# Uniformly Random Attractor for the Three-Dimensional Stochastic Nonautonomous Camassa-Holm Equations

Authors: Zhehao Huang; Zhengrong Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the uniformly random attractor for the three-dimensional stochastic nonautonomous
Camassa-Holm equations in the periodic box0,l3 in this paper. We associate with the concepts of uniform attractor and
random attractor and produce the concept of uniformly random attractor for a process. Then we establish the
existence of the uniformly random attractor in DA1/2 and DA for the equations.


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## Body

## 1. Introduction

The Camassa-Holm equation(1)ut-uxxt+3uux-2uxuxx-uuxxx=0
models the unidirectional propagation of shallow water waves over a flat bottom [1–4]. It has been paid a large number of attentions due to its rich nonlinear phenomenology. It is completely integrable [1], and it has stable solutions [5]. It possesses the peakons u=e|x-ct| which has been proved stable [6, 7]. It has been shown that (1) is locally well-posed for initial data u0∈Hs(ℝ)(s>3/2) [8, 9]. There are a rich variety of global solutions and blow-up solutions obtained in [8–10]. The global existence of weak solutions, conservative solutions, and disispative solutions was investigated in [6, 11, 12].Following the Camassa-Holm (1), some generalized types of the equation have been deeply considered by many authors, for instance [13–18]. The authors in [19] considered the three-dimensional Camassa-Holm equations subject to periodic boundary conditions:
(2)∂∂t(α02u-α12Δu)-νΔ(α02u-α12Δu)-u×(∇×(α02u-α12Δu))+1ρ0∇p=f(x),∇·u=0,u(x,0)=u0(x).
They established the global regularity of solutions of the equation and provided the estimates for the Hausdorff and fractal dimensions of the global attractor.In [20] the authors analyzed the effects produced by stochastic perturbations in the deterministic version of the three-dimensional Lagrangian averaged Navier-Stokes-α model:
(3)∂∂t(u-αΔu)+ν(Au-αΔ(Au))+(u·∇)(u-αΔu)-α∇u*·Δu+∇p=F(t,u)+G(t,u)W˙(t),∇·u=0,inD×(0,+∞),u=0,Au=0,on∂D×(0,+∞),u(0)=u0,
that is, the persistence of exponential stability as well as possible stabilization effects produced by the noise.In [21] the authors proved the existence of the pullback and forward attractors for three-dimensional Lagrangian averaged Navier-Stokes-α model with delay:
(4)∂∂t(u-αΔu)+ν(Au-αΔ(Au))+(u·∇)(u-αΔu)-α∇u*·Δu+∇p=f(t)+F(t,ut),∇·u=0,inD×(τ,+∞),u=0,Au=0,on∂D×(τ,+∞),u(t)=ϕ(t-τ),in(τ-h,τ),u(τ)=u0.In [22] the author investigated the existence of finite dimensional uniform attractor for three-dimensional nonautonomous Camassa-Holm equations with periodic boundary conditions:
(5)∂∂t(α02u-α12Δu)-νΔ(α02u-α12Δu)-u×(∇×(α02u-α12Δu))+1ρ0∇p=f(x,t),∇·u=0,u(x,τ)=uτ(x).In [23, 24] the author studied the existence of uniform attractor and convergence of the attractor as ɛ→0+ for a nonautonomous three-dimensional Lagrangian averaged Navier-Stokes-α model with singularly oscillating external force:
(6)∂∂t(u-αΔu)+ν(Au-αΔ(Au))+(u·∇)(u-αΔu)-α∇u*·Δu+∇p=gɛ,∇·u=0,inD×(τ,+∞),u=0,Au=0,on∂D×(τ,+∞),u(τ)=uτ,
where
(7)gɛ(x,y,z,t)={g0(x,y,z,t)+1ɛρg1(x,y,z,t),ɛ∈(0,1],ρ∈[0,1),g0(x,y,z,t),ɛ=0.Motivated by all their works, we initial our work to investigate the equations perturbed by an additive noise. We consider the following viscous version of three-dimensional stochastic nonautonomous Camassa-Holm equation in the periodic box[0,l]3:
(8)∂∂t(α02u-α12Δu)-νΔ(α02u-α12Δu)-u×(∇×(α02u-α12Δu))+1ρ0∇p=f(x,t)+QWt,∇·u=0,u(x,τ)=uτ(x),
where p/ρ0=π/ρ0+α02|u|2-α12(u·Δu) is the modified pressure, while π is the pressure, ν>0 is the constant viscosity, and ρ0>0 is a constant density. The function f is a given body forcing, and α0>0, α1≥0 are scale parameters. W(t) are two-sided real-valued Wiener processes on a probability space which will be specified later. Q:ℝn→L2([0,l]3)3 is a bounded linear operator. Also observing that at the limiting case α0=1, α1=0, we obtain the three-dimensional stochastic Navier-Stokes with periodic boundary conditions.Attractor is an important concept to describe the long-time behavior of solutions for a system in mathematical physics [25–27]. The notion of uniform attractor parallelling to that of the global autonomous systems has been systematically considered in [26]. In the approach presented in [27], to construct the uniform attractor, instead of the associated process {𝒰σ(t,τ),t≥τ,τ∈ℝ}, one should consider a family of processes {𝒰σ(t,s)}, σ∈Σ, in some Banach space E where the functional parameter σ0(s), s∈ℝ, is called the symbol and Σ is the symbol space including σ0(s). The approach implies that the structure of uniform attractor is described by the representation as a union of sections of all kernels of the family of processes. The kernel is the set of all complete trajectories of a process.While in the real world, a system is usually uncertain due to some external noise, which is random. The random effects are considered not only as compensations for the defects in some deterministic models but rather essential phenomena [28–32]. In order to capture the essential dynamics of random dynamical systems with wide fluctuations, the concept of pullback random attractor was introduced in [29, 33, 34], as an extension to stochastic systems of the theory of attractors for deterministic systems in [25, 27, 35–43]. A pullback random attractor 𝒜(ω) which can be constructed by a closed random absorbing set 𝒦(ω) for an asymptotically compact stochastic mapping 𝒮(t,s,ω) is given by
(9)𝒜(ω)=⋂s≥τ‍⋃t≥s‍𝒮(t,τ,θτ-tω,𝒦(θτ-tω))¯,
where θt is the metric process on probability space. The existence of random attractors for stochastic dynamical systems has been investigated extensively by many authors [29, 33, 34, 44–49]. In our paper, we associate with the concepts of uniform attractor and random attractor together and give the concept of uniformly random attractor. Then we consider (8) in an appropriate space and show that there is a uniformly (with respect to f) random attractor 𝒜(ω) which all solutions approach as t→∞. To our best knowledge, the long-time dynamical behavior of the three-dimensional stochastic nonautonomous Camassa-Holm equations has not been discussed, and we believe that it is a significant work to obtain a uniformly (with respect to f) random attractor for the system.The paper is organized as follows. In Section2, we present the abstract results describing the uniformly random attractor and some relevant definitions. In Section 3, we give some functional settings which are foundations for us to obtain the existence of uniformly (with respect to f) random attractor of (8). In Section 4, we convert (8) with an additive noise to deterministic equations with random parameters and define a process corresponding to the equations. In Section 5, we obtain the existence of uniformly (with respect to f) random attractors for (8) on the basis of the above preparations.


## 2. Abstract Results

In this section, we associate with the concepts of uniform attractor and random attractor and obtain the notion of uniformly random attractor. Let(E,∥·∥E) be a separable Hilbert space with the Borel σ-algebra ℬ(E), and let (Ω,ℱ,ℙ) be a probability space.Definition 1.
(
θ
t
)
t
∈
ℝ is called a measurable flow on probability space (Ω,ℱ) if θ:ℝ×Ω→Ω is (ℝ×ℱ,ℱ)-measurable, θ0 is the identity on Ω, θs+t=θt∘θs for all s, t∈ℝ, and θtℙ=ℙ for all t∈ℝ.Definition 2.
A random bounded set{B(ω)}ω∈Ω of E, is called tempered with respect to (θt)t∈ℝ if for ℙ a.e. ω∈Ω,
(10)limt→∞e-βtd(B(θ-tω))=0∀β>0,
where d(B)=supx∈E∥x∥E.Definition 3.
A random set{𝒦(ω)}ω∈Ω is called an absorbing set of a stochastic mapping 𝒮(t,s,ω) in E if for every random bounded set B and ℙ a.e. ω∈Ω, there exists tB(ω)>0 such that
(11)𝒮(t,s,θs-tω)B(θs-t(ω))⊆𝒦(ω)∀t≥tB(ω).Let{𝒰(t,τ,ω)}={𝒰(t,τ,ω),t≥τ,t,τ∈ℝ,ω∈Ω} be a three-parameter family of mappings acting on E:
(12)𝒰(t,τ,ω):E⟶E,t≥τ,t,τ∈ℝ,ω∈Ω.Definition 4.
A three-parameter family of random mappings{𝒰(t,τ,ω)} is said to be a process in E if it is (ℬ(ℝ+)×ℬ(ℝ+)×ℱ×ℬ(E),ℬ(E))-measurable and for ℙ a.e. ω∈Ω it satisfies
(13)𝒰(t,s,θs-τω)∘𝒰(s,τ,ω)=𝒰(t,τ,ω),∀t≥s≥τ,τ∈ℝ,𝒰(τ,τ,ω)=Id,τ∈ℝ.Definition 5.
A family of processes𝒰σ(t,τ,ω), σ∈Σ, acting in E is said to be (E×Σ,E)-continuous, if, for ℙ a.e. ω∈Ω and fixed t, τ, t≥τ, the random mapping (u,σ)↦𝒰σ(t,τ,ω)u is continuous from E×Σ into E.Definition 6.
A random curveu(s,ω), s∈ℝ, is said to be a complete trajectory of the process {𝒰(t,τ,ω)} if for ℙ a.e. ω∈Ω(14){𝒰(t,τ,ω)}u(τ,ω)=u(t,ω),∀t≥τ,t,τ∈ℝ.Definition 7.
The random kernel𝒩(ω) of the process {𝒰(t,τ,ω)} consists of all bounded complete trajectories of the process {𝒰(t,τ,ω)}:
(15)𝒩(ω)={u(·,ω)∣u(·,ω)satisfies Definition6and∥u(s,ω)∥E≤Mu(ω)fors∈ℝ}.Definition 8.
The random set(16)𝒩(s,ω)={u(s,ω)∣u(·,ω)∈𝒩(ω)}⊆E
is said to be the random kernel section at time t=s, s∈ℝ.LetBt(ω)=⋃σ∈Σ‍⋃s≥t‍𝒰σ(s,t,θt-sω)B(θt-sω), where B(ω) is a random set. Denote that B(ω)¯ is closure of the set B(ω) and ℝτ={t∈ℝ∣t≥τ}.Definition 9.
A random set𝒲τ,Σ(B)(ω)=⋂t≥τ‍Bt(ω)¯ is called the uniformly (with respect to σ∈Σ) random (pullback) omega-limit set of B(ω) which can be characterized as follows, analogously to that for semigroups,
(17)y∈𝒲τ,Σ(B)(ω)⟺therearesequences{xn}⊂B(θτ-tnω),{σn}⊂Σ,{tn}⊂ℝτ,such  thattn⟶+∞andsuch  that𝒰σn(tn,τ,θτ-tnω)xn⟶y(n⟶∞).LetB(ω)∈ℬ(E), and its Kuratowski measure of noncompactness κ(B) is defined by
(18)κ(B)=inf{δ>0∣Badmitsafinitcoveringbysetsofdiameter≤δ}.Definition 10.
A family of processes{𝒰σ(t,τ,ω)}, σ∈Σ, is said to be uniformly (with respect to σ∈Σ) random (pullback) omega-limit compact if, for ℙ a.e. ω∈Ω and any τ∈ℝ, the set Bt(ω) is bounded for every t and limt→∞κ(Bt(ω))=0.We now present a method to verify the uniformly (with respect toσ∈Σ) random (pullback) omega-limit compactness.Definition 11.
A family of processes{𝒰σ(t,τ,ω)}, σ∈Σ, is said to satisfy uniformly (with respect to σ∈Σ) condition (C) if, for ℙ a.e. ω∈Ω and any fixed τ∈ℝ, tempered set B(ω)∈ℬ(E), ɛ>0, there exist t0=t(τ,B,ɛ,ω)≥τ and a finite dimensional subspace E1 of E such that(i)
Π(⋃σ∈Σ‍⋃t≥t0‍𝒰σ(t,τ,θτ-tω)B(θτ-tω)) is bounded, and(ii)
∥(I-Π)(⋃σ∈Σ‍⋃t≥t0‍𝒰σ(t,τ,θτ-tω)x)∥≤ɛ, ∀x∈B(θτ-tω),where Π:E→E1 is a bounded projector.Therefore, we have the following results.Theorem 12.
LetΣ be a metric space, and let T(t) be a continuous invariant semigroup T(t)Σ=Σ on Σ. A family of processes {𝒰(t,τ,ω)}, σ∈Σ, acting in E is (E×Σ,E)-continuous (weakly) and possesses the compact uniformly (with respect to σ∈Σ) random attractor 𝒜Σ(ω) satisfying
(19)𝒜Σ(ω)=𝒲0,Σ(B0(ω))=𝒲τ,Σ(B0(ω))=⋃σ∈Σ‍𝒩σ(0,ω),∀τ∈ℝ,
if it(i)
has a bounded uniformly (with respect toσ∈Σ) random absorbing set B0(ω), and(ii)
satisfies uniformly (with respect toσ∈Σ) condition (C).Moreover, ifE is a uniformly convex Banach space, then the converse is true.


## 3. Functional Setting

We consider the probability space(Ω,ℱ,ℙ) where
(20)Ω={ω=(ω1,ω2,…,ωn)T∈C(ℝ,ℝn):ω(0)=0}.ℱ is the Borel σ-algebra induced by the compact-open topology of Ω and ℙ the corresponding Wiener measure on (Ω,ℱ). Then we will identify W(t) with
(21)W(t)=(w1(t),w2(t),…,wn(t))T=ω(t)fort∈ℝ.
Define the time shift by
(22)θtω(·)=ω(·+t)-ω(t),ω∈Ω,t∈ℝ.
Then {θt}t∈ℝ is a family of measure preserving transformations on probability space (Ω,ℱ,ℙ) in Definition 1.Next we define a symbol spaceΣ(σ0) for (8). We assume that the function f(·,t)=:f(t)∈Lloc2(ℝ;E) is translation bounded. That is, for f(s)∈Lb2(ℝ;E), we have
(23)∥f∥Lb22=∥f∥Lb2(ℝ;E)2=supt∈ℝ∫tt+1‍∥f(s)∥E2ds<∞.Definition 13 (cf. [41]).
A functionφ∈Lloc2(ℝ;E) is said to be normal if, for any ɛ>0, there exists δ>0 such that
(24)supt∈ℝ∫tt+δ‍∥φ(s)∥E2ds≤ɛ.We denote byLn2(ℝ;E) the set of all normal functions in Lloc2(ℝ;E). Obviously, Ln2(ℝ;E)⊂Lb2(ℝ;E), and it is proved in [41] that Ln2(ℝ;E) is a closed subset of Lb2(ℝ;E). Let a fixed symbol σ0(s)=f0(s)=f0(·,s) be normal functions in Lloc2(ℝ;E). That is, the family of translation {f0(s+η),η∈ℝ} forms a normal function set in Lloc2([T1,T2];E), where [T1,T2] is an arbitrary interval of the time axis ℝ. Therefore,
(25)Σ(σ0)=Σ(f0)={f0(x,s+η)∣η∈ℝ}Lloc2(ℝ;E).After integrating (8), one can easily see that
(26)ddt∫[0,l]3‍(α02u-α12Δu)dx=∫[0,l]3‍fdx+∫[0,l]3‍QWtdx.
On the other hand, because of the spatial periodicity of solution, we have ∫[0,l]3‍Δudx=0. Then we have
(27)ddt∫[0,l]3‍α02udx=∫[0,l]3‍fdx+∫[0,l]3QWtdx.
That is, the mean of solution is invariant provided that the means of the forcing term and the perturbing term are zero. In this paper, we will consider the forcing term, perturbing term and initial values with spatial means that are zero. That is, we assume ∫[0,l]3‍fdx=0, ∫[0,l]3‍QWtdx=0, and ∫[0,l]3‍uτdx=0.Next we introduce some essential functional spaces.(i)
We denote by𝒱={φ:φ a vector-valued trigonometric polynomial defined on [0,l]3, such that ∇·φ=0 and ∫[0,l]3‍φ(x)dx=0}, and we let H and V be the closure of 𝒱 in L2([0,l]3)3 and in H1([0,l]3)3, respectively. We can observe that H⊥, the orthogonal complement of H in L2([0,l]3)3, is {∇p:p∈H1([0,l]3)} (cf. [38, 41]).(ii)
We denote byP:L2([0,l]3)3→H the L2 orthogonal projection, usually referred to as Helmholtz-Leray projector, and by A=-PΔ the Stokes operator with domain D(A)=H2([0,l]3)3⋂‍V. Note that, in the case of periodic boundary condition, A=-Δ|D(A) is a self-adjoint positive operator with compact inverse. Hence, the space H has an orthogonal basis {ej}j∞ of eigenfunctions of A, that is, Aej=λjej, with
(28)0<λ1≤λ2≤⋯≤λj≤⋯,λj→+∞,asj→∞.
In fact, these eigenvalues have the form |k|24π/L2 with k∈ℤ3/{0}.(iii)
We denote by(·,·) the L2([0,l]3)3 inner product and by ∥·∥2 the corresponding L2([0,l]3)3 norm. By virtue of Poincaré inequality one can show that there is a constant c>0, such that
(29)c∥Au∥2≤∥u∥H2≤c-1∥Au∥2foreveryu∈D(A),c∥A1/2u∥2≤∥u∥H1≤c-1∥A1/2u∥2foreveryu∈V.
Moreover, one can show that V=D(A1/2) (cf. [38, 42]). We denote by V′ the dual of V. Hereafter c will denote a generic scale invariant positive constant which is independent of the physical parameters in the equation.(iv)
Following the notation for the Navier-Stokes equations we denote yhatB(u,v)=P[(u·∇)v], and we set B(v)u=B(u,v) for every u, v∈V. That is, for every fixed v∈V, B(v) is a linear operator acting on u. Note that
(30)(B(u,v),w)=-(B(u,w),v)foreveryu,v,w∈V.We also denote that B~(u,v)=-P(u×(∇×v)) for every u,v∈V. Using the identity
(31)(b·∇)a+∑j=13‍aj∇bj=-b×(∇×a)+∇(a·b),
one can easily show that
(32)(B~(u,v),w)=(B(u,v),w)-(B(w,v),u)=(B(v)u-B*(v)u,w)
for every u, v, w∈V, where B*(v) denotes the adjoint operator of the linear operator B(v) defined above. As a result we have
(33)B~(u,v)=(B(v)-B*(v))uforeveryu,v∈V.The next lemma will present some properties of the bilinear operatorB~.Lemma 14 (cf. [19]).
The operatorB~ can be extended continuously from V×V with values in V′, and it satisfies
(34)|〈B~(u,v),w〉V′|≤c∥u∥21/2∥A1/2u∥21/2∥A1/2v∥2∥A1/2w∥2,|〈B~(u,v),w〉V′|≤c∥A1/2u∥2∥A1/2v∥2∥w∥21/2∥A1/2w∥21/2
for every u, v, w∈V. Moreover,
(35)〈B~(u,v),w〉V′=-〈B~(w,v),u〉V′,foreveryu,v,w∈V
and in particular
(36)〈B~(u,v),u〉V′≡0foreveryu,v∈V.
Furthermore, one has
(37)|〈B~(u,v),w〉D(A)′|≤c∥u∥2∥A1/2v∥2∥A1/2w∥21/2∥Aw∥21/2
for every u∈H, v∈V, and w∈D(A) and by symmetry one has
(38)|〈B~(u,v),w〉D(A)′|≤c∥A1/2u∥21/2∥Au∥21/2∥A1/2v∥2∥w∥2
for every u∈D(A), v∈V, and w∈H. Also
(39)|〈B~(u,v),w〉D(A)′|≤c(∥u∥21/2∥A1/2u∥21/2∥v∥2∥Aw∥2+∥v∥2∥A1/2u∥2∥A1/2w∥21/2∥Aw∥21/2) for every u∈V, v∈H, and w∈D(A). In addition,
(40)|〈B~(u,v),w〉D(A)′|≤c(∥A1/2u∥21/2∥Au∥21/2∥v∥2∥A1/2w∥2+∥v∥2∥Au∥2∥w∥21/2∥A1/2w∥21/2)
for every u∈D(A), v∈H, and w∈V.


## 4. Stochastic Nonautonomous Camassa-Holm Equations

We now apply the result in Section2 to the stochastic nonautonomous Camassa-Holm equations. To associate a family of processes 𝒰σ(t,s,ω) with the stochastic equations over (Σ,T(t)) and (Ω,ℱ,ℙ,(θt)t∈ℝ), we need to convert the stochastic equations with a random additive term into deterministic equations with a random parameter.First we define the bounded linear operatorQ in (8) as follows:
(41)Qx=∑j=1n‍xjhj,x=(x1,x2,…,xn)T∈ℝn,
where hj∈D(A), j=1,2,…,n.Givenj=1,2,…,n, consider the Ornstein-Uhlenbeck equation:
(42)dyj+μyjdt=dwj(t).
One can easily check that a solution to (42) is given by
(43)yj(t)=yj(θtωj)=-μ∫-∞0‍eμτ(θtωj)(τ)dτ,t∈ℝ.
It is known that the random variable |yj(ωj)| is tempered and that yj(θtωj) is ℙ a.e is continuous. Now we put z(θtω)=(α02+α12A)-1∑j=1m‍hjyj(θtωj). By (42) we have
(44)d(α02z(θtω)+α12Az(θtω))+μ(α02z(θtω)dt+α12Az(θtω))dt=∑j=1n‍hjdwj(t).
Employing Cauchy-Schwarz’s inequality, we get
(45)∥z(θtω)∥22+∥A1/2z(θtω)∥22+∥Az(θtω)∥22≤c∑j=1n‍|yj(θtωj)|2,To show that (8) corresponds to a process {𝒰σ(t,τ,ω)}, we let v=u-z(θtω), where u is a solution of (8). Then for v(t,ω), we have
(46)∂∂t(α02v+α12Av)+νA(α02v+α12Av+α02z(θtω)+α12Az(θtω))-(v+z(θtω))×((α02v+α12Av+α02z(θtω)+α12Az(θtω))∇×(α02v+α12Av+α02z(θtω)+α12Az(θtω)))+1ρ0∇p=f(x,t)+μα02z(θtω)+μα12Az(θtω)
defined in the periodic box [0,l]3, satisfying
(47)∇·v=0,v(x,τ,ω)=vτ(x,ω)=uτ(x)-z(θτω)∈H.
We apply P to (46) and use the notation in Section 3 to obtain the equivalent system of equations
(48)∂∂t(α02v+α12Av)+νA(α02v+α12Av+α02z(θtω)+α12Az(θtω))+B~(v+z(θtω),α02v+α12Av+α02z(θtω)+α12Az(θtω))=Pf(x,t)+μα02z(θtω)+μα12Az(θtω)
satisfying the initial condition
(49)v(x,τ,ω)=vτ(x,ω).
By a Galerkin method as in [19], it can be proved that if f(x,t)∈Lloc2((0,T);H) and vτ∈V, for ℙ a.e. ω∈Ω, (48) has a unique solution satisfying for any T>τ(50)v(t,ω,vτ)∈C([τ,T);V)⋂‍L2([τ,T);D(A)),dvdt∈L2([τ,T);H)
and such that, for almost all t∈[τ,T) and for any w∈D(A),
(51)〈ddt(α02v+α12Av),w〉D(A)′+ν〈A(α02v+α12Av+α02z(θtω)+α12Az(θtω)),w〉D(A)′+〈B~(v+z(θtω),α02v+α12Av+α02z(θtω)+α12Az(θtω)),w〉D(A)′=(Pf(x,t)+μα02z(θtω)+μα12Az(θtω),w).Now for anyf(x,t)∈Σ(f0), (48) with f instead of f0 possesses a corresponding process {𝒰f(t,τ,ω)} acting on V. It is analogous to the proof in [27] to prove that, for ℙ a.e ω∈Ω, the family {𝒰f(t,τ,ω)∣f∈Σ(f0)} of processes is (V×Σ(f0);V)-continuous. Let
(52)𝒩f(ω)={vf(x,t,ω)fort∈ℝ∣vf(x,t,ω)issolutionof(48)satisfying∥vf(x,t,ω)∥H≤Mf(ω)∀t∈ℝ}
be the so-called kernel of the process {𝒰f(t,τ,ω)}.


## 5. Uniformly Random Attractor for Stochastic Nonautonomous Camassa-Holm Equation

In [19], the authors have shown that the semigroup corresponding to the autonomous system possesses a global attractor. In [21–24], the authors have proved that the deterministic version of nonautonomous system has a uniform attractor. The main objective of this section is to obtain the existence of uniformly (with respect to f∈Σ(f0)) random attractor for the stochastic nonautonomous Camassa-Holm equations in V and D(A).Lemma 15.
Let{B(ω)}ω∈Ω be tempered and vτ(ω)∈B(ω). Then the process {𝒰f(t,τ,ω)} corresponding to (48) possesses a uniformly (with respect to f∈Σ(f0)) random absorbing set 𝒦0(ω) in V.Proof.
Lettingw=v in (51), we have
(53)12ddt(α02∥v∥22+α12∥A1/2v∥22)+ν(α02∥A1/2v∥22+α12∥Av∥22)=-ν(∑j=1n‍Ahjyj(θtωj),v)-(B~(v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),v)+(Pf(x,t),v)+(∑j=1n‍μhjyj(θtωj),v).
Now we estimate the second term of (53) on the right-hand side. Applying Lemma 14 we have
(54)|(B~(v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),v)|=|(B~(z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),v)|≤c∥A1/2z(θtω)∥21/2∥Az(θtω)∥21/2×∥α02v+α12Av+∑j=1n‍hjyj(θtωj)∥2∥A1/2v∥2+c∥Az(θtω)∥2∥α02v+α12Av+∑j=1n‍hjyj(θtωj)∥2×∥v∥21/2∥A1/2v∥21/2≤c∥A1/2z(θtω)∥2∥v∥22+c∥Az(θtω)∥2∥A1/2v∥22+να124∥Av∥22+c∥A1/2z(θtω)∥2∥Az(θtω)∥2∥A1/2v∥22+c∥A1/2z(θtω)∥2∥Az(θtω)∥2+c∑j=1n‍|yj(θtωj)|2∥A1/2v∥22+c∥Az(θtω)∥2∥v∥22+να124∥Av∥22+c∥Az(θtω)∥22∥v∥22+c∥Az(θtω)∥22∥A1/2v∥22+c∥Az(θtω)∥22+c∑j=1n‍|yj(θtωj)|2∥A1/2v∥22+c∑j=1n‍|yj(θtωj)|2≤να122∥Av∥22+c∑j=1n‍|yj(θtωj)|(α02∥v∥22+α12∥A1/2v∥22)+c∑j=1n‍|yj(θtωj)|2(α02∥v∥22+α12∥A1/2v∥22)+c∑j=1n‍|yj(θtωj)|2.
By Poincaré’s inequality, we have
(55)ν|(∑j=1n‍Ahjyj(θtωj),v)|≤να026∥A1/2v∥22+c∑j=1n‍|yj(θtωj)|2,|(Pf(x,t),v)|≤∥f∥V′∥v∥2≤να026∥A1/2v∥22+c∥f∥V′2,|(∑j=1n‍μhjyj(θtωj),v)|≤να026∥A1/2v∥22+c∑j=1n‍|yj(θtωj)|2.
Associating with the above inequalities and employing Poincaré’s inequality, we have
(56)ddt(α02∥v∥22+α12∥A1/2v∥22)+νλ1(α02∥v∥22+α12∥A1/2v∥22)≤c∑j=1n‍|yj(θtωj)|(α02∥v∥22+α12∥A1/2v∥22)+c∑j=1n‍|yj(θtωj)|2(α02∥v∥22+α12∥A1/2v∥22)+c∑j=1n‍|yj(θtωj)|2+c∥f∥V′2.
Applying Gronwall’s lemma, we have
(57)α02∥v(t,ω)∥22+α12∥A1/2v(t,ω)∥22≤eJ1(τ,t,ω)(α02∥vτ(ω)∥22+α12∥A1/2vτ(ω)∥22)+∫τt‍(c∑j=1n‍|yj(θsωj)|2+c∥f(s)∥V′2)eJ1(s,t,ω)ds,
where
(58)J1(τ,t,ω)=-νλ1t+νλ1τ+c∑j=1n‍∫τt‍|yj(θsωj)|ds+c∑j=1n‍∫τt‍|yj(θsωj)|2ds.
Replacing ω by θ-tω in (57) and (58), we have
(59)α02∥v(t,θ-tω)∥22+α12∥A1/2v(t,θ-tω)∥22≤eJ1(τ-t,0,ω)(α02∥vτ(θ-tω)∥22+α12∥A1/2vτ(θ-tω)∥22)+∫τ-t0‍(c∑j=1n‍|yj(θsωj)|2+c∥f(s+t)∥V′2)eJ1(s,0,ω)ds.
Note that {|yj(θtωj)|}j=1n are stationary and ergodic (cf. [24]), then it follows from ergodic theorem that
(60)limt→∞1t∫-t0‍|yj(θτωj)|dτ=E(|yj(ωj)|).
On the other hand, we have
(61)limt→∞1t∫-t0‍|yj(θτωj)|dτ=E(|yj(ωj)|)=cμ,limt→∞1t∫-t0‍|yj(θτωj)|2dτ=E(|yj(ωj)|2)=cμ.
Employing (61), we have
(62)lims→-∞J1(s,0,ω)s=lims→-∞(νλ1+c∑j=1m‍1s∫s0‍|yj(θτωj)|dτlims→-∞+c∑j=1m‍1s∫s0‍|yj(θτωj)|2dτ)=(νλ1-cμ-cμ).
So for ℙ a.e. ω∈Ω, there are T0(ω)>0 and μ0>0 such that, for s≥T1(ω) and μ≥μ0,
(63)J1(s,0,ω)s=νλ1+c∑j=1m‍1s∫s0‍|yj(θτωj)|dτ+c∑j=1m‍1s∫s0‍|yj(θτωj)|2dτ<0,J1(s,0,ω)⟶-∞,ass⟶-∞.
Since {|yj(ωj)|}j=1n are tempered, the integral
(64)r1(ω)=∫-∞0‍∑j=1n‍|yj(θsωj)|2eJ1(s,0,ω)ds
is convergent. For any t>τ, there exists K∈ℕ such that -K<τ-t, and we have
(65)∫τ-t0‍∥f(s+t)∥V′2eJ1(s,0,ω)ds=∫τ-t0‍∥f0(s+t+η)∥V′2eJ1(s,0,ω)ds≤∫-K0‍∥f0(s+t+η)∥V′2eJ1(s,0,ω)ds.
Then by piecewise integration, we have
(66)∫-K0‍∥f0(s+t+η)∥V′2eJ1(s,0,ω)ds=∑j=1K‍∫-n+j-1-n+j‍∥f0(s+t+η)∥V′2eJ1(s,0,ω)ds.
By differential mean value theorem, for each 1≤j≤K, there exists sj∈[-n+j-1,-n+j], such that
(67)∑j=1K‍∫-n+j-1-n+j‍∥f0(s+t+η)∥V′2eJ1(s,0,ω)ds=∑j=1K‍eJ1(sj,0,ω)∫-n+j-1-n+j‍∥f0(s+t+η)∥V′2ds.
By the property of normal function, there exists Mf0, which just depends on f0, such that
(68)supt∈ℝ∫tt+1‍∥f0(s)∥V′2ds<Mf0.
It is easy to check that the series ∑j=1∞‍eJ1(sj,0,ω) is convergent. Associating with (65)–(68), for any t>τ, we have
(69)∫τ-t0‍∥f(s+t)∥V′2eJ1(s,0,ω)ds≤r2(ω),
where
(70)r2(ω)=Mf0∑j=1∞‍eJ1(sj,0,ω).
Therefore (64) and (69) imply that, for t>τ, the second term on the right-hand side of (59) can be bounded by
(71)∫τ-t0‍(c∑j=1n‍|yj(θsωj)|2+c∥f(s+t)∥V′2)eJ1(s,0,ω)ds≤r3(ω),
where r3(ω)=cr1(ω)+cr2(ω). For vτ(ω)∈B(ω) being tempered, there exists T1(ω)>0 such that, when t≥T2(ω),
(72)eJ1(τ-t,0,ω)(α02∥vτ(θ-tω)∥22+α12∥A1/2vτ(θ-tω)∥22)<r3(ω).
Letting T3(ω)=max{T1(ω),T2(ω)} and r4(ω)=2r3(ω), when t≥T3(ω), we have
(73)α02∥v(t,θ-tω)∥22+α12∥A1/2v(t,θ-tω)∥22≤r4(ω),∀t≥T3(ω),f∈Σ(f0).
We define(74)𝒦1(ω)={{∥A1/2v(ω)∥22}v(ω)∈V∣α02∥v(ω)∥22+α12∥A1/2v(ω)∥22≤r4(ω)}.
In conclusion, 𝒦1(ω) is a uniformly random absorbing set for {𝒰(t,τ,ω)} in V, which complete the proof.Lemma 16.
Let{B(ω)}ω∈Ω be tempered and vτ(ω)∈B(ω). Then the process {𝒰f(t,τ,ω)} corresponding to (48) possesses a uniformly (with respect to f∈Σ(f0)) random absorbing set 𝒦2(ω) in D(A).Proof.
Integrating (56) into [t,t+1] where t≥T3(ω), we have
(75)α02∥v(t+1,ω)∥22+α12∥A1/2v(t+1,ω)∥22+ν∫tt+1‍(α02∥A1/2v(s,ω)∥22+α12∥Av(s,ω)∥22)ds≤(α02∥v(t,ω)∥22+α12∥A1/2v(t,ω)∥22)+c∑j=1n‍∫tt+1‍|yj(θsωj)|+c∑j=1n‍∫t1‍×(α02∥v(s,ω)∥22+α12∥A1/2v(s,ω)∥22)ds+c∑j=1n‍∫tt+1‍|yj(θsωj)|2+c∑j=1n‍∫t=t1‍×(α02∥v(s,ω)∥22+α12∥A1/2v(s,ω)∥22)ds+c∑j=1n‍∫tt+1‍|yj(θsωj)|2ds.
Replacing ω by θ-t-1ω in (75), we have
(76)ν∫tt+1‍(α02∥A1/2v(s,θ-t-1ω)∥22+α12∥Av(s,θ-t-1ω)∥22)ds≤c∑j=1n‍∫tt+1‍|yj(θs-t-1ωj)|(α02∥v(s,θ-t-1ω)∥22≤c∑j=1n‍|yj(θs-t-1ωj)|∫tt‍+α12∥A1/2v(s,θ-t-1ω)∥22)ds+c∑j=1n‍∫tt+1‍|yj(θs-t-1ωj)|2(α02∥v(s,θ-t-1ω)∥22≤c∑j=1n‍|yj(θs-t-1ωj)|∫tt‍+α12∥A1/2v(s,θ-t-1ω)∥22)ds+c∑j=1n‍∫tt+1‍|yj(θs-t-1ωj)|2ds+(α02∥v(t,θ-t-1ω)∥22+α12∥A1/2v(t,θ-t-1ω)∥22).
As the previous consideration in Lemma 15, for s∈[t,t+1) where t≥T2(ω), we have
(77)α02∥v(s,θ-t-1ω)∥22+α12∥A1/2v(s,θ-t-1ω)∥22≤r5(ω),
where r5(ω)=eνλ1r4(ω).
Associating (76) with (77), we have
(78)∫tt+1‍(α02∥A1/2v(s,θ-t-1ω)∥22+α12∥Av(s,θ-t-1ω)∥22)ds≤r6(ω),
where
(79)r6(ω)=cr5(ω)∑j=1n‍∫-10‍|yj(θsωj)|ds+cr5(ω)∑j=1n‍∫-10‍|yj(θsωj)|2ds+c∑j=1n‍∫-10‍|yj(θsωj)|2ds+r5(ω).
Now we letw=Av in (51), and we have
(80)12ddt(α02∥A1/2v∥22+α12∥Av∥22)+ν(α02∥Av∥22+α12∥A3/2v∥22)=-ν(∑j=1n‍Ahjyj(θtωj),Av)-(B~({∑j=1n‍}v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),Av)+(Pf(x,t),Av)+(∑j=1n‍μhjyj(θtωj),Av).
Employing Lemma 14, the bilinear term in (80) can be bounded by
(81)(B~(v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),Av)≤c({∑j=1n‍}∥A1/2v+A1/2z(θtω)∥21/2∥Av+Az(θtω)∥21/2×∥α02v+α12Av+∑j=1n‍hjyj(θtωj)∥2∥A3/2v∥2)+c(∥Av+Az(θtω)∥2∥α02v+α12Av+∑j=1n‍hjyj(θtωj)∥2×∥Av∥21/2∥A3/2v∥21/2{∥α02v+α12Av+∑j=1n‍hjyj(θtωj∥2})≤να12∥A3/2v∥22+να024∥Av∥22+c(α02∥v∥22+α02∥A1/2v∥22)2+c(α02∥A1/2v∥22+α02∥Av∥22)2+c(α02∥v∥22+α02∥A1/2v∥22)(α02∥A1/2v∥22+α02∥Av∥22)+c∑j=1n‍|yj(θtωj)|2(α02∥v∥22+α02∥A1/2v∥22)+c∑j=1n‍|yj(θtωj)|2(α02∥A1/2v∥22+α02∥Av∥22)+c∑j=1n‍|yj(θtωj)|4.
The other terms are bounded by
(82)ν|(∑j=1n‍Ahjyj(θtωj),Av)|≤να024∥Av∥22+c∑j=1n‍|yj(θtωj)|2.|(Pf(x,t),Av)|≤∥f∥V′∥Av∥2≤να024∥Av∥22+c∥f∥V′2.μ|(∑j=1n‍hjyj(θtωj),Av)|≤να024∥Av∥22+c∑j=1n‍|yj(θtωj)|2.
Associating with all the above inequalities, we have
(83)ddt(α02∥A1/2v∥22+α12∥Av∥22)≤c(α02∥v∥22+α12∥A1/2v∥22)2+c(α02∥A1/2v∥22+α12∥Av∥22)2+c∑j=1n‍|yj(θtωj)|2+c∑j=1n‍|yj(θtωj)|4+c(α02∥v∥22+α12∥A1/2v∥22)×(α02∥A1/2v∥22+α12∥Av∥22)+c∑j=1n‍|yj(θtωj)|2(α02∥v∥22+α12∥A1/2v∥22)+c∥f∥V′2+c∑j=1n‍|yj(θtωj)|2(α02∥A1/2v∥22+α12∥Av∥22).
Applying Gronwall’s lemma on [s,t+1] where s∈(t,t+1) and t≥T3(ω), we have
(84)α02∥A1/2v(t+1,ω)∥22+α12∥Av(t+1,ω)∥22≤(α02∥A1/2v(s,ω)∥22+α12∥Av(s,ω)∥22)eJ2(s,t+1,ω)+c∫st+1‍∑j=1n‍|yj(θτωj)|2eJ2(τ,t+1,ω)dτ+c∫st+1‍∑j=1n‍|yj(θτωj)|4eJ2(τ,t+1,ω)dτ+c∫st+1‍∥f(τ)∥V′2eJ2(τ,t+1,ω)dτ+c∫st+1‍∑j=1n‍|yj(θτωj)|2c∫s=1‍×(α02∥v(τ,ω)∥22+α12∥A1/2v(τ,ω)∥22)eJ2(τ,t+1,ω)dτ+c∫st+1‍(α02∥v(τ,ω)∥22+α12∥A1/2v(τ,ω)∥22)2eJ2(τ,t+1,ω)dτ,
where
(85)J2(s,t,ω)=c∫st‍(α02∥v(τ,ω)∥22+α12∥A1/2v(τ,ω)∥22)dτ+c∫st‍(α02∥A1/2v(τ,ω)∥22+α12∥Av(τ,ω)∥22)dτ+c∫st‍∑j=1n‍|yj(θτωj)|2dτ.
Replacing ω by θ-t-1ω in (84) and (85), we have
(86)J2(τ,t+1,θ-t-1ω)≤c∫tt+1‍({∥A1/2v(τ,θ-t-1ω)∥22}α02∥v(τ,θ-t-1ω)∥22≤c∫t1‍+α12∥A1/2v(τ,θ-t-1ω)∥22)dτ+c∫tt+1‍(α02∥A1/2v(τ,θ-t-1ω)∥22+c∫t1‍+α12∥Av(τ,θ-t-1ω)∥22{α02∥A1/2v(τ,θ-t-1ω)∥22})dτ+c∑j=1n‍∫tt+1‍|yj(θτ-t-1ωj)|2dτ≤cr5(ω)+cr6(ω)+c∑j=1n‍∫-10‍|yj(θτωj)|2dτ=∆r7(ω).
And then
(87)α02∥A1/2v(t+1,θ-t-1ω)∥22+α12∥Av(t+1,θ-t-1ω)∥22≤(α02∥A1/2v(s,θ-t-1ω)∥22+α12∥Av(s,θ-t-1ω)∥22{α02∥A1/2v(s,θ-t-1ω)∥22})er7(ω)+cer7(ω)r5(ω)2+cer7(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ+cer7(ω)∫-10‍∑j=1n‍|yj(θτωj)|4dτ+cer7(ω)∫tt+1‍∥f(τ)∥V′2dτ+cer7(ω)r5(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ.
Integrating (87) with respect to s over (t,t+1) where t≥T3(ω) and employing the property of normal function in (68), we have
(88)α02∥A1/2v(t+1,θ-t-1ω)∥22+α12∥Av(t+1,θ-t-1ω)∥22≤r8(ω),∀t≥T3(ω),f∈Σ(f0),
where(89)r8(ω)=r6(ω)er7(ω)+cer7(ω)r5(ω)2+cer7(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ+cer7(ω)∫-10‍∑j=1n‍|yj(θτωj)|4dτ+cer7(ω)Mf0+cer7(ω)r5(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ.
We define(90)𝒦2(ω)={v(ω)∈D(A)∣α02∥A1/2v(ω)∥22+α12∥Av(ω)∥22≤r8(ω)}.
In conclusion, 𝒦2(ω) is a uniformly random absorbing set for {𝒰(t,τ,ω)} in D(A), which complete the proof.So much for that we have proved the existence of bounded uniformly (with respect tof∈Σ(f0)) random absorbing sets in V and D(A). Next we derive the existence of uniformly (with respect to f∈Σ(f0)) random attractors in V and D(A).Theorem 17.
Iff0(x,s) is a normal function in Lloc2(ℝ,V′), then the process {𝒰f0(t,τ,ω)} corresponding to (48) possess a compact uniformly (with respect to τ∈ℝ) random attractor 𝒜1(ω) in V which coincides with the uniformly (with respect to f∈Σ(f0)) random attractor 𝒜Σ(f0) of the family of processes {𝒰f(t,τ,ω)}, f∈Σ(f0):(91)𝒜1(ω)=𝒜Σ(f0)(ω)=𝒲0,Σ(f0)(𝒦1)=⋃f∈Σ(f0)‍𝒩f(0,ω),
where 𝒦1(ω) is the uniformly (with respect to f∈Σ(f0)) random absorbing set in V and 𝒩f(ω) is the kernel of the process {𝒰f(t,τ,ω)}. Furthermore, the kernel 𝒩f(ω) is nonempty for all f∈Σ(f0).Proof.
We only have to verify condition(C). As the previous section, for fixedm, let H1 be the subspace spanned by {ej}j=1m and H2 the orthogonal complement of H1 in H. We write(92)v=v1+v2,v1∈H1,v2∈H2foranyv∈H.
The proof of boundary of v1 is similar to the proof in Lemma 15. We need to estimate v2, where v=v1+v2 is a solution of (48). Letting w=v2 in (51), we have
(93)12ddt(α02∥v2∥22+α12∥A1/2v2∥22)+ν(α02∥A1/2v2∥22+α12∥Av2∥22)=-ν(∑j=1n‍Ahjyj(θtωj),v2)-(B~({∑j=1n‍}v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),v2)+(Pf(x,t),v2)+μ(∑j=1n‍hjyj(θtωj),v2).
Employing Lemma 14, the second term on the right-hand side of (93) can be bounded by
(94)|(B~({∑j=1n‍}v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),v2)|≤c∥v+z(θtω)∥21/2∥A1/2v+A1/2z(θtω)∥21/2×∥α02v+α12Av+∑j=1n‍hjyj(θtωj)∥2∥Av2∥2+c∥A1/2v+A1/2z(θtω)∥2×∥α02v+α12Av+∑j=1n‍hjyj(θtωj)∥2×∥A1/2v2∥21/2∥Av2∥21/2≤να028∥A1/2v2∥22+να122∥Av2∥22+c(α02∥v∥22+α12∥A1/2v∥22)×(α02∥A1/2v∥22+α12∥Av∥22)+c(α02∥v∥22+α12∥A1/2v∥22)2+c(α02∥A1/2v∥22+α12∥Av∥22)2+c∑j=1n‍|yj(θjωj)|2+c∑j=1n‍|yj(θjωj)|4+c∑j=1n‍|yj(θjωj)|2(α02∥v∥22+α12∥A1/2v∥22)+c∑j=1n‍|yj(θjωj)|2(α02∥A1/2v∥22+α12∥Av∥22).
By Poincaré’s inequality, note that
(95)ν|(∑j=1n‍Ahjyj(θjωj),v2)|≤να028∥A1/2v2∥22+c∑j=1n‍|yj(θtωj)|2,|(Pf,v2)|≤∥f∥V′∥v2∥≤να028∥A1/2v2∥22+c∥f∥V′2,μ|(∑j=1n‍hjyj(θtωj),v2)|≤να028∥A1/2v2∥22+c∑j=1n‍|yj(θjωj)|2.
Associating with (93)–(95) and applying Poincaré inequality, we have
(96)ddt(α02∥v2∥22+α12∥A1/2v2∥22)+νλm+1(α02∥v2∥22+α12∥A1/2v2∥22)≤c∥f∥V′2+G(t,ω),
where
(97)G(t,ω)=c(α02∥v∥22+α12∥A1/2v∥22)2+c(α02∥A1/2v∥22+α12∥Av∥22)2+c∑j=1n‍|yj(θtωj)|2+c(α02∥v∥22+α12∥A1/2v∥22)×(α02∥A1/2v∥22+α12∥Av∥22)+c∑j=1n‍|yj(θtωj)|4+c∑j=1n‍|yj(θtωj)|2(α02∥v∥22+α12∥A1/2v∥22)+c∑j=1n‍|yj(θtωj)|2(α02∥A1/2v∥22+α12∥Av∥22).
Applying Gronwall Lemma on [t,t+1] where t≥T3(ω), we have
(98)α02∥v2(t+1,ω)∥22+α12∥A1/2v2(t+1,ω)∥22≤e-νλm+1(α02∥v2(t,ω)∥22+α12∥A1/2v2(t,ω)∥22)+∫tt+1‍G(s,ω)e-νλm+1ds+c∫tt+1‍∥f(s)∥V′2e-νλm+1ds.
Replacingω by θ-t-1ω in (98), we have
(99)α02∥v2(t+1,θ-t-1ω)∥22+α12∥A1/2v2(t+1,θ-t-1ω)∥22≤e-νλm+1({∥A1/2v2(t,θ-t-1ω)∥22}α02∥v2(t,θ-t-1ω)∥22≤e-νλm+1+α12∥A1/2v2(t,θ-t-1ω)∥22)+∫tt+1‍G(s,θ-t-1ω)e-νλm+1ds+c∫tt+1‍∥f(s)∥V′2e-νλm+1ds.
Now we need to estimate the second term on the right-hand side of (99). As the previous consideration in Lemma 15, for s∈[t,t+1) where t≥T3(ω), we have
(100)α02∥v(s,θ-t-2)∥22+α12∥A1/2v(s,θ-t-2)∥22≤r9(ω),
where r9(ω)=e2νλ1r4(ω). As the previous consideration in Lemma 16, we have
(101)∫tt+1‍(α02∥A1/2v(s,θ-t-2ω)∥22∫tt+1‍+α12∥Av(s,θ-t-2ω)∥22{∥A1/2v(s,θ-t-2ω)∥22})ds≤r10(ω),
where
(102)r10(ω)=cr9(ω)∑j=1n‍∫-20‍|yj(θsωj)|ds+cr9(ω)∑j=1n‍∫-20‍|yj(θsωj)|2ds+c∑j=1n‍∫-20‍|yj(θsωj)|2ds+r9(ω).
Replacing ω by θ-t-2ω in (84), we have
(103)α02∥A1/2v(t+1,θ-t-2ω)∥22+α12∥Av(t+1,θ-t-2ω)∥22≤(α02∥A1/2v(s,θ-t-2ω)∥22+α12∥Av(s,θ-t-2ω)∥22{∥A1/2v(s,θ-t-2ω)∥22})eJ2(s,t+1,θ-t-2ω)+c∫tt+1‍∑j=1n‍|yj(θτ-t-2ωj)|2eJ2(τ,t+1,θ-t-2ω)dτ+c∫tt+1‍∑j=1n‍|yj(θτ-t-2ωj)|4eJ2(τ,t+1,θ-t-2ω)dτ+c∫tt+1‍∑j=1n‍|yj(θτ-t-2ωj)|2+c∫t1‍×({∥A1/2v(τ,θ-t-2ω)∥22}α02∥v(τ,θ-t-2ω)∥22+c∫tt+1‍+α12∥A1/2v(τ,θ-t-2ω)∥22)eJ2(τ,t+1,θ-t-2ω)dτ+c∫tt+1‍({∥A1/2v(τ,θ-t-2ω)∥22}α02∥v(τ,θ-t-2ω)∥22+c∫tt+1‍+α12∥A1/2v(τ,θ-t-2ω)∥22)2eJ2(τ,t+1,θ-t-2ω)dτ+c∫tt+1‍∥f(τ)∥V′2eJ2(τ,t+1,θ-t-2ω)dτ.
Analogously to the previous estimates, we have
(104)J2(τ,t+1,θ-t-2ω)≤c∫tt+1‍({∥A1/2v(τ,θ-t-2ω)∥22}α02∥v(τ,θ-t-2ω)∥22≤c∫tt+1‍+α12∥A1/2v(τ,θ-t-2ω)∥22)dτ+c∫tt+1‍(α02∥A1/2v(τ,θ-t-2ω)∥22+c∫tt+1‍+α12∥Av(τ,θ-t-2ω)∥22{∥A1/2v(τ,θ-t-2ω)∥22})dτ+c∑j=1n‍∫tt+1‍|yj(θτ-t-2ωj)|2dτ≤r11(ω),
where
(105)r11(ω)=cr9(ω)+cr10(ω)+c∑j=1n‍∫-20‍|yj(θτωj)|2dτ.
Integrating (103) with respect to s∈(t,t+1) associating (100), (101), and (104) with (68), we have
(106)α02∥A1/2v(t+1,θ-t-2ω)∥22+α12∥Av(t+1,θ-t-2ω)∥22≤r12(ω),
where
(107)r12(ω)=cr10(ω)er11(ω)+cer11(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ+cer11(ω)∫-10‍∑j=1n‍|yj(θτωj)|4dτ+cr9(ω)er11(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ+cr9(ω)2er11(ω)+cMf0er11(ω).
Replacing s, t+1, and ω by t, s, and θ-t-1ω, respectively, in (84) where s∈(t,t+1) and t≥T3(ω), we have
(108)α02∥A1/2v(s,θ-t-1ω)∥22+α12∥Av(s,θ-t-1ω)∥22≤(α02∥A1/2v(t,θ-t-1ω)∥22+α12∥Av(t,θ-t-1ω)∥22{∥A1/2v(t,θ-t-1ω)∥22})eJ2(t,t+1,θ-t-1ω)+c∫tt+1‍∑j=1n‍|yj(θτ-t-1ωj)|2eJ2(τ,t+1,θ-t-1ω)dτ+c∫tt+1‍∑j=1n‍|yj(θτ-t-1ωj)|4eJ2(τ,t+1,θ-t-1ω)dτ+c∫tt+1‍∑j=1n‍|yj(θτ-t-1ωj)|2+c∫t1‍×({∥A1/2v(τ,θ-t-1ω)∥22}α02∥v(τ,θ-t-1ω)∥22+α12∥A1/2v(τ,θ-t-1ω)∥22)eJ2(τ,t+1,θ-t-1ω)dτ+c∫tt+1‍({∥A1/2v(τ,θ-t-1ω)∥22}α02∥v(τ,θ-t-1ω)∥22+c∫tt+1‍+α12∥A1/2v(τ,θ-t-1ω)∥22)2eJ2(τ,t+1,θ-t-1ω)dτ+c∫tt+1‍∥f(τ)∥V′2eJ2(τ,t+1,θ-t-1ω)dτ≤r12(ω)er7(ω)+cer7(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ+cer7(ω)∫-10‍∑j=1n‍|yj(θτωj)|4dτ+cer7(ω)Mf0+cr5(ω)2er7(ω)+cr5(ω)er7(ω)∫-10‍∑j=1n‍|yj(θτωj)|2dτ=∆r13(ω).
Associating with the above inequalities, we have
(109)∫tt+1‍G(s,θ-t-1ω)ds≤r14(ω),
where
(110)r14(ω)=cr5(ω)2+cr13(ω)2+c∫-10‍∑j=1n‍|yj(θsωj)|2ds+cr5(ω)r13(ω)+c∫-10‍∑j=1n‍|yj(θsωj)|4ds+cr5(ω)∫-10‍∑j=1n‍|yj(θsωj)|2ds+cr13(ω)∫-10‍∑j=1n‍|yj(θsωj)|2ds.
According to the property of {λj}j=1∞ mentioned in Section 3, for ℙ a.e ω∈Ω and ∀ɛ>0, there exists M such that for m≥M(111)e-νλm+1({∥A1/2v2(t,θ-t-1ω)∥22}α02∥v2(t,θ-t-1ω)∥22+α12∥A1/2v2(t,θ-t-1ω)∥22)≤r5(ω)eνλm+1≤ɛr5(ω),∫tt+1‍G(s,θ-t-1ω)e-νλm+1ds≤r14(ω)eνλm+1≤ɛr14(ω),c∫tt+1‍∥f(s)∥V′2e-νλm+1ds≤cMf0eνλm+1≤ɛ.
Therefore, we deduce from (99) that, for ℙ a.e. ω∈Ω,
(112)α02∥v2(t,θ-tω)∥22+α12∥A1/2v2(t,θ-tω)∥22≤ɛ+ɛr5(ω)+ɛr14(ω),∀t≥T3(ω),f∈Σ(f0),
which indicates {𝒰f(t,τ,ω)}, f∈Σ(f0) satisfying uniform (with respect to f∈Σ(f0)) condition(C) in V. According to Theorem 12, the proof is completed.In the following we prove the existence of uniformly random attractor for the families of processes{𝒰f(t,τ,ω)}, f∈Σ(f0) corresponding to (48) in D(A).Theorem 18.
Iff0(x,s) is a normal function in Lloc2(ℝ,V′), then the process {𝒰f0(t,τ,ω)} corresponding to (48) possess a compact uniformly (with respect to τ∈ℝ) random attractor 𝒜2(ω) in D(A) which coincides with the uniformly (with respect to f∈Σ(f0)) random attractor 𝒜Σ(f0) of the family of processes {𝒰f(t,τ,ω)}, f∈Σ(f0):(113)𝒜2(ω)=𝒜Σ(f0)(ω)=𝒲0,Σ(f0)(𝒦2)=⋃f∈Σ(f0)‍𝒩f(0,ω),
where 𝒦2(ω) is the uniformly (with respect to f∈Σ(f0)) random absorbing set in D(A) and 𝒩f(ω) is the kernel of the process {𝒰f(t,τ,ω)}. Furthermore, the kernel 𝒩f(ω) is nonempty for all f∈Σ(f0).Proof.
In Lemma16, we have proved that the semigroup of processes {𝒰f(t,τ,ω)}, f∈Σ(f0), has a uniformly random absorbing set in D(A). Now we testify that the semigroup of processes corresponding to (48) satisfies uniform (with respect to f∈Σ(f0)) condition (C). Analogously to the proof in Lemma 16, we easily check that v1 is bounded in D(A). Letting w=Av2 in (51), we have
(114)12ddt(α02∥A1/2v2∥22+α12∥Av2∥22)+ν(α02∥Av2∥22+α12∥A3/2v2∥22)=-ν(∑j=1n‍Ahjyj(θtωj),Av2)-(B~({∑j=1n‍}v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),Av2)+(Pf(x,t),Av2)+μ(∑j=1n‍hjyj(θtωj),Av2).
Applying Lemma 14, the second term on the right-hand side of (114) can be bounded by
(115)|(B~({∑j=1n‍}v+z(θtω),α02v+α12Av+∑j=1n‍hjyj(θtωj)),Av2)|≤να028∥Av2∥22+να122∥A3/2v2∥22+c(α02∥v∥22+α12∥A1/2v∥22)×(α02∥A1/2v∥22+α12∥Av∥22)+c(α02∥v∥22+α12∥A1/2v∥22)2+c(α02∥A1/2v∥22+α12∥Av∥22)2+c∑j=1n‍|yj(θtωj)|2+c∑j=1n‍|yj(θtωj)|4+c∑j=1n‍|yj(θtωj)|2(α02∥v∥22+α12∥A1/2v∥22)2+c∑j=1n‍|yj(θtωj)|2(α02∥A1/2v∥22+α12∥Av∥22)2.
Note that
(116)ν|(∑j=1n‍hjyj(θtωj),Av2)|≤να028∥Av2∥22+c∑j=1n‍|yj(θtωj)|2,|(Pf(x,t),Av2)|≤∥f∥V′∥Av2∥2≤να028∥Av2∥22+c∥f∥V′2,μ|(∑j=1n‍hjyj(θtωj),Av2)|≤να028∥Av2∥22+c∑j=1n‍|yj(θtωj)|2.
Associating with (114)–(116) and applying Poincaré’s inequality, we have
(117)ddt(α02∥A1/2v2∥22+α12∥Av2∥22)+νλm+1(α02∥A1/2v2∥22+α12∥Av2∥22)≤G(t,ω)+c∥f∥V′2.
Applying Gronwall’s lemma over s∈[t,t+1] where t≥T2(ω), we have
(118)α02∥A1/2v2(t+1,ω)∥22+α12∥Av2(t+1,ω)∥22≤(α02∥A1/2v2(t,ω)∥22+α12∥Av2(t,ω)∥22)e-νλm+1+∫tt+1‍G(s,ω)e-νλm+1ds+c∫tt+1‍∥f∥V′2e-νλm+1ds.
Replacing ω by θ-t-1ω in (118), we have
(119)α02∥A1/2v2(t+1,θ-t-1ω)∥22+α12∥Av2(t+1,θ-t-1ω)∥22≤(α02∥A1/2v2(t,θ-t-1ω)∥22+α12∥Av2(t,θ-t-1ω)∥22{∥A1/2v2(t,θ-t-1ω)∥22})e-νλm+1+∫tt+1‍G(s,θ-t-1ω)e-νλm+1ds+c∫tt+1‍∥f∥V′2e-νλm+1ds.
Analogously to the consideration in Theorem 17, for ℙ a.e. ω∈Ω and ∀ɛ>0, there exists M such that, for m>M,
(120)e-νλm+1(α02∥A1/2v2(t,θ-t-1ω)∥22+α12∥Av2(t,θ-t-1ω)∥22{∥A1/2v2(t,θ-t-1ω)∥22})≤r12(ω)eνλm+1≤ɛr12(ω),∫tt+1‍G(s,θ-t-1ω)e-νλm+1ds≤r14(ω)eνλm+1≤ɛr14(ω),c∫tt+1‍∥f(s)∥V′2e-νλm+1ds≤cMf0eνλm+1≤ɛ.
Therefore, we deduce from (119) that, for ℙ a.e ω∈Ω,
(121)α02∥A1/2v2(t+1,θ-t-1ω)∥22+α12∥Av2(t+1,θ-t-1ω)∥22≤ɛ+ɛr12(ω)+ɛr14(ω),∀t≥T3(ω),f∈Σ(f0).
which indicates {𝒰f(t,τ,ω)}, f∈Σ(f0) satisfying uniform (with respect to f∈Σ(f0)) condition (C) in D(A). According to Theorem 12, the proof is completed.Now we introduce a homeomorphismΦ(θtω)u=u+z(θtω),u∈E, whose inverse homeomorphism Φ-1(θtω)u=u-z(θtω). Then the transformation
(122)𝒱σ(t,τ,ω)=Φ(θtω)∘𝒰σ(t,τ,ω)∘Φ-1(θτω)
generates a process corresponding to (8). Note that the two processes are equivalent by (122). It is easy to check that 𝒱σ(t,τ,ω) has a uniformly (with respect to σ∈Σ) random attractor provided 𝒰σ(t,τ,ω) possesses a uniformly (with respect to σ∈Σ) random attractor. As a result Theorems 17 and 18 imply that (8) has a uniformly (with respect to f∈Σ(f0)) random attractor in V and D(A).



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*Source: 492847-2013-09-09.xml*