∈(0,1 2], |yδ(θsω)|<|s|. Therefore, one can obtain (|β|+kβk2)Zs1 −∞ e−νλ1s|yδ(θsω)|2ds ⩽M1,(5.40) where M1>0 is independent of δ. It follows from (5.26) that there exists δ0∈(0,1 2] such that, for any 0 < δ ⩽δ0, (|β|+kβk2)Z0 s1 e−νλ1s|yδ(θsω)|2ds ⩽(|β|+kβk2)Z0 s1 e−νλ1s|z(θsω)|2ds. (5.41) Therefore, by (5.26) and (5.39)-(5.41), we have for any 0 < δ ⩽δ0, Rδ,H (τ, ω, 0) ⩽1+2e CM1+ 2|z(ω)|2+ 2 e CZ0 −∞ e−νλ1s|f(s+τ)|2ds 36 + (|β|+kβk2)Z0 s1 e−νλ1s|z(θsω)|2ds := e RH(τ, ω).(5.42) By using (5.26) again, we can deduce that there exists M2=M2(τ, ω)>0, independent of δ, such that Rδ,V (τ, ω)⩽M2.(5.43) Let nuko∞ k=1 ⊆[ 0<δ⩽δ0 Aδ(τ, ω). Then for each k, there is δk∈(0, δ0] such that uk∈Aδk(τ, ω). Let {tk}∞ k=1 be a sequence of numbers with tk→+∞. By the invariance of Aδk, there exists ˆuk∈Aδk(τ−tk, θ−tkω) such that uk=Uδk(tk, τ −tk, θ−tkω, ˆuk). It follows from (5.41) and Lemma 5.6 that ˆuk∈B0(τ−tk, θτ−tkω), where B0(τ, ω) = {u∈H:|u|2⩽ e RH(τ, ω)}. Then by (5.43) and Lemma 5.7, we find that ukis uniformly bounded in V, which, together with the compactness of embedding V ,→H, shows that the sequence ukis precompact in H. The proof is complete. Given δ∈(0,1 2], let IMδ(respectively, PIMδ) be the set of all invariant (respectively, T- periodic invariant) probability measures of (5.8) with parameter δsatisfying that the support of such measure is contained in the pullback attractor. Through the above analysis, we find that IMδ(respectively, PIMδ) is nonempty. The next result is concerned with the limiting behaviors of invariant measures of (5.8) with respect to δ. Theorem 5.3. Suppose (F1) and (F2) hold. Let δn∈(0,1 2]for all n∈Nsatisfying that δn→0 as n→+∞. If µδn t,ω ∈ IMδnfor every t∈Rand ω∈Ω, then there exist a subsequence {δnk}+∞ k=1 of {δn}+∞ n=1 depending on tand ω, and an invariant measure µ0 t,ω of U0such that µδnk t,ω converges weakly to µ0 t,ω as k→+∞. If, in addition, f(t)is a T-periodic function in tand µδn t,ω ∈ PIMδnfor every t∈Rand ω∈Ω, then there exist a subsequence {δnk}+∞ k=1 of {δn}+∞ n=1 depending on tand ω, and a T-periodic invariant measure µ0 t,ω of U0such that µδnk t,ω converges weakly to µ0 t,ω as k→+∞. Proof. The proof follows from Lemmas 5.8 and 5.9. The details are similar to Theorem 5.1 and hence omitted. Acknowledgements The work is partially supported by the NNSF of China (11471190, 11971260), the SDNSF (ZR2014AM002), the CSC (202006220197), the Spanish Ministerio de Ciencia e Innovaci´on under project PDI2021- 122991NB-C21, and Junta de Andaluc´ıa (Spain) under project P18-FR-4509. 37 Conflict of Interest The authors declare that they have no competing interests. References [1] Z. Brze´zniak, E. Motyl and M. Ondrejat, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, Ann. Probab.,45 (2017) 3145-3201. [2] T. Caraballo, B. Guo, N.H Tuan and R. 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