𝐻¹-Random Attractors and Asymptotic Smoothing Effect of Solutions for Stochastic Boussinesq Equations with Fluctuating Dynamical Boundary Conditions

Abstract

This work is concerned with the random dynamics of two-dimensional stochastic Boussinesq system with dynamical boundary condition. The white noises affect the system through a dynamical boundary condition. Using a method based on the theory of omega-limit compactness of a random dynamical system, we prove that the 𝐿²-random attractor for the generated random dynamical system is exactly the 𝐻¹-random attractor. This improves a recent conclusion derived by Brune et al. on the existence of the 𝐿²-random attractor for the same system.

1. Introduction

The Boussinesq equations are a coupled system of the Navier-Stokes equations and the scalar transport equation for fluid salinity, temperature, or density. These Boussinesq equations models various phenomena in environmental, geophysical and climate system, for example, oceanic density currents and the thermohaline circulation; see, for example, [1–3]. In this paper, the scalar quantity in the considered Boussinesq system is salinity, with dynamical boundary condition for the salinity.

Let𝐷⊂ℝ²be a bounded domain with the𝐶¹-smooth boundary𝜕𝐷=Γ, in the vertical plane. Let𝑊₀(𝑡),𝑊₁(𝑡), and𝑊₂(𝑡)be independent two-sided real-valued Wiener processes with values in appropriate function spaces. This paper is mainly concerned with the long time behavior of the solutions to the following Boussinesq equations with general additive noises and fluctuating dynamical boundary conditions [4]: 𝑑𝑢=1𝑑𝑡1ReΔ𝑢−𝑢⋅∇𝑢−∇𝑝−Fr²𝑈𝑒₂+𝑊̇₀on𝐷×ℝ⁺,div𝑢=0on𝐷×ℝ⁺,𝑢=0onΓ×ℝ⁺,𝑢(0)=𝑢₀,𝑑𝑈=1𝑑𝑡RePrΔ𝑈−𝑢⋅∇𝑈+𝑊̇₁on𝐷×ℝ⁺,𝑑𝑈_Γ=𝑑𝑡−𝜕_𝑛𝑈_Γ−𝑐𝑈_Γ+𝑓(𝑥)𝛼+𝑊̇₂onΓ×ℝ⁺,𝛾𝑈=𝑈_Γ,𝑈(0)=𝑈₀, (1) with velocity𝑢=𝑢(𝑥,𝑡)=(𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡))∈ℝ², salinity𝑈=𝑈(𝑥,𝑡)∈ℝ, and pressure𝑝=𝑝(𝑥,𝑡)∈ℝ. In (1),Fris the Froude number;Reis the Reynolds number; andPris the Prandtl number.Δis the Laplacian operator;∇is the gradient operator; div is the divergence operator;𝛾is the trace operator with respect to the boundaryΓ.𝑒₂∈ℝ²is a unit vector in the upward vertical direction (opposite to the gravity). Finally,𝑓(𝑥),𝑥∈Γis a given function describing the mean salinity flux through the boundary;𝑢₀and𝑈₀are the initial conditions;𝜕_𝑛𝑈_Γis the outer normal derivative. Without loss of generality, in this paper we takeFr,Re,Pr,𝑐and𝛼to be 1.

Qualitatively, the Boussinesq system (1) emphasizes the random dynamical boundary condition which modes the interaction of boundary and the domain. In particular, if𝑈describes the temperature then the heat exchange between physical domain and its boundary can be modeled; see, for example, [3]. For this Boussinesq system (1), the large deviation principle via a weak convergence approach is studied in [5], recently. In [4], they derived a priori estimates for the existence of random absorbing sets and showed that the random dynamical system (RDS) generated by the solution of (1) had a random attractor in𝐿²space. When the random dynamical boundary condition in (1) is replaced by the nonhomogeneous boundary condition, [6, 7] proved the existences of random attractors for this system with multiplicative noise and additive noises in𝐿²space, respectively. For the study of the random attractor about other models integrated the Navier-Stokes equations, we refer to [8] for the MHD equations.

In recent years, the theory of random attractors for some concrete dissipative stochastic partial differential equations has been studied by many authors; see, for example, [6, 8–12]; ever since [13, 14] launched their fundamental work on the RDS. Such an attractor, which generalizes nontrivially the global attractors well developed in [15–18] and so forth, is a compact invariant random set which attracts every orbit in the phase space. It is uniquely determined by attracting deterministic compact sets of phase space [19]. In order to obtain the existence of random attractors, one always need to show the existence of a compact random absorbing set in the sense of absorption [20]. This can be achieved by employing the standard Sobolev compact embedding of several functional spaces, when the generated RDSs are defined in some bounded domains; see, for example, [6–10, 20] and references cited there.

In this paper, we are interested in the existence of random attractors of the Boussinesq system (1) in𝐻¹space which is stronger than𝐿²space. It is pointed out that if the initial data belongs to𝐿², the solution to the system (1) enters into the space𝐿²∩𝐻¹and has no higher regularity, see [4]. Hence the Sobolev compact embedding cannot be employed in𝐻¹. Here, we try to overcome the obstacle of compact imbedding by using the notion of omega-limits compactness, which was initiated in [12, 21] in the framework of RDS. This type of compactness is equivalent to the asymptotic compactness [22, 23] in some spaces and can be proved by check the flattening condition; see [21]. More precisely, we prove the existence of the(𝐿²,𝐻¹)-random attractor for this RDS (for the bispaces random attractors; the reader is referred to [11, 16–18]). To this end, a so qualitatively new method is necessary, for instance, to derive a priori estimates of the solutions such that the flattening condition holds in𝐻¹space and then the necessary omega-limits compactness for the RDS in𝐻¹space is followed; see Lemma 12 in Section 4. The main advantage of this technique is that we need not to estimate the solutions in functional spaces of higher regularity to demonstrate the existence of compact random absorbing set which does not work in this case. This method has been used recently to obtain the existence of random attractors in𝐻¹space for the stochastic reaction-diffusion equations [24, 25].

The conclusion in this study shows that the𝐿²-random attractor is qualitatively an𝐻¹-random attractor. This implies that the solutions to (1) become eventually more smoothing than the initial data.

The outline of this paper is as follows. Section 2 presents some functional settings for the Boussinesq system. Section 3 lists the conditions and the main conclusion of this note. In Section 4, we prove some estimates for the solution orbits inℍand𝕍and then prove our main conclusion.

2. Functional Settings

We recall some function spaces and operators that we will be used in the following discussion.

Let 𝐿²∶=𝐿²(𝐷)²×𝐿²(𝐷)×𝐿²(Γ), (2) endowed with the scalar inner product(⋅,⋅)=(⋅,⋅)_{(𝐿2(𝐷))}²+(⋅,⋅)²_𝐿(𝐷)+(⋅,⋅)²_𝐿(Γ)and with norm denoted by‖⋅‖. This notation is also used to denote the norm in(𝐿²(𝐷))²and𝐿²(𝐷)without any confusion.

We define a functional space𝒱integrated the boundary and also the divergence free condition: 𝒱=𝑢,𝑈,𝑈_Γ∈𝐶^∞(𝐷)²×𝐶^∞(𝐷)×𝐶^∞(Γ);div𝑢=0. (3)

Define 𝐻¹_Γ=𝐻¹₀(𝐷)²×𝐻¹(𝐷)×𝐻^1/2(Γ), (4) where(𝐻¹₀(𝐷))²is the usual Sobolev space with equivalent norm‖∇⋅‖and𝐻¹(𝐷)is also the usual Sobolev space with the equivalent norm: ‖𝑈‖_𝐻¹=‖∇𝑈‖²_𝐻1+‖𝑈‖²_𝐿2(Γ)^1/2,𝑈∈𝐻¹(𝐷); (5) see [15, page 52], and𝐻^1/2(Γ)is given by𝛾(𝐻¹(𝐷))endowed by a norm‖𝜙‖_𝐻1/2=inf_{𝛾𝑈=𝜙}‖𝑈‖_𝐻¹; see [15, page 48].

Letℍthe closure of𝒱in𝐿²and𝕍be the closure of𝒱in𝐻¹_Γ, and the norm in𝒱being denoted by ‖⋅‖_𝕍. From the above argument, the space𝕍is equipped with the norm: ‖𝐔‖_𝕍=‖∇𝑢‖²+‖∇𝑈‖²+‖𝑈‖²_𝐿2(Γ)^1/2,𝐔=𝑢,𝑈,𝑈_Γ∈𝕍, (6) which is equivalent to the natural norm:(‖∇𝑢‖²+‖𝑈‖²_𝐻1(𝐷)+‖𝑈‖²_𝐻1/2(Γ))^1/2.

For𝐔=(𝑢,𝑈,𝑈_Γ),𝐕=(𝑣,𝑉,𝑉_Γ), we define the operators 𝐴₁𝐴𝑢=−Δ𝑢,₂𝑈,𝑈_Γ=−Δ𝑈,𝜕_𝑛𝑈_Γ+𝑈_Γ,𝐵₁()𝐵𝑢,𝑣=𝑢⋅∇𝑣,₂()𝐴𝑢,𝑉=𝑢⋅∇𝑉,𝐴𝐔=₁𝑢,𝐴₂𝑈,𝑈_Γ,𝐵(𝐔,𝐕)=𝐵₁(𝑢,𝑣),𝐵₂(𝑢,𝑉),0,𝐹(𝐔)=−𝑈𝑒₂,0,𝑓(𝑥),𝑓∈𝐿²(Γ). (7)(8)

By Lemma  2.2 in [4], the operator𝐴is a positive self-adjoint unbounded operator and has the Poincaré inequality: (𝐴𝐔,𝐔)≥𝜆‖𝐔‖²,𝜆>0. (9) Then𝐴⁻¹is also self-adjoint but compact operator inℍ, so we can utilize the elementary spectral theory in a Hilbert space. We infer that there exists a compete orthonormal family inℍ, also in𝕍,{𝐞_𝑗}^∞_𝑗=1of eigenvectors of𝐴. The corresponding spectrum of𝐴is discrete and denoted by{𝜆_𝑗}^∞_𝑗=1which are positive, increasing, and tend to infinity as𝑗→∞.

In particular, we also can use the spectrum theory to allow us to define the operator𝐴^𝑠, the power of𝐴. For𝑠>0, the operator𝐴^𝑠is also an strictly positive and self-adjoint unbounded operator inℍwith a dense domain𝐷(𝐴^𝑠)⊂ℍ. This allows us to introduce the function spaces, 𝐷𝐴^𝑠=𝐕=^∞_𝑗=1𝐕,𝐞_𝑗𝐞_𝑗∶‖𝐕‖²_𝐷(𝐴^𝑠₎=^∞_𝑗=1𝐕,𝐞_𝑗²𝜆_𝑗^2𝑠<+∞. (10) This norm‖⋅‖_𝐷(𝐴^𝑠₎on𝐷(𝐴^𝑠)is equivalent to the usual norm induced by𝐻^2𝑠; see Temam [15] for details. In particular,𝐷(𝐴⁰)=ℍand𝐷(𝐴^1/2)=𝕍.

Based on the orthonormal basis{𝐞_𝑗}^∞_𝑗=1of eigenfunctions of𝐴, we define the𝑚-dimensional subspace𝕍_𝑚=span{𝐞₁,𝐞₂,…,𝐞_𝑚}⊂𝕍and the canonical orthogonal projection𝑃_𝑚∶𝕍↦𝕍_𝑚such that for every𝐕∈𝕍,𝐕has a unique decomposition:𝐕=𝑃_𝑚𝐕+𝐕_𝑚, where 𝑃_𝑚𝐕=^𝑚_𝑗=1𝐕,𝐞_𝑗𝐞_𝑗∈𝕍_𝑚,𝐕_𝑚=𝐼−𝑃_𝑚𝐕=^∞_𝑗=𝑚+1𝐕,𝐞_𝑗𝐞_𝑗∈𝕍^⟂_𝑚; (11) that is,𝕍=𝕍_𝑚⊕𝕍^⟂_𝑚.

When the projection𝑃_𝑚operates on the first component of𝐕=(𝑣,𝑉,𝑉_Γ), one can easily show that 𝐴₁𝑃_𝑚𝑣²≤𝜆_𝑚+1𝐴₁^1/2𝑃_𝑚𝑣²,∀𝐕=𝑣,𝑉,𝑉_Γ∈𝐷(𝐴) (12) for𝑚∈𝑁⁺, the nature numbers set.

We also state the well-known Brezis-Gallouet’s inequality with Dirichlet boundary condition in two dimension case; see [26] or Proposition  3 in [27]; there exists a positive constant𝑐such that ‖𝑣‖_{(𝐿∞(𝐷))}²≤𝑐𝐴₁^1/2𝑣𝐴1+log¹𝑣²𝜆₁𝐴₁^1/2𝑣^21/2∀𝐕=𝑣,𝑉,𝑉_Γ∈𝐷(𝐴), (13) where𝜆₁is the first eigenvalue of the Stokes operator𝐴₁. By (12) we have 𝑃_𝑚𝑣_{(𝐿∞(𝐷))}²≤𝑐𝐴₁^1/2𝑃_𝑚𝑣𝐴1+log¹𝑃_𝑚𝑣²𝜆₁𝐴₁^1/2𝑃_𝑚𝑣^21/2≤𝑐‖𝑣‖_𝐻¹𝜆1+log^𝑚+1𝜆₁^1/2,∀𝐕=𝑣,𝑉,𝑉_Γ∈𝐷(𝐴). (14)

According to the above notations, we can write (1) as the following abstract evolution equation form: ()(𝐔)𝐔()𝑑𝐔+𝐴𝐔𝑑𝑡+𝐵𝐔,𝐔𝑑𝑡=𝐹𝑑𝑡+𝑑𝐖,𝑥,0=𝐔₀(𝜔)∈ℍ, (15) where𝐔=(𝑢,𝑈,𝑈_Γ).

3. Existence of Random Attractor in 𝕍

In order to model the noise in the initial problem (1), we need to define a metric dynamical system (MDS) which is a group of measure preserving transformations on a probability space. For the definition of the MDS we refer to [28, 29] and so forth.

A standard model for a spatially correlated noise is the generalized time derivative of a two-sidedBrownian motion𝜔=𝜔(𝑥,𝑡),𝑥∈ℝ². Letℍbe the separable Hilbert space defined in Section 2. As usual, we introduce the spatially valued Brownian motion MDS 𝜃=(Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ), where Ω={𝜔∈𝐶₀(ℝ,ℍ)∶𝜔(0)=0}with compact open topology. This topology is metrizable by the complete metric: 𝑑𝜔₁,𝜔₂=^∞_𝑛=112^𝑛𝑑_𝑛𝜔₁,𝜔₂1+𝑑_𝑛𝜔₁,𝜔₂, (16) where𝑑_𝑛(𝜔₁,𝜔₂)=max_{|𝑡|≤𝑛}|𝜔₁−𝜔₂|for𝜔₁and𝜔₂inΩ.ℱ=ℬ(𝐶₀(ℝ,ℍ))is the Borel-𝜎-algebra induced by the compact open topology ofΩ. Suppose the Wiener process𝜔has covariance operator𝑄. Letℙbe the Wiener measure with respect to𝑄. The Wiener shift is defined by 𝜃_𝑠𝜔(𝑡)=𝜔(𝑡+𝑠)−𝜔(𝑠),𝜔∈Ω,𝑡,𝑠∈ℝ. (17) Then the measureℙis ergodic and invariant with respect to the shift𝜃.

The associated probability space defines a canonical Wiener process𝑊. We also note that such a Wiener process𝑊generates a filtration(ℱ_𝑡)_𝑡∈ℝ: ℱ_𝑡≡{𝑊(𝜏)∣𝜏≤𝑡}⊂ℱ. (18)

We introduce the following stochastic partial differential equation on𝐷: 𝑑𝐙+𝐴𝐙𝑑𝑡=𝑑𝑊. (19) Because𝐴is a positive and self-adjoint operator, there exists a mild solution to this stochastic equation with the form 𝐙(𝑡)=𝐙(0)+^𝑡₀𝑒^{−(𝑡−𝜏)𝐴}𝑑𝑊,𝑡>0, (20) which is called an Ornstein-Uhlenbeck process; see [30]. For the Ornstein-Uhlenbeck process we have the regularity hypothesis; see also [4].

Lemma 1. Suppose that the covariance operator𝑄of the Wiener process𝜔has a finite trace; that is,𝑄satisfies that tr_ℍ𝑄𝐴^{2𝑠−1+𝛿}=tr_ℍ𝐴^{𝑠−1/2+𝛿/2}𝑄𝐴^{𝑠−1/2+𝛿/2}<+∞, (21) for some𝑠≥0and some (arbitrary small)𝛿>0, wheretr_ℍdenotes the trace of the covariance operator. Then anℱ₀-measurable Gaussian variable𝐙=(𝑧,𝑍)∈𝐷(𝐴^𝑠)exists, and the process (𝑡,𝜔)→𝐙(𝜃_𝑡𝜔)is a continuous stationary solution to the stochastic equation (19). Furthermore, the random variable‖𝐙(𝜔)‖²_𝐷(𝐴^𝑠₎is tempered and the expectation 𝔼‖𝐙‖²_𝐷(𝐴^𝑠₎=12tr_ℋ𝐴^𝑠−1/2𝑄𝐴^𝑠−1/2<+∞. (22)

Let𝐕=𝐔−𝐙(𝜃_𝑡𝜔); by (15) and (19) we have the following deterministic equation with a parameter𝜔: 𝑑𝜃𝑑𝑡𝐕+𝐴𝐕+𝐵𝐕+𝐙_𝑡𝜔,𝐕+𝐙𝜃_𝑡𝜔=𝐹𝐕+𝐙𝜃_𝑡𝜔,𝐕(𝑥,0)=𝐕₀(𝜔)=𝐔₀(𝜔)−𝐙(𝜔)∈ℍ. (23)

Here we denote the solution to (23) by𝐕(𝑡,𝜔,𝐕₀(𝜔)),𝐕(𝑡,𝜔)or more briefly𝐕(𝑡). By Lemma  4.7 in [4], the solution of the evolution equation (23) generates a continuous measurable RDS𝜓inℍgiven by 𝜓(𝑡,𝜔)𝐕₀(𝜔)=𝐕(𝑡,𝜔,𝐕₀(𝜔)). Put𝐔(𝑡,𝜔,𝐔₀(𝜔))=𝐕(𝑡,𝜔,𝐔₀(𝜔)−𝐙(𝜔))+𝐙(𝜃_𝑡𝜔). Then 𝐔(𝑡,𝜔,𝐔₀(𝜔))or briefly𝐔(𝑡)is a solution to (23) with initial value𝐔₀(𝜔).

Given 𝜑(𝑡,𝜔)𝐔₀(𝜔)=𝐔𝑡,𝜔,𝐔₀(𝜔)=𝐕𝑡,𝜔,𝐔₀(𝜔)(𝜔)𝜃−𝐙+𝐙_𝑡𝜔,𝜔∈Ω, (24) then𝜑is also a continuous measurable RDS inℍfor the original equation (15), that is, (1).

As for the general theory of random dynamical systems we may refer to [28, 31] for details.

The main conclusion of this study states the following.

Theorem 2. One supposes that (21) holds. Set that 𝑀(𝜔)=𝐾^𝜖‖𝑍(𝜔)‖²_𝐻2+𝐾^𝜖‖𝑧(𝜔)‖²_𝐻1−𝜆, (25) where𝐾^𝜖is a generic constant depending on the data of the problem and some𝜖that has to be chosen sufficiently small and𝜆is the same as in (9). Assume that the mathematical expectation of𝑀𝔼𝑀<0. (26) Then the RDS𝜑generated by (1) admits a unique random attractor{𝒜_𝕍(𝜔)}_𝜔∈Ωin𝕍in the sense that forℙ-a.s.𝜔∈Ω, 𝒜_𝕍(𝜔)𝑖𝑠𝑐𝑜𝑚𝑝𝑎𝑐𝑡𝑖𝑛𝕍,𝜑(𝑡,𝜔)𝒜_𝕍(𝜔)=𝒜_𝕍𝜃_𝑡𝜔,𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡≥0,lim_𝑡→∞dist_𝕍𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔,𝒜_𝕍(𝜔)∀{𝐵(𝜔=0,)}_𝜔∈Ω∈𝒟, (27)(28)(29) where dist_𝕍 denotes the Hausdorff semiistance in𝕍and𝒟is the collection of tempered subsets ofℍas in [4]. Furthermore,{𝒜_𝕍(𝜔)}_𝜔∈Ωis identical with the random attractor{𝒜_ℍ(𝜔)}_𝜔∈Ωinℍ.

By the well-known abstract result of Theorem  3.5 in [20], the existence of a random attractor{𝒜_ℍ(𝜔)}_𝜔∈Ωin the following sense that 𝒜_ℍ(𝜔)iscompactin𝜑(ℍ,t,𝜔)𝒜_ℍ(𝜔)=𝒜_ℍ𝜃_𝑡𝜔,forevery𝑡≥0,lim_𝑡→∞dist_ℍ𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔,𝒜_ℍ(𝜔)=0forall{𝐵(𝜔)}_𝜔∈Ω∈𝒟, (30) has been obtained in [4]. However, Our Theorem 2 shows the existence of random attractor which is compact in the space𝕍, which is stronger thanℍ. Thus the Sobolev compact embedding theorem is not available and it seems impossible to obtain a compact random absorbing set in the space𝕍when the initial data belongs toℍ. Then Theorem  3.5 in [20] is unapplicable to the proof of our Theorem 2.

Fortunately, Theorem 2 can be proved by checking the omega-limit compactness in the space𝕍, which is based on the viewpoint of Kuratowski measure of noncompactness. This kind of compactness can be easily obtained by showing the flattening condition, see [21].

For clarification, we state some general concepts used in the sequel. Let𝑋be a Banach space with norm ‖⋅‖_𝑋and𝒟be the collection of all random subsets of𝑋.

Definition 3 (see [21]). An RDS𝜑on𝑋over an MDS is said to be omega-limit compact if for every𝜀>0and𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟, there exists𝑇=𝑇(𝜀,𝐵,𝜔)such that for all𝑡≥𝑇, 𝑘_𝑡≥𝑇𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔≤𝜀,ℙ-a.s.𝜔∈Ω, (31) where𝑘(𝐵)is the Kuratowski measure of noncompactness of𝐵defined by 𝑘(𝐵)=inf{𝑑>0∶𝐵admitsafinitecoverbysetsofdiameter≤𝑑. (32)

Definition 4 (see [21]). An RDS𝜑on𝑋over an MDS is said to possess the flattening condition if forℙ-a.s.𝜔∈Ωand every𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟, there exist𝑇=𝑇(𝐵,𝜔,𝜀)>0and a finite dimensional space𝑋₁of𝑋such that for a bounded projector𝑃∶𝑋→𝑋₁, 𝑃_𝑡≥𝑇𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔isboundedin)𝑋,(𝐼−𝑃_𝑡≥𝑇𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔_𝑋≤𝜀. (33)

Definition 5 (see [29]). (i) A random variable𝑋∈ℝ⁺over an MDS is tempered if forℙ-a.s.𝜔∈Ω, lim_𝑡→±∞log⁺𝑋𝜃_𝑡𝜔|𝑡|=0. (34)
(ii) A random set𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟is called tempered if𝑅(𝜔)=sup_{𝑥∈𝐵(𝜔)}‖𝑥‖_𝑋is a tempered random variable.

Definition 6. A random set{𝐾(𝜔)}_𝜔∈Ωis an𝒟-random absorbing set for RDS𝜑over an MDS if forℙ-a.s.𝜔∈Ωand every𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟, there exists𝑇=𝑇(𝐵,𝜔)>0such that for all𝑡≥𝑇, 𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔⊂𝐾(𝜔), (35) where𝜑(𝑡,𝜃_−𝑡𝜔)𝐵(𝜃_−𝑡𝜔)=⋃_{𝑥∈𝐵(𝜃−𝑡𝜔)}𝜑(𝑡,𝜃_−𝑡𝜔)𝑥.

For our problem,𝑋=𝕍and𝒟is the collection of tempered subsets ofℍ. The finite dimensional space 𝑋₁=𝕍_𝑚=span{𝐞₁,𝐞₂,…,𝐞_𝑚}and the bounded operator𝑃=𝑃_𝑚for a sufficient large𝑚, where 𝐞_𝑖(𝑖=1,2,…,𝑚)and𝑃_𝑚are defined in Section 2.

4. The Proofs of Main Result

To prove Theorem 2, we need a series of lemmas. First we give a useful lemma similar to the classic Gronwall lemma (see [15]).

Lemma 7. Assume that𝑦,𝑦^′, andℎare three locally integrable functions and𝑦,ℎnonnegative on[𝑎,∞)such that for𝑠≥𝑎, 𝑦^′(𝑠)+𝑏𝑦(𝑠)≤ℎ(𝑠). (36) Then for every𝑡≥𝑎and any positive constant𝑟, one has 𝑦(𝑡+𝑟)≤𝑒^−𝑏𝑟_𝑡^𝑡+𝑟𝑦(𝑠)𝑒^{𝑏(𝑠−𝑡)}𝑑𝑠+_𝑡^𝑡+𝑟ℎ(𝑠)𝑒^{𝑏(𝑠−𝑡)}𝑑𝑠. (37)

Proof. Let𝑡≤𝑠≤𝑡+𝑟. We multiply (36) by𝑒^{𝑏(𝑠−𝑡)}and the resulting inequality reads 𝑑𝑦(𝑠)𝑒𝑑𝑠^{𝑏(𝑠−𝑡)}≤ℎ(𝑠)𝑒^{𝑏(𝑠−𝑡)}. (38) Hence, by integration from𝑠to𝑡+𝑟, 𝑦(𝑡+𝑟)𝑒^𝑏𝑟≤𝑦(𝑠)𝑒^{𝑏(𝑠−𝑡)}+_𝑡^𝑡+𝑟ℎ(𝑠)𝑒^{𝑏(𝑠−𝑡)}𝑑𝑠. (39) Integrating the above inequality with respect to𝑠between𝑡and𝑡+𝑟, we get the desired inequality.

Lemma 8. There exist positive constants𝐾and𝐾^𝜖such that the followings hold: ‖𝐕(𝑡)‖²≤𝑒^{∫𝑡0𝑀(𝜃𝜏𝜔)𝑑𝜏}𝐾^𝜖𝐕₀(𝜔)²+𝐾^𝜖𝑡₀𝐺𝜃_𝑠𝜔𝑒^{∫𝑡𝑠𝑀(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠,𝑑‖𝐕‖𝑑𝑡²+𝐾‖𝐕‖²_𝕍≤𝐾^𝜖𝑍𝜃_𝑡𝜔²_𝐻2+𝑧𝜃_𝑡𝜔²_𝐻1‖𝑣‖²+𝐾^𝜖‖𝑉‖²+𝐺𝜃_𝑡𝜔,𝑑‖𝐕‖𝑑𝑡^2𝕍≤𝑔(𝑡,𝜔)‖𝐕‖²_𝕍+ℎ(𝑡,𝜔), (40)(41)(42) where 𝑀(𝜔)=𝐾^𝜖‖𝑍(𝜔)‖²_𝐻2+𝐾^𝜖‖𝑧(𝜔)‖²_𝐻1𝐺(𝜔)−𝜆,=𝐾^𝜖+𝐾^𝜖‖𝑧(𝜔)‖²‖𝑍(𝜔)‖²_𝐻2+𝐾^𝜖‖𝑧(𝜔)‖²_𝐻1‖𝑧(𝜔)‖²_𝐻2+𝐾^𝜖‖𝑍(𝜔)‖²_𝐻2,𝑔(𝑡,𝜔)=𝐾‖𝑣(𝑡,𝜔)‖²‖𝑣(𝑡,𝜔)‖²_𝐻1+𝐻₁𝜃_𝑡𝜔,ℎ(𝑡,𝜔)=𝐾‖𝑉(𝑡,𝜔)‖²+𝐻₂𝜃_𝑡𝜔,𝐻₁(𝜔)=𝐾^𝜖‖𝑧(𝜔)‖²_𝐻2+𝐾^𝜖‖𝑍(𝜔)‖²_𝐻2,𝐻₂(𝜔)=𝐾^𝜖‖𝑧(𝜔)‖²_𝐻2‖𝑍(𝜔)‖²_𝐻1+𝐾^𝜖‖𝑧(𝜔)‖²_𝐻1‖𝑧(𝜔)‖²_𝐻2+𝐾‖𝑍(𝜔)‖²+𝐾^𝜖. (43)(44)

Proof. The inequality (40) is the same as the inequality(18)at page  1112 in [4]. The inequality (41) is a combination of the formulas(15)and(16)at pages 1110-1111 in [4] with a tiny modification. Following a same calculation as page  1114 in [4] we can obtain (42).

Lemma 9. Assume that𝔼𝑀<0. Let𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟. Then forℙ-a.s.𝜔∈Ω, there exist random radii𝜚₁(𝜔)and𝜚₁(𝜔)and constant𝑇=𝑇(𝐵,𝜔)>0such that for all𝑡≥𝑇, the solution 𝐕(𝑡,𝜔,𝐔₀(𝜔)−𝐙(𝜔))of problem (23) with𝐔₀∈𝐵(𝜔)satisfies that for every𝑙∈[𝑡,𝑡+1]: 𝐕𝑙,𝜃_−𝑡−1𝜔,𝐔₀(𝜔)−𝐙(𝜔)≤𝜚₁(𝜔),_𝑡𝑡+1𝐕𝑠,𝜃_−𝑡−1𝜔,𝐔₀(𝜔)−𝐙(𝜔)²_𝕍𝑑𝑠≤𝜚₂(𝜔), (45)(46) where𝒟is the collection of tempered random subsets ofℍ.

Proof. We replace𝑡with𝑙, and𝜔with𝜃_−𝑡−1𝜔in (40) to produce that for𝑙∈[𝑡,𝑡+1], 𝐕𝑙,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²≤𝑒^{∫𝑙0𝑀(𝜃𝜏−𝑡−1𝜔)𝑑𝜏}𝐾^𝜖𝐕₀(𝜃_−𝑡−1𝜔)²+𝐾^𝜖𝑙₀𝐺𝜃_{𝑠−𝑡−1}𝜔𝑒^{∫𝑙𝑠𝑀(𝜃𝜏−𝑡−1𝜔)𝑑𝜏}𝑑𝑠=𝑒^{∫𝑙−𝑡−1−𝑡−1𝑀(𝜃𝜏𝜔)𝑑𝜏}𝐾^𝜖𝐕₀𝜃_−𝑡−1𝜔²+𝐾^{𝜖𝑙−𝑡−1}_−𝑡−1𝐺𝜃_𝑠𝜔𝑒^{∫𝑠𝑙−𝑡−1𝑀(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠≤𝑒^{∫𝑙−𝑡−1−𝑡−1𝑀(𝜃𝜏𝜔)𝑑𝜏}𝐾^𝜖×𝐕₀𝜃_−𝑡−1𝜔²+𝐙₀𝜃_−𝑡−1𝜔²+𝐾^𝜖0_−𝑡−1𝐺𝜃_𝑠𝜔𝑒^{∫𝑠𝑙−𝑡−1𝑄(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠. (47) By noting that for𝑙∈[𝑡,𝑡+1],𝑙−𝑡−1∈[−1,0], then we find that for𝑠∈[−𝑡−1,0], 𝑒^{∫𝑠𝑙−𝑡−1𝑀(𝜃𝜏𝜔)𝑑𝜏}=𝑒^{−𝜆1(𝑙−𝑡−1)+𝜆1𝑠}×𝑒^{∫𝑠𝑙−𝑡−1(𝐾𝜖‖𝑍(𝜃𝜏𝜔)‖2𝐻2+𝐾𝜖‖𝑧(𝜃𝜏𝜔)‖2𝐻1)𝑑𝜏}≤𝑒^{𝜆1+𝜆1𝑠}𝑒^{∫0𝑠(𝐾𝜖‖𝑍(𝜃𝜏𝜔)‖2𝐻2+𝐾𝜖‖𝑧(𝜃𝜏𝜔)‖2𝐻1)𝑑𝜏}≤𝑒^𝜆1𝑒^{∫0𝑠𝑀(𝜃𝜏𝜔)𝑑𝜏}, (48) from which and (47) it follows that for every𝑙∈[𝑡,𝑡+1], 𝐕𝑙,𝜃_−𝑡−1𝜔,𝐕₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²≤𝑒^{∫0−𝑡−1𝑀(𝜃𝑠𝜔)𝑑𝑠}𝐾^𝜖𝐕₀𝜃_−𝑡−1𝜔²+𝐙𝜃_−𝑡−1𝜔²+𝐾^𝜖0_−𝑡−1𝐺𝜃_𝑠𝜔𝑒^{∫0𝑠𝑀(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠. (49) By the Birkhoff’s ergodic theorem and along with our assumption (26), it yields that lim_𝑡→±∞1𝑡+1⁰_−𝑡−1𝑀𝜃_𝜏𝜔𝑑𝜏=𝔼𝑀(𝜔)<0, (50) and then it implies that 𝑒^{∫0−𝑡−1𝑀(𝜃𝜏𝜔)𝑑𝜏}≈𝑒^{(𝑡+1)𝔼𝑀}⟶0,as𝑡⟶+∞. (51) As𝐺(𝜃_𝜏)is subexponential growth, so ⁰_−∞𝐺𝜃_𝑠𝜔𝑒^{∫0𝑠𝑀(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠<+∞. (52) Note that‖𝐙(𝜔)‖²is also tempered random variable and𝐔₀(𝜔)∈𝐵(𝜔)with𝐵∈𝒟, whence there exists constant𝑇=𝑇(𝐵,𝜔)>0such that for all𝑡≥𝑇with𝑙∈[𝑡,𝑡+1], 𝐕𝑙,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²≤𝜚²₁(𝜔)∶=𝐾^𝜖1+⁰_−∞𝐺𝜃_𝑠𝜔𝑒^{∫0𝑠𝑀(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠. (53) Furthermore,𝜚₁(𝜔)is tempered; see [4]. This completes the proof of (45).
Next, we show (46). Integrating (41) from𝑡to𝑡+1with𝑡≥𝑇, where𝑇is in (53), we get 𝐾_𝑡^𝑡+1𝐕(𝑠,𝜔,𝐔₀(𝜔)−𝐙(𝜔))²_𝕍𝑑𝑠≤𝐾^𝜖_𝑡^𝑡+1𝑍𝜃_𝑠𝜔²_𝐻2+𝑧𝜃_𝑠𝜔²_𝐻1×𝑣𝑠,𝜔,𝑣₀(𝜔)−𝑧(𝜔)²𝑑𝑠+𝐾^𝜖_𝑡^𝑡+1𝑉𝑠,𝜔,𝑉₀(𝜔)−𝑍(𝜔)²+𝑑𝑠_𝑡𝑡+1𝐺𝜃_𝑠𝜔𝑑𝑠+𝐾𝐕𝑡,𝜔,𝑉₀(𝜔)−𝑍(𝜔)². (54) Then by (45) we get that for all𝑡≥𝑇, _𝑡^𝑡+1𝐕𝑠,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍𝑑𝑠≤𝐾^𝜖𝜚²₁(𝜔)_𝑡^𝑡+1𝑍𝜃_{𝑠−𝑡−1}𝜔²_𝐻2+𝑧𝜃_{𝑠−𝑡−1}𝜔²_𝐻1+𝑑𝑠_𝑡𝑡+1𝐺𝜃_{𝑠−𝑡−1}𝜔𝑑𝑠+𝐾^𝜖𝜚²₁(𝜔)=𝐾^𝜖𝜚²₁(𝜔)⁰₋₁𝑍𝜃_𝑠𝜔²_𝐻2+𝑧𝜃_𝑠𝜔²_𝐻1+𝑑𝑠₀₋₁𝐺𝜃_𝑠𝜔𝑑𝑠+𝐾^𝜖𝜚²₁(𝜔)∶=𝜚₂(𝜔), (55) which gives an expression for𝜚₂(𝜔).

Lemma 10. Assume that𝔼𝑀<0. Let𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟. Then forℙ-a.s.𝜔∈Ω, there exist random radium𝑅(𝜔)and constant𝑇=𝑇(𝐵,𝜔)>0such that for all𝑡≥𝑇, the solution𝐕(𝑡,𝜔,𝐔₀(𝜔)−𝐙(𝜔))of problem (23) with𝐔₀∈𝐵(𝜔)satisfies that for every𝑙∈[𝑡,𝑡+1], 𝐕𝑙,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔_𝕍≤𝑅(𝜔), (56) where𝒟is the collection of tempered random subsets ofℍ.

Proof. Using the classic Gronwall’s lemma (see [15]) to the inequality (42) on the interval[𝑠,𝑙]with 𝑡≤𝑠≤𝑙≤𝑡+1, we get that 𝐕𝑙,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍≤𝑒^{∫𝑡𝑡+1𝑔(𝜏,𝜃−𝑡−1𝜔)𝑑𝜏}×_𝑡^𝑡+1𝐕𝑠,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍+𝑑𝑠_𝑡𝑡+1ℎ𝑠,𝜃_−𝑡−1𝜔𝑑𝑠. (57) Note that by Lemma 9 there exists𝑇=𝑇(𝐵,𝜔)such that for all𝑡≥𝑇, _𝑡^𝑡+1𝑔𝜏,𝜃_−𝑡−1𝜔=𝑑𝜏_𝑡𝑡+1𝐾𝑣𝑠,𝜃_−𝑡−1𝜔²𝑣𝑠,𝜃_−𝑡−1𝜔²_𝐻1+𝐻₁𝜃_{𝑠−𝑡−1}𝜔𝑑𝑠≤𝐾𝜚²₁(𝜔)𝜚²₂(𝜔)+⁰₋₁𝐻₁𝜃_𝑠𝜔𝑑𝑠∶=𝐶₁(𝜔),_𝑡𝑡+1ℎ𝑠,𝜃_−𝑡−1𝜔=𝑑𝑠_𝑡𝑡+1𝐾𝑉𝑠,𝜃_−𝑡−1𝜔²+𝐻₂𝜃_{𝑠−𝑡−1}𝜔𝑑𝑠≤𝐾𝜚²₁(𝜔)+⁰₋₁𝐻₂𝜃_𝑠𝜔𝑑𝑠∶=𝐶₂(𝜔). (58) Then by (57) and (58) it gives that for all𝑡≥𝑇and𝑙∈[𝑡,𝑡+1], 𝐕𝑙,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍≤𝑒^𝐶1(𝜔)𝐾𝜚₂(𝜔)+𝐶₂(𝜔)∶=𝑅²(𝜔), (59) which gives an expression for𝑅(𝜔).

Lemma 11. Assume that𝔼𝑀<0. Let𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟. Then forℙ-a.s.𝜔∈Ωand every𝜀>0, there are𝑁=𝑁(𝜀,𝜔),𝐾=𝐾(𝜔), and𝑇=𝑇(𝜀,𝐵,𝜔)>0such that for all𝑡≥𝑇and𝑚≥𝑁, the solution 𝐕(𝑡,𝜔,𝐔₀(𝜔)−𝐙(𝜔))of problem (23) with𝐔₀∈𝐵(𝜔)satisfies that 𝑃_𝑚𝐕𝑡,𝜃_−𝑡𝜔,𝐔₀𝜃_−𝑡𝜔−𝐙𝜃_−𝑡𝜔_𝕍𝐕≤𝐾,_𝑚𝑡,𝜃_−𝑡𝜔,𝐔₀𝜃_−𝑡𝜔−𝐙𝜃_−𝑡𝜔_𝕍≤𝜀, (60) where𝒟is the collection of tempered random subsets ofℍ.

Proof . Multiplying (23) by𝐴𝐕_𝑚with respect to the𝐿²inner product leads us to 12𝑑𝐕𝑑𝑡^𝑚2𝕍+𝐴𝐕_𝑚²=𝐵₁𝑣+𝑧𝜃_𝑡𝜔,𝑣+𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚+𝐵₂𝑣+𝑧𝜃_𝑡𝜔,𝑉+𝑍𝜃_𝑡𝜔,𝐴₂𝑉_𝑚+𝑒₂𝑉+𝑍𝜃_𝑡𝜔,𝐴₁𝑣_𝑚+𝑓,𝐴₂𝑉_𝑚Γ=𝐼₁+𝐼₂+𝐼₃, (61) where 𝐼₁=𝐵₁𝑣+𝑧𝜃_𝑡𝜔,𝑣+𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚,𝐼₂=𝐵₂𝑣+𝑧𝜃_𝑡𝜔,𝑉+𝑍𝜃_𝑡𝜔,𝐴₂𝑉_𝑚,𝐼₃=𝑒₂𝑉+𝑍𝜃_𝑡𝜔,𝐴₁𝑣_𝑚+𝑓,𝐴₂𝑉_𝑚Γ. (62) In order to estimate𝐼₁, we rewrite it as 𝐼₁=𝐵₁(𝑣,𝑣),𝐴₁𝑣_𝑚+𝐵₁𝑣,𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚+𝐵₁𝑧𝜃_𝑡𝜔,𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚+𝐵₁𝑧𝜃_𝑡𝜔,𝑣,𝐴₁𝑣_𝑚, (63) where by utilizing the inequality (14) and Agmon’s inequality inℝ²(see [15]), it gives that 𝐵₁(𝑣,𝑣),𝐴₁𝑣_𝑚≤𝐵₁𝑃_𝑚𝑣,𝑣,𝐴₁𝑣_𝑚+𝐵₁𝑣_𝑚,𝑣,𝐴₁𝑣_𝑚≤𝑃_𝑚𝑣_𝐿∞‖𝑣‖_𝐻¹𝐴₁𝑣_𝑚+𝑣_𝑚𝐿∞‖𝑣‖_𝐻¹𝐴₁𝑣_𝑚𝜆≤𝐾1+log^𝑚+1𝜆₁^1/2‖𝑣‖²_𝐻1𝐴₁𝑣_𝑚+𝐾^𝜖𝑣_𝑚^1/2𝐴₁𝑣_𝑚^3/2‖𝑣‖_𝐻¹≤𝜖𝐴16¹𝑣_𝑚²+𝐾^𝜖𝜆1+log^𝑚+1𝜆₁‖𝑣‖⁴_𝐻1+𝐾^𝜖‖𝑣‖²‖𝑣‖⁴_𝐻1,𝐵₁𝑣,𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚≤𝐵₁𝑃_𝑚𝑣,𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚+𝐵₁𝑣_𝑚,𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚≤𝑃_𝑚𝑣_𝐿∞𝑧(𝜃_𝑡𝜔)_𝐻1𝐴₁𝑣_𝑚+𝑣_𝑚𝐿∞𝑧(𝜃_𝑡𝜔)_𝐻1𝐴₁𝑣_𝑚𝜆≤𝐾1+log^𝑚+1𝜆₁^1/2‖𝑣‖_𝐻¹𝑧(𝜃_𝑡𝜔)_𝐻1𝐴₁𝑣_𝑚+𝐾𝑣_𝑚^1/2𝐴₁𝑣_𝑚^3/2𝑧(𝜃_𝑡𝜔)²_𝐻1≤𝜖𝐴16¹𝑣_𝑚²+𝐾^𝜖𝜆1+log^𝑚+1𝜆₁‖𝑣‖²_𝐻1𝑧(𝜃_𝑡𝜔)²_𝐻1+𝐾^𝜖‖𝑣‖²𝑧𝜃_𝑡𝜔⁴_𝐻1. (64)(65) Similarly by utilizing the Agmon’s inequality, we deduce that 𝐵₁𝑧𝜃_𝑡𝜔,𝑧𝜃_𝑡𝜔,𝐴₁𝑣_𝑚≤𝜖𝐴16¹𝑣_𝑚²+𝐾^𝜖𝑧(𝜃_𝑡𝜔)²_𝐻2𝑧(𝜃_𝑡𝜔)²_𝐻1,𝐵₁𝑧𝜃_𝑡𝜔,𝑣,𝐴₁𝑣_𝑚≤𝜖𝐴16¹𝑣_𝑚²+𝐾^𝜖𝑧𝜃_𝑡𝜔²_𝐻2‖𝑣‖²_𝐻1. (66) Then by (63)–(66) we get that 𝐼₁≤𝜖4𝐴₁𝑣_𝑚²+𝐾^𝜖𝜆1+log^𝑚+1𝜆₁×‖𝑣‖²_𝐻1𝑧𝜃_𝑡𝜔²_𝐻1+‖𝑣‖⁴_𝐻1+𝐾^𝜖‖𝑣‖²‖𝑣‖⁴_𝐻1+𝑧(𝜃_𝑡𝜔)²_𝐻2‖𝑣‖²_𝐻1+‖𝑣‖²𝑧(𝜃_𝑡𝜔)⁴_𝐻1+𝑧(𝜃_𝑡𝜔)²_𝐻2𝑧(𝜃_𝑡𝜔)²_𝐻1. (67) We then estimate𝐼₂in (61). We first have 𝐼₂=𝐵₂𝑣,𝑍𝜃_𝑡𝜔,𝐴₂𝑉_𝑚+𝐵₂(𝑣,𝑉),𝐴₂𝑉_𝑚+𝐵₂𝑧𝜃_𝑡𝜔,𝑍𝜃_𝑡𝜔,𝐴₂𝑉_𝑚+𝐵₂𝑧𝜃_𝑡𝜔,𝑉,𝐴₂𝑉_𝑚, (68) where 𝐵₂(𝑣,𝑉),𝐴₂𝑉_𝑚≤𝐵₂𝑃_𝑚𝑣,𝑉,𝐴₂𝑉_𝑚+𝐵₂𝑣_𝑚,𝑉,𝐴₂𝑉_𝑚≤𝑃_𝑚𝑣_𝐿∞‖𝑉‖_𝐻¹𝐴₂𝑉_𝑚+𝑣_𝑚𝐿∞‖𝑉‖_𝐻¹𝐴₂𝑉_𝑚≤𝜖𝐴16²𝑉_𝑚²+𝐾^𝜖𝜆1+log^𝑚+1𝜆₁‖𝑣‖²_𝐻1‖𝑉‖²_𝐻1+𝐾^𝜖‖𝑣‖𝐴₁𝑣_𝑚‖𝑉‖²_𝐻1≤𝜖𝐴16²𝑉_𝑚²+𝐾^𝜖𝜆1+log^𝑚+1𝜆₁‖𝑣‖²_𝐻1‖𝑉‖²_𝐻1+𝜖𝐴16¹𝑣_𝑚²+𝐾^𝜖‖𝑣‖²‖𝑉‖⁴_𝐻1,𝐵₂𝑣,𝑍𝜃_𝑡𝜔,𝐴₂𝑉_𝑚≤𝜖𝐴16²𝑉_𝑚²+𝐾^𝜖𝜆1+log^𝑚+1𝜆₁‖𝑣‖²_𝐻1𝑍(𝜃_𝑡𝜔)²_𝐻1+𝜖𝐴16¹𝑣_𝑚²+𝐾^𝜖‖𝑣‖²𝑍(𝜃_𝑡𝜔)⁴_𝐻1,𝐵₂𝑧𝜃_𝑡𝜔,𝑍𝜃_𝑡𝜔,𝐴₂𝑉_𝑚≤𝜖𝐴16²𝑉_𝑚²+𝐾^𝜖𝑧𝜃_𝑡𝜔²_𝐻2𝑍(𝜃_𝑡𝜔)²_𝐻1,𝐵₂𝑧𝜃_𝑡𝜔,𝑉,𝐴₂𝑉_𝑚≤𝜖𝐴16²𝑉_𝑚²+𝐾^𝜖𝑧(𝜃_𝑡𝜔)²_𝐻2‖𝑉‖²_𝐻1. (69) Then it follows from (68) and (69) that 𝐼₂≤𝜖4𝐴₂𝑉_𝑚²+𝜖8𝐴₁𝑣_𝑚²+𝐾^𝜖𝜆1+log^𝑚+1𝜆₁×‖𝑣‖²_𝐻1‖𝑉‖²_𝐻1+‖𝑣‖²_𝐻1𝑍𝜃_𝑡𝜔²_𝐻1+𝐾^𝜖‖𝑣‖²‖𝑉‖⁴_𝐻1+‖𝑣‖²𝑍(𝜃_𝑡𝜔)⁴_𝐻1+𝐾^𝜖𝑧(𝜃_𝑡𝜔)²_𝐻2‖𝑉‖²_𝐻1+𝑧𝜃_𝑡𝜔²_𝐻2𝑍𝜃_𝑡𝜔²_𝐻1. (70) Moreover, 𝐼₃≤𝜖8𝐴₁𝑣_𝑚²+𝐾^𝜖‖𝑉‖²+𝐾^𝜖𝑍𝜃_𝑡𝜔²+𝜖4𝐴₂(𝑉_𝑚,𝑉_𝑚)_Γ²+𝐾^𝜖𝑓². (71) Then by (67), (70), and (71), formula (61) becomes 𝑑𝐕𝑑𝑡^𝑚2𝕍+𝜆_𝑚+1𝐕_𝑚²_𝕍≤𝜆1+log^𝑚+1𝜆₁𝑃(𝑡,𝜔)+𝑄(𝑡,𝜔), (72) where 𝑃(𝑡,𝜔)=𝐾^𝜖‖𝑣‖⁴_𝐻1+‖𝑣‖²_𝐻1𝑧𝜃_𝑡𝜔²_𝐻1+‖𝑣‖²_𝐻1‖𝑉‖²_𝐻1+‖𝑣‖²_𝐻1𝑍𝜃_𝑡𝜔²_𝐻1,𝑄(𝑡,𝜔)=𝐾^𝜖‖𝑣‖²‖𝑣‖⁴_𝐻1+‖𝑣‖²𝑧𝜃_𝑡𝜔⁴_𝐻1+‖𝑣‖²‖𝑉‖⁴_𝐻1+‖𝑣‖²𝑍𝜃_𝑡𝜔⁴_𝐻1+𝑧𝜃_𝑡𝜔²_𝐻2‖𝑉‖²_𝐻1+𝑧𝜃_𝑡𝜔²_𝐻2𝑍𝜃_𝑡𝜔²_𝐻1+𝑧(𝜃_𝑡𝜔)²_𝐻2𝑧(𝜃_𝑡𝜔)²_𝐻1+𝑧𝜃_𝑡𝜔²_𝐻2‖𝑣‖²_𝐻1+𝑍(𝜃_𝑡𝜔)²+𝑓². (73) Then by utilizing Lemma 7 to (72) we deduce that 𝐕_𝑚𝑡+1,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍≤𝜆1+log^𝑚+1𝜆₁×𝑒^{−𝜆𝑚+1}_𝑡^𝑡+1𝑒^{𝜆𝑚+1(𝑠−𝑡)}𝑃𝑠,𝜃_−𝑡−1𝜔𝑑𝑠+𝑒^{−𝜆𝑚+1}_𝑡^𝑡+1𝑒^{𝜆𝑚+1(𝑠−𝑡)}𝑄𝑠,𝜃_−𝑡−1𝜔𝑑𝑠+𝑒^{−𝜆𝑚+1}×_𝑡^𝑡+1𝑒^{𝜆𝑚+1(𝑠−𝑡)}𝐕_𝑚𝑠,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍𝑑𝑠. (74) By Lemma 11, there exists𝑇=𝑇(𝐵,𝜔)>0such that for all𝑡≥𝑇, _𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑃𝑠,𝜃_−𝑡−1𝜔𝑑𝑠≤𝐾^𝜖_𝑡^𝑡+1𝑒^{𝜆𝑚+1(𝑠−𝑡)}×𝑅⁴(𝜔)+𝑅²(𝜔)𝑧(𝜃_{𝑠−𝑡−1}𝜔)²_𝐻1+𝑅⁴(𝜔)+𝑅²(𝜔)×𝑍(𝜃_{𝑠−𝑡−1}𝜔)²_𝐻1≤𝐾𝑑𝑠_𝜖𝜆_𝑚+1×𝑒^𝜆𝑚+12𝑅⁴(𝜔)+𝑅²(𝜔)sup_{−1≤𝑠≤0}𝑧𝜃_𝑠𝜔²_𝐻1+𝑅²(𝜔)sup_{−1≤𝑠≤0}𝑍(𝜃_𝑠𝜔)²_𝐻1𝑅(𝜔)∶=𝜆_𝑚+1𝑒^𝜆𝑚+1, (75) where 𝑅(𝜔)=𝐾^𝜖2𝑅⁴(𝜔)+𝑅²(𝜔)sup_{−1≤𝑠≤0}𝑧(𝜃_𝑠𝜔)²_𝐻1+𝑅²(𝜔)sup_{−1≤𝑠≤0}𝑍(𝜃_𝑠𝜔)²_𝐻1 (76) is independent of𝜆_𝑚+1. By a similar calculation as (75), we find that there exist an random variable𝑅(𝜔)such that for all𝑡≥𝑇, _𝑡^𝑡+1𝑒^{𝜆𝑚+1(𝑠−𝑡)}𝑄𝑠,𝜃_−𝑡−1𝜔𝑅(𝜔)𝑑𝑠≤𝜆_𝑚+1𝑒^𝜆𝑚+1,_𝑡𝑡+1𝑒^{𝜆𝑚+1(𝑠−𝑡)}×𝐕_𝑚𝑠,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍≤𝑅(𝜔)𝑑𝑠₂𝜆_𝑚+1𝑒^𝜆𝑚+1, (77) where𝑅(𝜔)is in Lemma 9. Then (74) together with (75) and (77) implies that for all𝑡≥𝑇, 𝐕_𝑚𝑡+1,𝜃_−𝑡−1𝜔,𝐔₀𝜃_−𝑡−1𝜔−𝐙𝜃_−𝑡−1𝜔²_𝕍≤1𝜆_𝑚+1𝜆1+log^𝑚+1𝜆₁𝑅(𝜔)+1𝜆_𝑚+1𝑅(𝜔)+𝑅(𝜔)²⟶0, (78) as𝑚→+∞. Consequently, for every𝜀>0, there exists a integer𝑁and positive constants𝐾=𝐾(𝜔)such that for all𝑡≥𝑇+1and𝑚≥𝑁, 𝐼−𝑃_𝑚𝐕𝑡,𝜃_−𝑡𝜔,𝐔₀𝜃_−𝑡𝜔−𝐙𝜃_−𝑡𝜔_𝕍𝑃≤𝜀,_𝑚𝐕𝑡,𝜃_−𝑡𝜔,𝐔₀𝜃_−𝑡𝜔−𝐙𝜃_−𝑡𝜔_𝕍≤𝐾. (79) This leads to the desirable conclusion.

Lemma 12. Assume that𝔼𝑀<0. Then the RDS𝜑corresponding to the Boussinesq system (1) is omega-limit compact in𝕍; that is, for every𝜀>0and an arbitrary𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟, there is an 𝑇=𝑇(𝜀,𝐵,𝜔)>0such that forℙ-a.s.𝜔∈Ω, 𝑘_𝑡≥𝑇𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔≤𝜀, (80) where𝒟is the collection of tempered random subsets ofℍ.

Proof. By Lemma 11, for every𝐔₀(𝜔)∈𝐵(𝜔), there exist constants𝐾(𝜔)and𝑇=𝑇(𝜀,𝐵,𝜔)and 𝑁₁∈ℕsuch that for all𝑡≥𝑇and𝑚≥𝑁₁,‖𝑃_𝑚𝐕(𝑡,𝜃_−𝑡𝜔,𝐔₀(𝜃_−𝑡𝜔)−𝐙(𝜃_−𝑡𝜔))‖_𝕍≤𝐾(𝜔)and 𝐼−𝑃_𝑚𝐕𝑡,𝜃_−𝑡𝜔,𝐔₀𝜃_−𝑡𝜔−𝐙𝜃_−𝑡𝜔_𝕍≤𝜀2. (81) Note that 𝑃_𝑚𝐙(𝜔)_𝕍≤‖𝐙(𝜔)‖_𝕍,𝐼−𝑃_𝑚𝐙(𝜔)_𝕍≤1𝜆_𝑚+1‖𝐙(𝜔)‖_𝐻²⟶0 (82) as𝑚→∞, and then there exists𝑁₂∈ℕsuch that for every𝑚≥𝑁₂, 𝑃_𝑚𝐙(𝜔)_𝕍≤‖𝐙(𝜔)‖_𝕍,𝐼−𝑃_𝑚𝐙(𝜔)_𝕍≤𝜀2. (83) Put𝑁=max{𝑁₁,𝑁₂}. By the definition of the RDS𝜑, along with (81) and (82), we find that there exist 𝐾(𝜔)=𝐾(𝜔)+‖𝐙(𝜔)‖_𝕍such that for all𝑡≥𝑇𝑃_𝑁𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔_𝕍≤𝐾(𝜔),(𝐼−𝑃_𝑁)𝜑(𝑡,𝜃_−𝑡𝜔)𝐵(𝜃_−𝑡𝜔)_𝕍≤𝜀, (84) where𝑇=𝑇(𝜀,𝐵,𝜔)>0is the same as (81). That is to say, the RDS𝜑satisfies the flattening conditions in𝕍. By utilizing the additive property of Kuratowski measure of noncompactness; see Lemma  2.5 (iii) in [12]; it follows from (84) that forℙ-a.s.𝜔∈Ω, 𝑘_𝑡≥𝑇𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔≤𝑘𝑃_{𝑁𝑡≥𝑇}𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔+𝑘𝐼−𝑃_{𝑁𝑡≥𝑇}𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔≤0+𝑘𝐵_𝕍(0,𝜀)=2𝜀, (85) where𝐵_𝕍(0,𝜀)is the𝜀-neighborhood at centre0in𝕍. This completes the proof.

Proof of Theorem 2. By Theorem  5.1 in [4], the RDS𝜑associated with the Boussinesq system (1) admits a unique compact random attractor{𝒜_𝐻(𝜔)}_𝜔∈Ωinℍ. Furthermore{𝒜_𝐻(𝜔)}_𝜔∈Ω∈𝒟. Put 𝐾₀(𝜔)=𝐔=𝑢,𝑈,𝑈_Γ∈ℍ;‖𝐔‖≤𝜚₁(𝜔)+‖𝐙(𝜔,},)‖𝜔∈Ω (86) where𝜚₁(𝜔)is in (42). Observe that‖𝐙(𝜔)‖is tempered, and then Lemma 9 implies that{𝐾₀(𝜔)}_𝜔∈Ωis a random absorbing set for the RDS𝜑in𝒟. By Theorem  3.5 in [20], we know that{𝒜_𝐻(𝜔)}_𝜔∈Ωis the𝛀-limit set of{𝐾₀(𝜔)}_𝜔∈Ω(see [19]); that is, 𝒜_ℍ(𝜔)=_{𝑠≥0𝑡≥𝑠}𝜑𝑡,𝜃_−𝑡𝜔𝐾₀(𝜃_−𝑡𝜔)^ℍ,𝜔∈Ω. (87)
For𝜔∈Ω, the following is given: 𝒜_𝕍(𝜔)=_{𝑠≥0𝑡≥𝑠}𝜑𝑡,𝜃_−𝑡𝜔𝐾₀𝜃_−𝑡𝜔^𝕍. (88) In the sequel we will show that{𝒜_𝕍(𝜔)}_𝜔∈Ωis a random attractor in𝕍in the sense that{𝒜_𝕍(𝜔)}_𝜔∈Ωsatisfies (27)–(29).
We divide the proof into three steps.
Step  1 (compactness). By Lemma  4.5 (v) in [12] and the omega-limit compactness of𝜑in𝕍(from Lemma 12), we have 𝑘_𝑡≥𝑇𝜑𝑡,𝜃_−𝑡𝜔𝐾₀𝜃_−𝑡𝜔^𝕍=𝑘_𝑡≥𝑇𝜑𝑡,𝜃_−𝑡𝜔𝐾₀𝜃_−𝑡𝜔⟶0,as𝑇⟶∞. (89) Since⋃_𝑡≥𝑇𝜑(𝑡,𝜃_−𝑡𝜔)𝐾₀(𝜃_−𝑡𝜔)^𝕍is norm-closed in𝕍, thanks to the nested property of the Kuratowski measure of noncompactness (see again Lemma  2.5 (iv) in [12]), we know that for𝜔∈Ω,𝒜_𝕍(𝜔)is nonempty and compact as required, which shows (27).
Step  2 (invariant property). By (87) and (88) it is easy to see that for𝜔∈Ω, 𝑥∈𝒜_ℍ(𝜔)⟺∃𝑡_𝑛⟶∞,𝑥_𝑛∈𝐾₀𝜃_−𝑡𝜔suchthat𝜑𝑡_𝑛,𝜃_−𝑡𝑛𝜔𝑥−−−→_𝑛^‖⋅‖ℍ𝑥,𝑦∈𝒜_𝕍(𝜔)⟺∃𝑡_𝑛⟶∞,𝑥_𝑛∈𝐾₀𝜃_−𝑡𝜔suchthat𝜑𝑡_𝑛,𝜃_−𝑡𝑛𝜔𝑥−−−→_𝑛^{‖⋅‖𝕍}𝑦. (90)(91) It is obvious that𝒜_𝕍(𝜔)⊆𝒜_ℍ(𝜔)for𝜔∈Ω. Conversely, if𝑥∈𝒜_ℍ(𝜔), by the equivalent regime (90), there exist two sequences𝑡_𝑛→∞and𝑥_𝑛∈𝐾₀(𝜃_−𝑡𝜔)such that 𝜑𝑡_𝑛,𝜃_−𝑡𝑛𝜔𝑥−−−→_𝑛^‖⋅‖ℍ𝑥. (92) Note that𝜑(𝑡_𝑛,𝜃_−𝑡𝑛𝜔)𝑥_𝑛∈𝕍and𝜑is omega-limit compact in𝕍(from Lemma 12). Then there exist an 𝑦∈𝕍and a subsequence{𝑛_𝑘}with𝑡_𝑛𝑘→∞such that 𝜑𝑡_𝑛𝑘,𝜃_{−𝑡𝑛𝑘}𝜔𝑥_𝑛𝑘⟶𝑦inthespace𝕍, (93) and by the nested relation of𝕍andℍ, we have𝑥=𝑦, whence connection with (91) and (93) we have𝑥∈𝒜_𝕍(𝜔). Then𝒜_ℍ(𝜔)⊆𝒜_𝕍(𝜔)for𝜔∈Ω, which indicates that{𝒜_𝕍(𝜔)}_𝜔∈Ωpossesses the invariance property (28).
Step  3 (attracting property). Suppose that (29) does not hold. Then there exists𝛿>0,𝑥_𝑛∈𝐵(𝜃_−𝑡𝑛𝜔)and𝑡_𝑛→∞such that dist_𝕍𝜑𝑡_𝑛,𝜃_−𝑡𝑛𝜔𝑥_𝑛,𝒜_𝕍(𝜔)≥𝛿. (94) According to Lemma 12,𝜑is omega-limit compact in𝕍. Then we can extract a subsequence from the sequence{𝜑(𝑡_𝑛,𝜃_−𝑡𝑛𝜔)𝑥_𝑛}_𝑛(denoted by its original form) satisfying that there exists an𝑦∈𝕍such that when𝑡_𝑛→∞, 𝜑𝑡_𝑛,𝜃_−𝑡𝑛𝜔𝑥_𝑛⟶𝑦inthespace𝕍. (95) We then need to show that𝑦∈𝒜_𝕍(𝜔)for𝜔∈Ω. Indeed, since{𝐾₀(𝜔)}_𝜔∈Ωis a random absorbing set for𝜑inℍ, then for𝑥_𝑛∈𝐵(𝜃_−𝑡𝑛𝜔), there exists𝑇=𝑇(𝐵,𝜔)>0such that for all𝑡≥𝑇, 𝜑𝑡,𝜃_−𝑡𝜃_{−(𝑡𝑛−𝑡)}𝜔𝑥_𝑛𝜃_−𝑡𝜃_{−(𝑡𝑛−𝑡)}𝜔∈𝐾₀𝜃_{−(𝑡𝑛−𝑡)}𝜔. (96) At the same time, by the cocycle property of the RDS𝜑, for𝑡_𝑛≥𝑡we have that 𝜑𝑡_𝑛,𝜃_−𝑡𝑛𝜔𝑥_𝑛𝜃_−𝑡𝑛𝜔=𝜑𝑡_𝑛−𝑡+𝑡,𝜃_−𝑡𝑛𝜔𝑥_𝑛𝜃_−𝑡𝑛𝜔=𝜑𝑡_𝑛−𝑡,𝜃_{−(𝑡𝑛−𝑡)}𝜔𝜑𝑡,𝜃_−𝑡𝑛𝜔𝑥_𝑛𝜃_−𝑡𝑛𝜔=𝜑𝑡_𝑛−𝑡,𝜃_{−(𝑡𝑛−𝑡)}𝜔𝜑𝑡,𝜃_−𝑡𝜃_{−(𝑡𝑛−𝑡)}𝜔×𝑥_𝑛𝜃_−𝑡𝜃_{−(𝑡𝑛−𝑡)}𝜔. (97) We now choose fixed𝑡≥𝑇such that𝑡_𝑛≥𝑡. Denote𝑡^′_𝑛=𝑡_𝑛−𝑡and𝑦_𝑛=𝜑(𝑡,𝜃_−𝑡𝜃_{−(𝑡′𝑛)}𝜔)𝑥_𝑛(𝜃_−𝑡𝜃_{−(𝑡′𝑛)}𝜔).Then (96) indicates that𝑦_𝑛∈𝐾₀(𝜃_{−𝑡′𝑛}𝜔). It is obvious that if𝑡_𝑛→∞then𝑡^′_𝑛→∞. Thus it follows from (97) that the limit in (95) can be rewritten as 𝜑𝑡^′_𝑛,𝜃_{−𝑡′𝑛}𝜔𝑦_𝑛𝜃_{−𝑡′𝑛}𝜔⟶𝑦inthespace𝕍, (98) as𝑡^′_𝑛→∞, whereas by (91) we get that𝑦∈𝒜_𝕍(𝜔)for𝜔∈Ω, which contradicts (94). This proves that{𝒜_𝕍(𝜔)}_𝜔∈Ωpossesses attracting property (29).
Note that the collection𝒟considered also includes all deterministic bounded subsets in𝕍. Then the uniqueness for{𝒜_𝕍(𝜔)}_𝜔∈Ωis followed by Theorem  4.3 in [19].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the China Natural Science Fund Grant (no. 11071199) and the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ120515).