# H

Authors: Yijin Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## 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 theL2-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.


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## Body

## 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]:
(1)dudt=(1ReΔu-u·∇u-∇p-1Fr2Ue2)+W˙0onD×ℝ+,divu=0onD×ℝ+,u=0onΓ×ℝ+,u(0)=u0,dUdt=(1RePrΔU-u·∇U)+W˙1onD×ℝ+,dUΓdt=(-∂nUΓ-cUΓ+f(x)α)+W˙2onΓ×ℝ+,γU=UΓ,U(0)=U0,
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;∂nUΓis 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. Section2 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(2)L2:=(L2(D))2×L2(D)×L2(Γ),
endowed with the scalar inner product(·,·)=(·,·)(L2(D))2+(·,·)L2(D)+(·,·)L2(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:
(3)𝒱={(u,U,UΓ)∈(C∞(D))2×C∞(D)×C∞(Γ);divu=0}.Define(4)HΓ1=(H01(D))2×H1(D)×H1/2(Γ),
where(H01(D))2is the usual Sobolev space with equivalent norm∥∇·∥andH1(D)is also the usual Sobolev space with the equivalent norm:
(5)∥U∥H1=(∥∇U∥H12+∥U∥L2(Γ)2)1/2,U∈H1(D);
see [15, page 52], andH1/2(Γ)is given byγ(H1(D))endowed by a norm∥ϕ∥H1/2=infγU=ϕ∥U∥H1; see [15, page 48].Letℍthe closure of𝒱inL2and𝕍be the closure of𝒱inHΓ1, and the norm in𝒱being denoted by ∥·∥𝕍. From the above argument, the space𝕍is equipped with the norm:
(6)∥U∥𝕍=(∥∇u∥2+∥∇U∥2+∥U∥L2(Γ)2)1/2,U=(u,U,UΓ)∈𝕍,
which is equivalent to the natural norm:(∥∇u∥2+∥U∥H1(D)2+∥U∥H1/2(Γ)2)1/2.ForU=(u,U,UΓ),V=(v,V,VΓ), we define the operators
(7)A1u=-Δu,A2(U,UΓ)=(-ΔU,∂nUΓ+UΓ),B1(u,v)=u·∇v,B2(u,V)=u·∇V,AU=(A1u,A2(U,UΓ)),B(U,V)=(B1(u,v),B2(u,V),0),(8)F(U)=(-Ue2,0,f(x)),f∈L2(Γ).By Lemma  2.2 in [4], the operatorAis a positive self-adjoint unbounded operator and has the Poincaré inequality:
(9)(AU,U)≥λ∥U∥2,λ>0.
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,{ej}j=1∞of eigenvectors ofA. The corresponding spectrum ofAis discrete and denoted by{λj}j=1∞which are positive, increasing, and tend to infinity asj→∞.In particular, we also can use the spectrum theory to allow us to define the operatorAs, the power ofA. Fors>0, the operatorAsis also an strictly positive and self-adjoint unbounded operator inℍwith a dense domainD(As)⊂ℍ. This allows us to introduce the function spaces,
(10)D(As)={V=∑j=1∞‍(V,ej)ej:∥V∥D(As)2=∑j=1∞‍(V,ej)2λj2s<+∞}.
This norm∥·∥D(As)onD(As)is equivalent to the usual norm induced byH2s; see Temam [15] for details. In particular,D(A0)=ℍandD(A1/2)=𝕍.Based on the orthonormal basis{ej}j=1∞of eigenfunctions ofA, we define them-dimensional subspace𝕍m=span{e1,e2,…,em}⊂𝕍and the canonical orthogonal projectionPm:𝕍↦𝕍msuch that for everyV∈𝕍,Vhas a unique decomposition:V=PmV+Vm, where
(11)PmV=∑j=1m‍(V,ej)ej∈𝕍m,Vm=(I-Pm)V=∑j=m+1∞‍(V,ej)ej∈𝕍m⊥;
that is,𝕍=𝕍m⊕𝕍m⊥.When the projectionPmoperates on the first component ofV=(v,V,VΓ), one can easily show that
(12)∥A1Pmv∥2≤λm+1∥A11/2Pmv∥2,∀V=(v,V,VΓ)∈D(A)
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
(13)∥v∥(L∞(D))2≤c∥A11/2v∥(1+log∥A1v∥2λ1∥A11/2v∥2)1/2∀V=(v,V,VΓ)∈D(A),
whereλ1is the first eigenvalue of the Stokes operatorA1. By (12) we have
(14)∥Pmv∥(L∞(D))2≤c∥A11/2Pmv∥(1+log∥A1Pmv∥2λ1∥A11/2Pmv∥2)1/2≤c∥v∥H1(1+logλm+1λ1)1/2,∀V=(v,V,VΓ)∈D(A).According to the above notations, we can write (1) as the following abstract evolution equation form:
(15)dU+AUdt+B(U,U)dt=F(U)dt+dW,U(x,0)=U0(ω)∈ℍ,
whereU=(u,U,UΓ).


## 3. Existence of Random Attractor in𝕍

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,(θt)t∈ℝ), where Ω={ω∈C0(ℝ,ℍ):ω(0)=0}with compact open topology. This topology is metrizable by the complete metric:
(16)d(ω1,ω2)=∑n=1∞‍12ndn(ω1,ω2)1+dn(ω1,ω2),
wheredn(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
(17)θsω(t)=ω(t+s)-ω(s),ω∈Ω,t,s∈ℝ.
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(ℱt)t∈ℝ:
(18)ℱt≡{W(τ)∣τ≤t}⊂ℱ.We introduce the following stochastic partial differential equation onD:
(19)dZ+AZdt=dW.
BecauseAis a positive and self-adjoint operator, there exists a mild solution to this stochastic equation with the form
(20)Z(t)=Z(0)+∫0t‍e-(t-τ)AdW,t>0,
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
(21)trℍ(QA2s-1+δ)=trℍ(As-1/2+δ/2QAs-1/2+δ/2)<+∞,
for somes≥0and some (arbitrary small)δ>0, wheretrℍdenotes the trace of the covariance operator. Then anℱ0-measurable Gaussian variableZ=(z,Z)∈D(As)exists, and the process (t,ω)→Z(θtω)is a continuous stationary solution to the stochastic equation (19). Furthermore, the random variable∥Z(ω)∥D(As)2is tempered and the expectation
(22)𝔼∥Z∥D(As)2=12trℋ(As-1/2QAs-1/2)<+∞.LetV=U-Z(θtω); by (15) and (19) we have the following deterministic equation with a parameterω:
(23)ddtV+AV+B(V+Z(θtω),V+Z(θtω))=F(V+Z(θtω)),V(x,0)=V0(ω)=U0(ω)-Z(ω)∈ℍ.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(θtω). Then U(t,ω,U0(ω))or brieflyU(t)is a solution to (23) with initial valueU0(ω).Given(24)φ(t,ω)U0(ω)=U(t,ω,U0(ω))=V(t,ω,U0(ω)-Z(ω))+Z(θtω),ω∈Ω,
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
(25)M(ω)=Kϵ∥Z(ω)∥H22+Kϵ∥z(ω)∥H12-λ,
whereKϵis 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 ofM(26)𝔼M<0.
Then the RDSφgenerated by (1) admits a unique random attractor{𝒜𝕍(ω)}ω∈Ωin𝕍in the sense that forℙ-a.s.ω∈Ω,
(27)𝒜𝕍(ω)iscompactin𝕍,(28)φ(t,ω)𝒜𝕍(ω)=𝒜𝕍(θtω),foreveryt≥0,(29)limt→∞dist𝕍(φ(t,θ-tω)B(θ-tω),𝒜𝕍(ω))=0,∀{B(ω)}ω∈Ω∈𝒟,
where dist𝕍 denotes the Hausdorff semiistance in𝕍and𝒟is the collection of tempered subsets ofℍas in [4]. Furthermore,{𝒜𝕍(ω)}ω∈Ωis identical with the random attractor{𝒜ℍ(ω)}ω∈Ωinℍ.By the well-known abstract result of Theorem  3.5 in [20], the existence of a random attractor{𝒜ℍ(ω)}ω∈Ωin the following sense that
(30)𝒜ℍ(ω)iscompactinℍ,φ(t,ω)𝒜ℍ(ω)=𝒜ℍ(θtω),foreveryt≥0,limt→∞distℍ(φ(t,θ-tω)B(θ-tω),𝒜ℍ(ω))=0forall{B(ω)}ω∈Ω∈𝒟,
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, Theorem2 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,
(31)k(⋃t≥T‍φ(t,θ-tω)B(θ-tω))≤ε,ℙ-a.s.ω∈Ω,
wherek(B)is the Kuratowski measure of noncompactness ofBdefined by
(32)k(B)=inf{d>0:Badmitsafinitecoverbysetsofdiameter≤d}.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,
(33)P(⋃t≥T‍φ(t,θ-tω)B(θ-tω))isboundedinX,∥(I-P)(⋃t≥T‍φ(t,θ-tω)B(θ-tω))∥X≤ε.Definition 5 (see [29]).
(i) A random variableX∈ℝ+over an MDS is tempered if forℙ-a.s.ω∈Ω,
(34)limt→±∞log+X(θtω)|t|=0.
(ii) A random setB={B(ω)}ω∈Ω∈𝒟is called tempered ifR(ω)=supx∈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,
(35)φ(t,θ-tω)B(θ-tω)⊂K(ω),
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=𝕍m=span{e1,e2,…,em}and the bounded operatorP=Pmfor a sufficient largem, where ei(i=1,2,…,m)andPmare defined in Section 2.


## 4. The Proofs of Main Result

To prove Theorem2, 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,
(36)y′(s)+by(s)≤h(s).
Then for everyt≥aand any positive constantr, one has
(37)y(t+r)≤e-br(∫tt+r‍y(s)eb(s-t)ds+∫tt+r‍h(s)eb(s-t)ds).Proof.
Lett≤s≤t+r. We multiply (36) byeb(s-t)and the resulting inequality reads
(38)dds(y(s)eb(s-t))≤h(s)eb(s-t).
Hence, by integration fromstot+r,
(39)y(t+r)ebr≤y(s)eb(s-t)+∫tt+r‍h(s)eb(s-t)ds.
Integrating the above inequality with respect tosbetweentandt+r, we get the desired inequality.Lemma 8.
There exist positive constantsKandKϵsuch that the followings hold:
(40)∥V(t)∥2≤e∫0t‍M(θτω)dτKϵ∥V0(ω)∥2+Kϵ∫0t‍G(θsω)e∫st‍M(θτω)dτds,(41)ddt∥V∥2+K∥V∥𝕍2≤Kϵ(∥Z(θtω)∥H22+∥z(θtω)∥H12)∥v∥2+Kϵ∥V∥2+G(θtω),(42)ddt∥V∥𝕍2≤g(t,ω)∥V∥𝕍2+h(t,ω),
where
(43)M(ω)=Kϵ∥Z(ω)∥H22+Kϵ∥z(ω)∥H12-λ,G(ω)=Kϵ+Kϵ∥z(ω)∥2∥Z(ω)∥H22+Kϵ∥z(ω)∥H12∥z(ω)∥H22+Kϵ∥Z(ω)∥H22,(44)g(t,ω)=K∥v(t,ω)∥2∥v(t,ω)∥H12+H1(θtω),h(t,ω)=K∥V(t,ω)∥2+H2(θtω),H1(ω)=Kϵ∥z(ω)∥H22+Kϵ∥Z(ω)∥H22,H2(ω)=Kϵ∥z(ω)∥H22∥Z(ω)∥H12+Kϵ∥z(ω)∥H12∥z(ω)∥H22+K∥Z(ω)∥2+Kϵ.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]:
(45)∥V(l,θ-t-1ω,U0(ω)-Z(ω))∥≤ϱ1(ω),(46)∫tt+1‍∥V(s,θ-t-1ω,U0(ω)-Z(ω))∥𝕍2ds≤ϱ2(ω),
where𝒟is the collection of tempered random subsets ofℍ.Proof.
We replacetwithl, andωwithθ-t-1ωin (40) to produce that forl∈[t,t+1],
(47)∥V(l,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥2≤e∫0l‍M(θτ-t-1ω)dτKϵ∥V0(θ-t-1ω)∥2+Kϵ∫0l‍G(θs-t-1ω)e∫sl‍M(θτ-t-1ω)dτds=e∫-t-1l-t-1‍M(θτω)dτKϵ∥V0(θ-t-1ω)∥2+Kϵ∫-t-1l-t-1‍G(θsω)e∫sl-t-1‍M(θτω)dτds≤e∫-t-1l-t-1‍M(θτω)dτKϵ×(∥V0(θ-t-1ω)∥2+∥Z0(θ-t-1ω)∥2)+Kϵ∫-t-10‍G(θsω)e∫sl-t-1‍Q(θτω)dτds.
By noting that forl∈[t,t+1],l-t-1∈[-1,0], then we find that fors∈[-t-1,0],
(48)e∫sl-t-1‍M(θτω)dτ=e-λ1(l-t-1)+λ1s×e∫sl-t-1‍(Kϵ∥Z(θτω)∥H22+Kϵ∥z(θτω)∥H12)dτ≤eλ1+λ1se∫s0‍(Kϵ∥Z(θτω)∥H22+Kϵ∥z(θτω)∥H12)dτ≤eλ1e∫s0‍M(θτω)dτ,
from which and (47) it follows that for everyl∈[t,t+1],
(49)∥V(l,θ-t-1ω,V0(θ-t-1ω)-Z(θ-t-1ω))∥2≤e∫-t-10‍M(θsω)dsKϵ(∥V0(θ-t-1ω)∥2+∥Z(θ-t-1ω)∥2)+Kϵ∫-t-10‍G(θsω)e∫s0‍M(θτω)dτds.
By the Birkhoff’s ergodic theorem and along with our assumption (26), it yields that
(50)limt→±∞1t+1∫-t-10‍M(θτω)dτ=𝔼M(ω)<0,
and then it implies that
(51)e∫-t-10‍M(θτω)dτ≈e(t+1)𝔼M⟶0,ast⟶+∞.
AsG(θτ)is subexponential growth, so
(52)∫-∞0‍G(θsω)e∫s0‍M(θτω)dτds<+∞.
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],
(53)∥V(l,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥2≤ϱ12(ω):=Kϵ(1+∫-∞0‍G(θsω)e∫s0‍M(θτω)dτds).
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
(54)K∫tt+1‍∥V(s,ω,U0(ω)-Z(ω))∥𝕍2ds≤Kϵ∫tt+1‍(∥Z(θsω)∥H22+∥z(θsω)∥H12)×∥v(s,ω,v0(ω)-z(ω))∥2ds+Kϵ∫tt+1‍∥V(s,ω,V0(ω)-Z(ω))∥2ds+∫tt+1‍G(θsω)ds+K∥V(t,ω,V0(ω)-Z(ω))∥2.
Then by (45) we get that for allt≥T,
(55)∫tt+1‍∥V(s,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2ds≤Kϵϱ12(ω)∫tt+1‍(∥Z(θs-t-1ω)∥H22+∥z(θs-t-1ω)∥H12)ds+∫tt+1‍G(θs-t-1ω)ds+Kϵϱ12(ω)=Kϵϱ12(ω)∫-10‍(∥Z(θsω)∥H22+∥z(θsω)∥H12)ds+∫-10‍G(θsω)ds+Kϵϱ12(ω):=ϱ2(ω),
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],
(56)∥V(l,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍≤R(ω),
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
(57)∥V(l,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2≤e∫tt+1‍g(τ,θ-t-1ω)dτ×(∫tt+1‍∥V(s,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2ds+∫tt+1‍h(s,θ-t-1ω)ds).
Note that by Lemma 9 there existsT=T(B,ω)such that for allt≥T,
(58)∫tt+1‍g(τ,θ-t-1ω)dτ=∫tt+1‍(K∥v(s,θ-t-1ω)∥2∥v(s,θ-t-1ω)∥H12+H1(θs-t-1ω)∥v(s,θ-t-1ω)∥2)ds≤Kϱ12(ω)ϱ22(ω)+∫-10‍H1(θsω)ds:=C1(ω),∫tt+1‍h(s,θ-t-1ω)ds=∫tt+1‍(K∥V(s,θ-t-1ω)∥2+H2(θs-t-1ω))ds≤Kϱ12(ω)+∫-10‍H2(θsω)ds:=C2(ω).
Then by (57) and (58) it gives that for allt≥Tandl∈[t,t+1],
(59)∥V(l,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2≤eC1(ω)(Kϱ2(ω)+C2(ω)):=R2(ω),
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
(60)∥PmV(t,θ-tω,U0(θ-tω)-Z(θ-tω))∥𝕍≤K,∥Vm(t,θ-tω,U0(θ-tω)-Z(θ-tω))∥𝕍≤ε,
where𝒟is the collection of tempered random subsets ofℍ.Proof .
Multiplying (23) byAVmwith respect to theL2inner product leads us to
(61)12ddt∥Vm∥𝕍2+∥AVm∥2=(B1(v+z(θtω),v+z(θtω)),A1vm)+(B2(v+z(θtω),V+Z(θtω)),A2Vm)+(e2(V+Z(θtω)),A1vm)+(f,A2(Vm)Γ)=I1+I2+I3,
where
(62)I1=(B1(v+z(θtω),v+z(θtω)),A1vm),I2=(B2(v+z(θtω),V+Z(θtω)),A2Vm),I3=(e2(V+Z(θtω)),A1vm)+(f,A2(Vm)Γ).
In order to estimateI1, we rewrite it as
(63)I1=(B1(v,v),A1vm)+(B1(v,z(θtω)),A1vm)+(B1(z(θtω),z(θtω)),A1vm)+(B1(z(θtω),v),A1vm),
where by utilizing the inequality (14) and Agmon’s inequality inℝ2(see [15]), it gives that
(64)|B1(v,v),A1vm|≤|B1(Pmv,v),A1vm|+|B1(vm,v),A1vm|≤∥Pmv∥L∞∥v∥H1∥A1vm∥+∥vm∥L∞∥v∥H1∥A1vm∥≤K(1+logλm+1λ1)1/2∥v∥H12∥A1vm∥+Kϵ∥vm∥1/2∥A1vm∥3/2∥v∥H1≤ϵ16∥A1vm∥2+Kϵ(1+logλm+1λ1)∥v∥H14+Kϵ∥v∥2∥v∥H14,(65)|(B1(v,z(θtω)),A1vm)|≤|B1(Pmv,z(θtω)),A1vm|+|(B1(vm,z(θtω)),A1vm)|≤∥Pmv∥L∞∥z(θtω)∥H1∥A1vm∥+∥vm∥L∞∥z(θtω)∥H1∥A1vm∥≤K(1+logλm+1λ1)1/2∥v∥H1∥z(θtω)∥H1∥A1vm∥+K∥vm∥1/2∥A1vm∥3/2∥z(θtω)∥H12≤ϵ16∥A1vm∥2+Kϵ(1+logλm+1λ1)∥v∥H12∥z(θtω)∥H12+Kϵ∥v∥2∥z(θtω)∥H14.
Similarly by utilizing the Agmon’s inequality, we deduce that
(66)|B1(z(θtω),z(θtω)),A1vm|≤ϵ16∥A1vm∥2+Kϵ∥z(θtω)∥H22∥z(θtω)∥H12,|B1(z(θtω),v),A1vm|≤ϵ16∥A1vm∥2+Kϵ∥z(θtω)∥H22∥v∥H12.
Then by (63)–(66) we get that
(67)I1≤ϵ4∥A1vm∥2+Kϵ(1+logλm+1λ1)×(∥v∥H12∥z(θtω)∥H12+∥v∥H14)+Kϵ(∥v∥2∥v∥H14+∥z(θtω)∥H22∥v∥H12+∥v∥2∥z(θtω)∥H14+∥z(θtω)∥H22∥z(θtω)∥H12).
We then estimateI2in (61). We first have
(68)I2=(B2(v,Z(θtω)),A2Vm)+(B2(v,V),A2Vm)+(B2(z(θtω),Z(θtω)),A2Vm)+(B2(z(θtω),V),A2Vm),
where
(69)|B2(v,V),A2Vm|≤|B2(Pmv,V),A2Vm|+|B2(vm,V),A2Vm|≤∥Pmv∥L∞∥V∥H1∥A2Vm∥+∥vm∥L∞∥V∥H1∥A2Vm∥≤ϵ16∥A2Vm∥2+Kϵ(1+logλm+1λ1)∥v∥H12∥V∥H12+Kϵ∥v∥∥A1vm∥∥V∥H12≤ϵ16∥A2Vm∥2+Kϵ(1+logλm+1λ1)∥v∥H12∥V∥H12+ϵ16∥A1vm∥2+Kϵ∥v∥2∥V∥H14,|B2(v,Z(θtω)),A2Vm|≤ϵ16∥A2Vm∥2+Kϵ(1+logλm+1λ1)∥v∥H12∥Z(θtω)∥H12+ϵ16∥A1vm∥2+Kϵ∥v∥2∥Z(θtω)∥H14,|B2(z(θtω),Z(θtω)),A2Vm|≤ϵ16∥A2Vm∥2+Kϵ∥z(θtω)∥H22∥Z(θtω)∥H12,|B2(z(θtω),V),A2Vm|≤ϵ16∥A2Vm∥2+Kϵ∥z(θtω)∥H22∥V∥H12.
Then it follows from (68) and (69) that
(70)I2≤ϵ4∥A2Vm∥2+ϵ8∥A1vm∥2+Kϵ(1+logλm+1λ1)×(∥v∥H12∥V∥H12+∥v∥H12∥Z(θtω)∥H12)+Kϵ(∥v∥2∥V∥H14+∥v∥2∥Z(θtω)∥H14+Kϵ∥z(θtω)∥H22∥V∥H12+∥z(θtω)∥H22∥Z(θtω)∥H12).
Moreover,
(71)I3≤ϵ8∥A1vm∥2+Kϵ∥V∥2+Kϵ∥Z(θtω)∥2+ϵ4∥A2(Vm,(Vm)Γ)∥2+Kϵ∥f∥2.
Then by (67), (70), and (71), formula (61) becomes
(72)ddt∥Vm∥𝕍2+λm+1∥Vm∥𝕍2≤(1+logλm+1λ1)P(t,ω)+Q(t,ω),
where
(73)P(t,ω)=Kϵ(∥v∥H14+∥v∥H12∥z(θtω)∥H12+∥v∥H12∥V∥H12+∥v∥H12∥Z(θtω)∥H12),Q(t,ω)=Kϵ(∥v∥2∥v∥H14+∥v∥2∥z(θtω)∥H14+∥v∥2∥V∥H14+∥v∥2∥Z(θtω)∥H14+∥z(θtω)∥H22∥V∥H12+∥z(θtω)∥H22∥Z(θtω)∥H12+∥z(θtω)∥H22∥z(θtω)∥H12+∥z(θtω)∥H22∥v∥H12+∥Z(θtω)∥2+∥f∥2).
Then by utilizing Lemma 7 to (72) we deduce that
(74)∥Vm(t+1,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2≤(1+logλm+1λ1)×e-λm+1∫tt+1‍eλm+1(s-t)P(s,θ-t-1ω)ds+e-λm+1∫tt+1‍eλm+1(s-t)Q(s,θ-t-1ω)ds+e-λm+1×∫tt+1‍eλm+1(s-t)∥Vm(s,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2ds.
By Lemma 11, there existsT=T(B,ω)>0such that for allt≥T,
(75)∫tt+1‍eλm+1sP(s,θ-t-1ω)ds≤Kϵ∫tt+1‍eλm+1(s-t)×(R4(ω)+R2(ω)∥z(θs-t-1ω)∥H12+R4(ω)+R2(ω)×∥Z(θs-t-1ω)∥H12)ds≤Kϵλm+1×eλm+1(+R2(ω)sup-1≤s≤0∥Z(θsω)∥H122R4(ω)+R2(ω)sup-1≤s≤0∥z(θsω)∥H12+R2(ω)sup-1≤s≤0∥Z(θsω)∥H12):=R^(ω)λm+1eλm+1,
where
(76)R^(ω)=Kϵ(2R4(ω)+R2(ω)sup-1≤s≤0∥z(θsω)∥H12+R2(ω)sup-1≤s≤0∥Z(θsω)∥H12)
is independent ofλm+1. By a similar calculation as (75), we find that there exist an random variableR^^(ω)such that for allt≥T,
(77)∫tt+1‍eλm+1(s-t)Q(s,θ-t-1ω)ds≤R^^(ω)λm+1eλm+1,∫tt+1‍eλm+1(s-t)×∥Vm(s,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2ds≤R(ω)2λm+1eλm+1,
whereR(ω)is in Lemma 9. Then (74) together with (75) and (77) implies that for allt≥T,
(78)∥Vm(t+1,θ-t-1ω,U0(θ-t-1ω)-Z(θ-t-1ω))∥𝕍2≤1λm+1(1+logλm+1λ1)R^(ω)+1λm+1(R^^(ω)+R(ω)2)⟶0,
asm→+∞. Consequently, for everyε>0, there exists a integerNand positive constantsK=K(ω)such that for allt≥T+1andm≥N,
(79)∥(I-Pm)V(t,θ-tω,U0(θ-tω)-Z(θ-tω))∥𝕍≤ε,∥PmV(t,θ-tω,U0(θ-tω)-Z(θ-tω))∥𝕍≤K.
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,
(80)k(⋃t≥T‍φ(t,θ-tω)B(θ-tω))≤ε,
where𝒟is the collection of tempered random subsets ofℍ.Proof.
By Lemma11, for everyU0(ω)∈B(ω), there exist constantsK^(ω)andT=T(ε,B,ω)and N1∈ℕsuch that for allt≥Tandm≥N1,∥PmV(t,θ-tω,U0(θ-tω)-Z(θ-tω))∥𝕍≤K^(ω)and
(81)∥(I-Pm)V(t,θ-tω,U0(θ-tω)-Z(θ-tω))∥𝕍≤ε2.
Note that
(82)∥PmZ(ω)∥𝕍≤∥Z(ω)∥𝕍,∥(I-Pm)Z(ω)∥𝕍≤1λm+1∥Z(ω)∥H2⟶0
asm→∞, and then there existsN2∈ℕsuch that for everym≥N2,
(83)∥PmZ(ω)∥𝕍≤∥Z(ω)∥𝕍,∥(I-Pm)Z(ω)∥𝕍≤ε2.
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≥T(84)∥PNφ(t,θ-tω)B(θ-tω)∥𝕍≤K(ω),∥(I-PN)φ(t,θ-tω)B(θ-tω)∥𝕍≤ε,
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,
(85)k(⋃t≥T‍φ(t,θ-tω)B(θ-tω))≤k(PN(⋃t≥T‍φ(t,θ-tω)B(θ-tω)))+k((I-PN)(⋃t≥T‍φ(t,θ-tω)B(θ-tω)))≤0+k(B𝕍(0,ε))=2ε,
whereB𝕍(0,ε)is theε-neighborhood at centre0in𝕍. This completes the proof.Proof of Theorem2.
By Theorem  5.1 in [4], the RDSφassociated with the Boussinesq system (1) admits a unique compact random attractor{𝒜H(ω)}ω∈Ωinℍ. Furthermore{𝒜H(ω)}ω∈Ω∈𝒟. Put
(86)K0(ω)={U=(u,U,UΓ)∈ℍ;∥U∥≤ϱ1(ω)+∥Z(ω)∥,ω∈Ω},
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{𝒜H(ω)}ω∈Ωis theΩ-limit set of{K0(ω)}ω∈Ω(see [19]); that is,
(87)𝒜ℍ(ω)=⋂s≥0‍⋃t≥s‍φ(t,θ-tω)K0(θ-tω)¯ℍ,ω∈Ω.
Forω∈Ω, the following is given:
(88)𝒜𝕍(ω)=⋂s≥0‍⋃t≥s‍φ(t,θ-tω)K0(θ-tω)¯𝕍.
In the sequel we will show that{𝒜𝕍(ω)}ω∈Ωis a random attractor in𝕍in the sense that{𝒜𝕍(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
(89)k(⋃t≥T‍φ(t,θ-tω)K0(θ-tω)¯𝕍)=k(⋃t≥T‍φ(t,θ-tω)K0(θ-tω))⟶0,asT⟶∞.
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,𝒜𝕍(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,
(90)x∈𝒜ℍ(ω)⟺∃tn⟶∞,xn∈K0(θ-tω)suchthatφ(tn,θ-tnω)xn→∥·∥ℍx,(91)y∈𝒜𝕍(ω)⟺∃tn⟶∞,xn∈K0(θ-tω)suchthatφ(tn,θ-tnω)xn→∥·∥𝕍y.
It is obvious that𝒜𝕍(ω)⊆𝒜ℍ(ω)forω∈Ω. Conversely, ifx∈𝒜ℍ(ω), by the equivalent regime (90), there exist two sequencestn→∞andxn∈K0(θ-tω)such that
(92)φ(tn,θ-tnω)xn→∥·∥ℍx.
Note thatφ(tn,θ-tnω)xn∈𝕍andφis omega-limit compact in𝕍(from Lemma 12). Then there exist an y∈𝕍and a subsequence{nk}withtnk→∞such that
(93)φ(tnk,θ-tnkω)xnk⟶yinthespace𝕍,
and by the nested relation of𝕍andℍ, we havex=y, whence connection with (91) and (93) we havex∈𝒜𝕍(ω). Then𝒜ℍ(ω)⊆𝒜𝕍(ω)forω∈Ω, which indicates that{𝒜𝕍(ω)}ω∈Ωpossesses the invariance property (28).
Step  3 (attracting property). Suppose that (29) does not hold. Then there existsδ>0,xn∈B(θ-tnω)andtn→∞such that
(94)dist𝕍(φ(tn,θ-tnω)xn,𝒜𝕍(ω))≥δ.
According to Lemma 12,φis omega-limit compact in𝕍. Then we can extract a subsequence from the sequence{φ(tn,θ-tnω)xn}n(denoted by its original form) satisfying that there exists any∈𝕍such that whentn→∞,
(95)φ(tn,θ-tnω)xn⟶yinthespace𝕍.
We then need to show thaty∈𝒜𝕍(ω)forω∈Ω. Indeed, since{K0(ω)}ω∈Ωis a random absorbing set forφinℍ, then forxn∈B(θ-tnω), there existsT=T(B,ω)>0such that for allt≥T,
(96)φ(t,θ-tθ-(tn-t)ω)xn(θ-tθ-(tn-t)ω)∈K0(θ-(tn-t)ω).
At the same time, by the cocycle property of the RDSφ, fortn≥twe have that
(97)φ(tn,θ-tnω)xn(θ-tnω)=φ(tn-t+t,θ-tnω)xn(θ-tnω)=φ(tn-t,θ-(tn-t)ω)φ(t,θ-tnω)xn(θ-tnω)=φ(tn-t,θ-(tn-t)ω)φ(t,θ-tθ-(tn-t)ω)×xn(θ-tθ-(tn-t)ω).
We now choose fixedt-≥Tsuch thattn≥t-. Denotetn′=tn-t-andyn=φ(t-,θ-t-θ-(tn′)ω)xn(θ-t-θ-(tn′)ω).Then (96) indicates thatyn∈K0(θ-tn′ω). It is obvious that iftn→∞thentn′→∞. Thus it follows from (97) that the limit in (95) can be rewritten as
(98)φ(tn′,θ-tn′ω)yn(θ-tn′ω)⟶yinthespace𝕍,
astn′→∞, whereas by (91) we get thaty∈𝒜𝕍(ω)forω∈Ω, which contradicts (94). This proves that{𝒜𝕍(ω)}ω∈Ωpossesses attracting property (29).
Note that the collection𝒟considered also includes all deterministic bounded subsets in𝕍. Then the uniqueness for{𝒜𝕍(ω)}ω∈Ωis followed by Theorem  4.3 in [19].



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