Global Mild Solutions and Attractors for Stochastic Viscous Cahn-Hilliard Equation

Volume 2011, Article ID 670786,22pages doi:10.1155/2011/670786

Research Article

Global Mild Solutions and Attractors for Stochastic Viscous Cahn-Hilliard Equation

Xuewei Ju,

Hongli Wang,

Desheng Li,

and Jinqiao Duan

1Department of Mechanic, Mechanical College, Tianjin University, Tianjin 300072, China

2Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China

3Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

Correspondence should be addressed to Xuewei Ju,xwjumath@hotmail.com Received 22 March 2011; Accepted 19 May 2011

Academic Editor: Nicholas D. Alikakos

Copyrightq2011 Xuewei Ju et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn-Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.

1. Introduction

This paper is devoted to the existence of mild solutions and global asymptotic behavior for the following stochastic viscous Cahn-Hilliard equation:

d1−αu−αΔu

Δ²u−Δfu

dtdW, x, t∈G×t0,∞, 1.1

subjected to homogeneous Dirichlet boundary conditions

ux, t 0, x, t∈∂G×t0,∞, 1.2

in dimensionn 1,2 or 3, whereG _n

i10, LiinRⁿ, andα∈ 0,1is a parameter,f is a polynomial of odd degree with a positive leading coefficient

fx ^2p−1

k1

akx^k, a2p−1>0. 1.3

In deterministic case, the model was first introduced by Novick-Cohen1to describe the dynamics of viscous first order phase transitions, which has been extensively studied in the past decades. The existence of global solutions and attractors are well known; moreover, the global attractorAα of the system has the same finite Hausdorffdimension for different parameter valuesα. One can also show thatAαis continuous asαvaries in0,1. See2for details and1for recent development.

While the deterministic model captures more intrinsic nature of phase transitions in binary, it ignores some random effects such as thermal fluctuations which are present in any material. In recent years, there appeared many interesting works on stochastic Cahn-Hilliard equations. Cardon-Weber3proved the existence of solution as well as its density for a class of stochastic Cahn-Hilliard equations with additive noise using an appropriate convolution semigroupin the sense of that in4posed on cubic domains. The authors in5derived the existence for a generalized stochastic Cahn-Hilliard equation in general convex or Lipschitz domains. The main novelty was the derivation of space-time H ¨older estimates for the Greens kernel of the stochastic problem, by using the domains geometry, which can be very useful in many other circumstances. In6, the asymptotic behavior for a generalized Cahn-Hilliard equation was studied, which can also act as a very good toy model for treating the stochastic case.

Instead of deterministic viscous Cahn-hilliard equation, here, we consider the general stochastic equation 1.1 which is affected by a space-time white noise. In such a case, new difficulties appear, and the resulting stochastic model must be treated in a different way. Fortunately, the rapidly growing theory of random dynamical systems provides an appropriate tool. Crauel and Flandoli7 see also Schmalfuss8introduced the concept of a random attractor as a proper generalization of the corresponding deterministic global attractor which turns out to be very helpful in the understanding of the long-time dynamics for stochastic differential equations. In this present work, we first establish some existence results on mild solutions. Then, by applying the abstract theory on stochastic attractors mentioned above, we show that the system has global attractors in appropriate phase spaces.

In caseα0,1.1reduces to the stochastic Cahn-Hilliard equation which was studied in9, where the authors obtain the existence and uniqueness of the weak solutions to the initial and Neumann boundary value problem in some phase spaces under appropriate assumptions on noise. Here, we make slightly stronger assumptions on noise and prove existence and uniqueness of mild solutions with higher regularity. Furthermore, we show the existence of random attractors in appropriate phase spaces.

This paper is organized as follows. In Section2, we first make some preliminary works, then we state our main results. In Section3, we consider the solutions of the the linear part of the system 1.1-1.2 and stochastic convolution. Regularities of solutions will also be addressed in this part. Section4consists of some investigations on the Stochastic Lyapunov functional of the system. The proofs on the existence results for mild solutions and global attractors will be given in Sections5and6, respectively. Finally, the last section stands as an appendix for some basic knowledge of random dynamical systemRDS.

2. Preliminaries and Main Results

In this section, we first make some preliminary works, then we state explicitly our main results.

2.1. Functional Spaces

Let·,· and| · |denote respectively the inner product and norm ofHL²G. We define the linear operatorA −Δwith domainDA H²G

H₀¹G.Ais positive and selfadjoint.

By spectral theory, we can define the powersA^sand spacesHs DA^s/2with norms|u|s

|A^s/2u|for reals. Note that H0 L²G. It is well known that Hs is a subspace of H^sG and| · |_sis onH^sGa norm equivalent to the usual one. Moreover, we have the following Poincare inequality and interpolation inequality:

|u|_s1 ≤λ^−s₁ ^2−s1/2|u|_s2, ∀s1, s2∈R, s1< s2, ∀u∈Hs2, 2.1

|u|_σs11−σs2≤ |u|^σs1|u|^1−σ_s2 , σ∈0,1, 2.2

whereλ1is the first eigenvalue ofA.

We can defineA⁻¹:H → DAto be the Green’s operator forA. Thus,

vA⁻¹w⇐⇒Avw. 2.3

By Rellich’s Theorem, we know thatA⁻¹ is compact, and A : DA → H is a linear and bounded operator. Finally, we introduce the invertible operatorBα:Hs → Hs,s∈Rdefined by

Bα:αI 1−αA⁻¹. 2.4

For eachα∈ 0,1andβ ≥0, we know thatBα^β :Hs → Hsis bounded and has a bounded inversesee10,11. We also define the operatorAα:B⁻¹_α Awith domain

DAα

⎧⎨

⎩

DA ifα >0,

DA0 H4. 2.5

By definition, it is clear thatDA^s/2α Hsin caseα >0.

Lemma 2.1. Forα >0, there existM1, M2, andM3such that

α^1/2|v| ≤ |v|_Bα ≤M₁^1/2|v|, v∈H, 2.6 α^1/2|v|₁≤ |v|_1,Bα ≤M^1/2₂ |v|₁, v∈H1, 2.7 λ1

αλ11−α _1/2

|v| ≤ |v|_Bα⁻¹ ≤M₃^1/2|v|, v∈H, 2.8

where

|v|_Bα : v, Bαv^1/2,

|v|_1,Bα :

A^1/2v, BαA^1/2v1/2

,

|v|_Bα⁻¹ :

v, B_α⁻¹v_1/2 .

2.9

Proof. Here, we only verify2.8is valid; the proofs of2.6and2.7can be found in11.

SinceB^−1/2α is bounded, there existsM3≥0, such that|B^−1/2α |²≤M3. Then, for anyv∈H, we have

v, B⁻¹_α v

B_α^−1/2v, B_α^−1/2v

B_α^−1/2u²≤M3|v|², 2.10 which completes the right part of2.8.

Now, we proof the left part of2.8let

0< λ1≤λ2 ≤ · · · ≤λk≤ · · · 2.11 denote the eigenvalues ofA, repeated with the respective multiplicity, and the corresponding unit eigenvector is denoted by{wk}^∞_k1, which forms an orthonormal basis forH. We have

wk, B⁻¹_α wk

λk

αλk1−α ≥ λ1

αλ11−α, k∈Z. 2.12 Sincev∈H, there exist{bk}^∞_k1⊂R, such thatv_∞

k1bkwk. Consequently,

v, B⁻¹_α v

_∞

k1

bkwk, B_α⁻¹ ∞ k1

bkwk

^∞

k1

bkwk, B_α⁻¹bkwk

^∞

k1

λk

αλk1−αb²_k≥ λ1

αλ11−α ∞ k1

b²_k

λ1

αλ11−α|v|²,

2.13

which finishes the proof.

2.2. Assumptions on the Noise

The stochastic processWt, defined on a probability space Ω,F,P, is a two-side in time Wiener process onHwhich is given by the expansions

Wt ^∞

k0

√αkβktwk, 2.14

where {wk}^∞_k1 is a basis ofH consisting of unit eigenvectors of A,{αk}^∞_k1 is a bounded sequence of nonnegative numbers, and

βkt 1

√αkWt, wk, k∈N 2.15

is a sequence of mutually independent real valued standard Brownian motions in a fixed probability spaceΩ,F,Padapted to a filtration{Ft}_t≥0.

For convenience, we will define the covariance operatorQonHas follows:

Qwkαkwk, k∈N. 2.16

The processWtwill be called as theQ-Wiener process. We need to impose onQone of the following assumptions:

Q1TrB^−1−δ_α A^−2δQ<∞for some 0< δ≤1, Q1^∗TrB⁻²_α A⁻¹Q<∞,and TrB_α⁻²A⁻²Q≤2D,

Q2TrB^−1−δ_α A^−1δQ<∞for some 0< δ≤1, TrB⁻²_α Q<∞, and TrB_α⁻²A⁻²Q≤2D, Q2^∗TrB^−1−δ_α A^−1δQ < ∞, TrB⁻²_α A^σQ < ∞for some 0 < δ ≤ 1 and σ > 0, and

TrB⁻²_α A⁻²Q≤2D,

whereDis given in Section4. It is obvious that

Q2^∗ ⇒Q2, Q1^∗ ⇒Q1. 2.17

2.3. Main Results

We will assume throughout the paper that the space dimensionnand the integerpin1.3 satisfy the following growth condition:

p

⎧⎨

⎩

any positive integer, ifn1 or 2,

2, ifn3. 2.18

Under the above assumptions on the noise, we can now put the original problem1.1- 1.2in an abstract form

du

AαuB⁻¹_α fu

dtB⁻¹_α A⁻¹dW, 2.19

with which we will also associate the following initial condition:

ut0 u0. 2.20

Note that sinceB_α⁻¹ is bounded fromHsinto itself for each α > 0,2.19is qualitatively of second order in space forα >0 although it also has a nonlocal character. In contrast, forα0

the equation is of fourth-order in space and local in character. Thus,α0 is a singular limit for the equation.

Definition 2.2. Let I : t0, t0 τ be an interval in R. We say that a stochastic process ut, ω;t0, u0is a mild solution of the system2.19-2.20inHs, if

u·, ω;t0, u0∈CI; Hs, P-a.s. ω∈Ω, 2.21

moreover, it satisfies inHsthe following integral equation:

ut, ω;t0, u0 e^−Aαt−t0v0− _t

t0

e^−Aαt−s

B⁻¹_α fu−βWAs

dsWAt, P-a.s. ω∈Ω, 2.22 whereWAtis called stochastic convolutionsee Section3for details,βis a positive constant chosen in Section3andv0 u0−WAt0.

The main results of the paper are contained in the following two theorems.

Theorem 2.3. iLetα0, and, the hypothesisQ2be satisfied. Then for everyu0∈H2, there is a unique maximally defined mild solutionut, ω;t0, u0of 2.19-2.20inH2for allt∈t0,∞.

iiLetα ∈0,1, and, the hypothesisQ1be satisfied. Then for everyu0 ∈H1, there is a unique maximally defined mild solutionut, ω;t0, u0of 2.19-2.20inH1for allt∈t0,∞.

Theorem 2.4. (i) Letα0, and, the hypothesisQ2^∗be satisfied. Then the stochastic flow associated with2.19-2.20has a compact stochastic attractorA0ω⊂H2at time 0, which pullback attracts every bounded deterministic setB⊂H2.

(ii) Letα∈ 0,1, and, the hypothesisQ1^∗be satisfied. Then the stochastic flow associated with2.19-2.20has a compact stochastic attractorAαω⊂H1at time 0, which pullback attracts every bounded deterministic setB⊂H1.

3. Stochastic Convolution

LetWAtbe the unique solution of linear equation du

Aαβ

udtB⁻¹_α A⁻¹dW, 3.1

where β is a positive constant to be further determined. Then, WAt is an ergodic and stationary process9,12called the stochastic convolution. Moreover,

WAt _t

−∞e^{−t−sAαβ}B⁻¹_α A⁻¹dWs. 3.2 Some regularity properties satisfied byWAtare given below.

Lemma 3.1. Assume thatQ1holds. Then,∇WAthas a version which is γ-H¨older continuous with respect tot, x∈R×Gfor anyγ∈0, δ/2.

Proof. We only consider the casen 3. For the sake of simplicity, we also assume thatG ₃

i10, π. The eigenvectors ofAcan be given explicitly as follows:

wkx 2

π 3/2

cosk1x1cosk2x2cosk3x3, x x1, x2, x3∈R³, 3.3

with corresponding eigenvalues

μkk²₁k₂²k²₃|k|², k∈Z³, 3.4

wherek k1, k2, k3varies inZ³. Using2.14, we find that

WAt, x

k∈Z³

√ αk

_t

−∞e^{−t−sηkβ} 1

αμk1−αdβks

wkx, 3.5

whereηkμ²_k/αμk1−α, and hence,

∇WAt, x− ∇WA

t, y

k∈Z³

√ αk

_t

−∞e^{−t−sηkβ1/αμk1−α}dβks

∇wkx− ∇wk

y ,

E∇WAt, x− ∇WA

t, y²

≤

k∈Z³

αk

αμk1−α_{2 t}

−∞e^{−2t−sηkβ}ds∇wkx− ∇wk

y².

3.6

For anyγ ∈0,1, one trivially verifies that there is a constantcγ >0 independent ofk such that for anyk∈Z³andx, y∈G

∇wkx− ∇wk

y≤cγμ^1γ/2_k x−y^γ. 3.7

Thus, we have

E∇WAt, x− ∇WA

t, y²

≤ cγ²

2x−y^2γ

k∈Z³

αk

αμk1−α2η⁻¹_k μ^1γ_k

c_γ²

2x−y^2γ

k∈Z³

αk

αμk1−α₂αμk1−α μ²_k μ^1γ_k

c_γ²

2x−y^2γ

k∈Z³

αk

μk

αμk1−αμ^−2γ_k .

3.8

Now, lett, s∈R. We may assume thatt≥s. Then, E

|∇WAt, x− ∇WAs, x|²

k∈Z³

αk

αμk1−α2

× _t

s

e^{−2ηkβt−σ}dσ _s

−∞

e^{−ηkβt−σ}−e^{−ηkβs−σ}2

dσ

|∇wkx|²

k∈Z³

αk

αμk1−α₂ 1 2

ηkβ

1−e^{−2ηkβt−s}

· |∇wkx|².

3.9

Let 0≤γ≤1/2, and let

c_γ sup

r1,r2≥0

|e^−r1−e^−r2|

|r1−r2|^2γ . 3.10

Since the function gr e^−ris a Lipschitzoneon0,∞, we always havec_γ<∞. Observe that E

|∇WAt, x− ∇WAs, x|²

≤ 4^γ

π³c_γ|t−s|^2γ

k∈Z³

αk

αμk1−α₂

ηkβ_2γ−1 μk.

≤ 4^γ

π³c_γ|t−s|^2γ

k∈Z³

αk

αμk1−α₂η^2γ−1_k μk

4^γ

π³c_γ|t−s|^2γ

k∈Z³

αk

μk

αμk1−α _2γ1

μ^−22γ_k .

3.11

By Q1, we know that TrB_α^−1−δA^−2δQ < ∞for some 0 < δ ≤ 1. Therefore, by 3.8and 3.11, one deduces that there exists a constantc_γ>0 such that

E∇WAt, x− ∇WA

s, y²

≤c_γx−y²|t−s|²_γ

, ∀t, x, s, y

∈R×G. 3.12

AsWAt, x−WAs, yis a Gaussian process, we find that for eachm∈Z, there is a constant c^m_γ >0 such that

E∇WAt, x− ∇WA

s, y^2m

≤c^m_γx−y²|t−s|²_mγ

. 3.13

Now, thanks to the well-known Kolmogorov test, one concludes thatWAt, xisγ−2/m- H ¨older continuous int, x. Becauseγ ∈ 0,1/2andm ∈ Z are arbitrary, we see that the conclusion of the lemma holds true. The proof is complete.

Lemma 3.2. AssumeQ2holds. Then, for anyM >0, there exists aβ0such that for allβ≥β0,

E

|WAt|²₂

≤M. 3.14

Proof.

E

|ΔWAt|² E

⎛

⎝

k∈Z³

√αk

_t

−∞e^{−ηkβt−s} 1

αμk1−αdβksΔwkx

⎞

⎠

2

k∈Z³

αk

αμk1−α2

_t

−∞e^{−2ηkβt−s}ds|Δwkx|²

≤

k∈Z³

αk

αμk1−α₂ 1 2

ηkβ|Δwkx|²

≤ 1

2

η1β

k∈Z³

αk

μk

αμk1−α 2

.

3.15

Since TrB_α⁻²Q<∞, one can now easily choose aβlarge enough so thatE|ΔWAt|²≤M, and the proof is complete.

Similarly, we can verify the following basic fact.

Lemma 3.3. Assume Q2 holds. Then, ΔWA has a version which is γ-H¨older continuous with respect tot, x∈R×Gfor anyγ∈0, δ/2.

Lemma 3.4. Assume thatQ2^∗holds. Then, for anyM >0, there existsβ0such that for allβ≥β0,

E

|WAt|²_2σ

≤M. 3.16

4. Stochastic dissipativeness in H

It is well known that in the deterministic case without forcing terms,

Ju 1 2|∇u|²

G

Fudx 4.1

is a Lyapunov functional of the system i.e.d/dtJu ≤ 0, where Fu is the primitive function offuwhich vanishes at zero. In this section, we will prove a similar property for the stochastic equation by adapting some argument in9.

Assume thatusatisfies2.19-2.20. As usual, we may assume in advance thatuis sufficiently regular so that all the computations can be performed rigorously. Applying the It ˆo formula toJu, we obtain

dJu Juu, du 1 2Tr

JuuuB⁻²_α A⁻²Q dt

Juu, B⁻¹_α A⁻¹dW

−

Juu, B_α⁻¹AuB_α⁻¹fu dt1

2Tr

JuuuB_α⁻²A⁻²Q dt,

4.2

whereJu, Juudenote, respectively, the first and second derivative ofJ. Since

Juu Aufu, 4.3

there existsC1 >0 such that forα0,

Juu, B⁻¹_α AuB⁻¹_α fu

Aufu²

1 ≥λ²₁Aufu²₋₁ λ²₁

Aufu, uA⁻¹fu λ²₁

|u|²₁fu²

−12

fu, u

≥dλ²₁

|u|²₁

G

Fudx

−C1 dλ²₁Ju−C1,

4.4

wheredmin{1, 4pa_2p−1}. And for 0< α≤1, Juu, B_α⁻¹AuB_α⁻¹fu

Aufu, B_α⁻¹AuB_α⁻¹fu Aufu²

B⁻¹α

≥ λ²₁

αλ11−αAufu²

−1

≥ dλ²₁

αλ11−αJu−C1,

4.5

where we have used2.8. Simple computations show that

Juuu Afu, 4.6

and hence,

Tr

JuuuB⁻²_α A⁻²Q Tr

AB⁻²_α A⁻²Q ^∞

i1

Di

G

fuw_i²dx

Tr

B⁻²_α A⁻¹Q ^∞

i1

Di

G

fuw_i² dx

,

4.7

where{wi}^∞_i1is the orthonormal basis ofHas in2.14, andDiαi/αλi1−α². We infer from3.3that

|wi|_L∞≤C2, 4.8

whereC2>0 depends only onG. Therefore,

G

fuw²_i dx ≤C²₂

G

fudx. 4.9

SetC3such that

fs≤2 2p−1

a2p−1s^2p−2C3, s∈R, 4.10

then

G

fuw_i²dx ≤C²₂

2

2p−1 a2p−1

G

u^2p−2dxC3|G|

≤ 1 4pa2p−1

G

u^2pdxC4,

4.11

whereC4depends onf,p, andG. LetC5satisfy

Fs≥ 1

4pa_2p−1s^2p− C5

|G|, s∈R, 4.12

then

G

fuw²_i dx

≤Ju C4C5. 4.13

Finally,

Tr

JuuuB⁻²_α A⁻²Q

≤Tr

B_α⁻²A⁻¹Q Tr

B_α⁻²A⁻²Q

Ju C4C5. 4.14

Since

E

Juu, B⁻¹_α A⁻¹dW

0, 4.15

we have from4.2that d

dtEJu E

Juu,−B⁻¹_α Au−B⁻¹_α u 1

2E Tr

JuuuB⁻²_α A⁻²Q

. 4.16

Further, by4.4,4.5and4.14, it holds that d

dt

EJu

≤ −

D−1 2Tr

B_α⁻²A⁻²Q

EJu Tr

B⁻²_α A⁻¹Q Tr

B⁻²_α A⁻²Q

C4C5 C1, 4.17

whereD min{dλ²₁, dλ²₁/αλ11−α}. This is precisely what we promised.

Now, by directly applying the classical Gronwall Lemma, we have the following lemma.

Lemma 4.1. LetWbe a H-valued Q-Wiener process with

Tr

B_α⁻²A⁻¹Q

<∞, Tr

B_α⁻²A⁻²Q

≤2D, 4.18

and letutbe the mild solution to2.19-2.20. Then,

EJut≤EJu0 CQ, t∈t0,∞, 4.19

where

CQ Tr

B_α⁻²A⁻¹Q Tr

B⁻²_α A⁻²Q

C4C5 C1

D−1/2Tr

B⁻²α A⁻²Q . 4.20

As a consequence, we immediately obtain the following basic result.

Corollary 4.2. LetWbe a H-valued Q-Wiener process with

Tr

B_α⁻²A⁻¹Q

<∞, Tr

B_α⁻²A⁻²Q

≤2D. 4.21

Then, there exists a continuous nonnegative functionΨrsuch that for any solutionutof 2.19-2.20, one has

E

|ut|²₁

≤Ψ E

|u0|²₁

, ∀t∈t0,∞. 4.22

5. The Existence and Unique of Global Mild Solutions

In this section, we study the existence and unique of global mild solutions of the problem 2.19-2.20. The basic idea is to transform the original problem into a nonautonomous one by using the simple variable change below:

vt ut−WAt. 5.1

We observe thatvtsatisfies the following system:

dv dt

Aα−β

vB⁻¹_α fvWA 0, vt0 u0−WAt0.

5.2

Let

Gv, t −B⁻¹_α fvWA βWA, v0u0−WAt0. 5.3

Then,5.2reads

dv

dt AαvGv, t, vt0 v0.

5.4

To prove Theorem 2.3, it suffices to establish some corresponding existence results for the nonautonomous system5.4.

Definition 5.1. Let I : t0, t0 τ be an interval in R. We say that a stochastic process vt, ω;t0, v0is a mild solution of the system5.4inHs, if

v·, ω;t0, v0∈CI; Hs, P-a.s.ω∈Ω, 5.5

and satisfies inHsthe following integral equation:

vt, ω;t0, v0 e^−Aαt−t0v0− _t

t0

e^−Aαt−s

B⁻¹_α fu−βWAs

ds, P-a.s.ω∈Ω. 5.6

Theorem 5.2. Letα0. Suppose that the Hypothesis(Q2) is satisfied.

Then, for everyu0∈H2, there is a unique globally defined mild solutionvt, ω;t0, v0of5.4 inH2with

vt, ω;t0, v0∈Ct0,∞; H2∩C_loc^0,1−rt0,;H4r∩Ct0,∞;H4, 5.7

for all 0≤r <1.

Proof. We only consider the case wheren3. First, it is easy to verify thatP-a.s.

Gv, t∈CLip;γH2×t0,∞, H. 5.8

Indeed, by Lemma3.3, we see thatWAt∈H2isγ-H ¨older continuous with respect tot∈R P-a.s. Recall thatfis a polynomial of degree 2p−1 withp 2in casen3. One deduces that there existC1, C2ω>0 such that

|Gv1, t1−Gv2, t2| ≤C1|v1−v2|₂|WAt1−WAt2|₂

≤C2ω

|v1−v2|₂|t1−t2|^γ

, P-a.s. 5.9

It then follows from11, Lemma 47.4that there is a unique maximally defined mild solution vof5.4inH2ont0, TsatisfyingP-a.s.

vt, ω;t0, v0 e^−A2t−t0v0− _t

t0

e^−A2t−s

Afus−βWAs ds, vt, ω;t0, v0∈Ct0, T; H2∩C^0,1−r_loc t0, T;H4r∩Ct0, T;H4,

5.10

for all 0 ≤ r < 1. Furthermore, we also know that vis a strong solution in H2. Hence, it satisfies in the strong sense that

dv

dt A²vAfu−βWA 0, vt0 v0. 5.11 In what follows, we showT ∞, thus proving the theorem.

Simple computations yields

Δfu≤fu

L^∞|Δu|fu

L^∞|∇u|²_L4. 5.12

Sincefis a polynomial of degree 3, there existκ1andκ2such that fs≤κ1

1|s|²

, fs≤κ21|s|, ∀s∈R. 5.13

Therefore,

fu

L^∞|Δu| ≤κ1

1|u|²_L∞

|Δu|

≤2κ1

1|v|²_L∞|WA|²_L∞

|Δv||ΔWA|. 5.14

By the Nirenberg-Gagiardo inequality, there existC3, C4, C5 >0 such that

|u|²_L∞ ≤C3|Δu|², u∈H2,

|u|²_L∞ ≤C4Δ²u^1/3|u|^5/3_L6 , u∈H4,

|Δu| ≤C5Δ²u^1/2|∇u|^1/2, u∈H4.

5.15

Hence, fu

L^∞|Δu| ≤2κ1

1|v|²_L∞|WA|²_L∞

|Δv||ΔWA|

≤2κ1

1C4Δ²v^1/3|v|^5/3_L6 C3|ΔWA|²

C5|∇v|^1/2Δ²v^1/2|ΔWA|

. 5.16 ByQ2and Lemma3.2, we know that forP-a.s.ω ∈Ω, there exists anR1ω>0 such that

|ΔWAt| ≤R1ω for allt∈R. On the other hand, by Lemma3.2and Corollary4.2, we find thatP-a.s.vis bounded inH1. Thus, forP-a.s.ω∈Ω, there existC6ω, C7ω>0 such that

|v|^5/3_L6 ≤C6ω, |∇v|^1/2≤C7ω, 5.17

where the continuous imbeddingH¹→L⁶is used. Consequently, we have fu

L^∞|Δu| ≤C8ω

1Δ²v^1/3R1ω

Δ²v^1/2R1ω

, P-a.s.ω∈Ω. 5.18

Similarly forP-a.s.ω∈Ω, one easily deduces that there existsC9ω>0 such that fu

L^∞|Δu| ≤C9ω

1Δ²v^1/6R1ω

Δ²v^1/4R1ω

. 5.19

It then follows from5.12that forP-a.s.ω∈Ω, Δfu≤C8ω

1Δ²v^1/3R1ω

Δ²v^1/2R1ω

C9ω

1L3Δ²v^1/6R1ω

Δ²v^1/4R1ω

≤C10ω

1Δ²v^5/6

.

5.20

Now, taking theL²inner-product of equation5.11withΔ²v, one obtains 1

2 d

dt |Δv|²Δ²v²≤

G

ΔfuΔ²vdx β

G

WAΔ²vdx

≤ 1 4

Δ²v²Δfu² 1 4

Δ²v²β²|WA|²

≤ 1 2

Δ²v²Δfu²β²λ⁻²₁ |ΔWA|².

5.21

By5.20, we deduce thatP-a.s.

d

dt |Δv|²Δ²v²≤C11ω

1Δ²v^5/3

. 5.22

Furthermore, by Young’s inequality and|Δ²v|²≥λ²₁|Δv|², we know that there existsC12ω>

0 such thatP-a.s.

d

dt |Δv|²≤ −λ²₁

2 |Δv|²C12ω. 5.23

Applying the gronwall lemma on5.23, one gets

|Δv|²≤ 2C12ω

λ²₁ , P-a.s.ω∈Ω. 5.24

This implies that the weak solution solutionvdoes not blow up in finite time in the spaceH2. Hence,Tv0 ∞, for allu0∈H2.

Theorem 5.3. Letα∈0,1, and let Hypothesis(Q1) be satisfied. Then, for everyu0 ∈H1, there is a unique maximally defined mild solutionvt, ω;t0, v0of 5.4inH1for allt∈t0,∞with

vt, ω;t0, v0∈Ct0,∞;H1∩C^0,1−r_loc t0,∞;H2r∩Ct0,∞;H2, 5.25

for 0≤r <1.

Proof. As noted above, −Aα is a positive selfadjoint linear operator on H with compact resolvent. The negative operator−Aαgenerate an analytic semigroupe^−Aαt. It is easy to verify by Lemma3.1thatP-a.s.

Gv, t∈CLip;γH1×t0,∞, H. 5.26

It then follows from11, Lemma 47.4that there is a unique maximally defined mild solution vof5.4inH1ont0, Twith

vt, ω;t0, v0∈Ct0, T;H1∩C^0,1−r_loc t0, T;H2r∩Ct0, T;H2, 5.27

wheret0< T Tv0≤ ∞, and 0≤r <1. Furthermore,vis a strong solution inH1and hence solves5.4in the strong sense. To complete the proof of the theorem, there remains to check thatTv0 ∞.

Equation5.4is equivalent to

Bαdv

dt Av−fvWA βBαWA, vt0 v0. 5.28 Multiplying5.28byAv, one gets

1 2

d

dt|v|²_1,Bα|v|²₂

fvWA, Av

βBαWA, Av 0. 5.29

We observe that

fvWA, Av

G

∇fvWA∇v dx

G

fvWA|∇v|²dx

G

fvWA∇WA∇v dx.

5.30

We takeC₁andC₂such that

fx≥ 2p−1

2 a2p−1x^2p−2−C₁, fx≤2

2p−1

a2p−1 x^2p−2C2,

5.31

for allx∈R. Then, fvWA, Av

≥ 2p−1 2 a_2p−1

G

|vWA|^2p−2|∇v|²dx−C₁

G

|∇v|²dx

−2 2p−1

a2p−1

G

|vWA|^2p−2|∇v||∇WA|dx−C₂

G

|∇v||∇WA|dx

≥ 1 4

2p−1 a2p−1

G

|vWA|^2p−2|∇v|²dx−2C₁

G

|∇v|²dx

−C₃

G

|∇WA|^2pdx

G

|∇WA|²dx

,

5.32 where we have used H ¨older’s inequality, Young’s inequality, and the appropriate imbeddings H¹G → L^rGin dimension n 1 or 2 and 3. We also know by2.6that there exists α≤Cα≤M1such that

BαWA, Av≤Cα

G

|∇WA|²dx

G

|∇v|²dx

. 5.33

Combining the last two inequalities together, we deduce that there exists constantsC₄, C₅>0 such that

1 2

d

dt|v|²_1,Bα|v|²₂1 4

2p−1 a2p−1

G

|vWA|^2p−2|∇v|²dx

≤2C₄

G

|∇v|²dxC₅

G

|∇WA|^2pdx

G

|∇WA|²dx

.

5.34

In view of2.7, there existsα≤C_α≤M1such that 1

2C_αd

dt|v|²₁|v|²₂ 1 4

2p−1 a2p−1

G

|vWA|^2p−2|∇v|²dx

≤2C₄|v|²₁C₅

G

|∇WA|^2pdx

G

|∇WA|²dx

.

5.35

Using the gronwall lemma on5.35, the following inequality holds:

|v|²₁≤2e^4C4/CαC4|v0|²₁ 2e^4C4/Cα _t

t0

C₅

G

|∇WAs|^2pdx

G

|∇WAs|²dx

ds.

5.36

Lemma3.1guarantees thatP-a.s.

_t

t0

G

|∇WAs|^2pdxds <∞, _t

t0

G

|∇WAs|²dxds <∞. 5.37

This and5.36implies that the mild solutionvdoes not blow up in finite time in the space H1. It follows thatTv0 ∞. The proof is complete.

Remark 5.4. The conclusions in Theorem2.3are readily implied in the above two theorems.

6. Attractors for Stochastic Viscous Cahn-Hilliard Equation

For convenience of the reader, some basic knowledge of RDS are summarized in the Appendix at the end of this paper.

6.1. Stochastic Flows

Thanks to Theorem2.3, the mappingu0→ut, ω;t0, u0defines a stochastic flowSαt, s;ω, Sαt, s;ωu0ut, ω;s, u0, α∈0,1. 6.1 Notice thatP-a.s.

iSαt, s;ω Sαt, r;ω◦Sαr, s;ω, for alls≤r≤t,

iiS0t, s;ωis continuous inH2, andSαt, s;ωis continuous inH1for 0< α≤1.

6.2. Compactness Properties of Stochastic FlowSαt, s;ω

Lemma 6.1. (i) Under AssumptionQ2^∗, the stochastic flow S0t, s;ωis uniformly compact at time 0. More precisely, for allB⊂H2bounded and eacht0 <0,S00, t0;ωBis v relatively compact inH2.

(ii) Under AssumptionQ1^∗, the flowSαt, s;ω, 0< α≤1, is uniformly compact at time 0.

More precisely, for allB⊂H1bounded and eacht0 <0,Sα0, t0;ωBisP-a.s. relatively compact in H1.

Proof. i LetB ⊂ H2 be a given bounded deterministic set. By Lemma3.4, we know that forP-a.s. ω ∈ Ω, there existsR2ω > 0, such that |WAt|_2σ ≤ R2ω,t ∈ R. DefineB B∪B2σ0, R2ω, whereB2σ0, R2ωdenotes the open ball centered at 0 with radiusR2ω inH_2σ. Then,B⊂H2isP-a.s. bounded, and

S00, t0;ωB⊂

e^A2t0v0− ₀

t0

e^A2sGvs, sdsWA0, v0∈B

⊂N1N2N3N4, 6.2 P-a.s., where

N1e^A2t0B, N2

₀

−δe^A2sGvs, sds, v0∈B

,

N3e^−A2δ _−δ

t0

e^A2sδGvs, sds, v0∈B

, N4B2σ0, R2ω,

6.3

andδis an arbitrary constant satisfying 0< δ <−t0.

Since fort > 0 fixed the operatore^−A2t is compact, we see thatN1,N3, andN4 are relatively compact sets inH2. Now, we show thatP-a.s.S00, t0;ωBis relatively compact. To this end, we first give an estimate on the Kuratowski measure ofN2⊂H2.

Forv0 ∈B, one has

₀

−δe^A2s−t0Gvs, sds

2

₀

−δAe^A2s−t0Gvs, sds

. 6.4