Research Article | Open Access
Global Mild Solutions and Attractors for Stochastic Viscous Cahn-Hilliard Equation
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.
This paper is devoted to the existence of mild solutions and global asymptotic behavior for the following stochastic viscous Cahn-Hilliard equation: subjected to homogeneous Dirichlet boundary conditions in dimension or 3, where in , and is a parameter, is a polynomial of odd degree with a positive leading coefficient
In deterministic case, the model was first introduced by Novick-Cohen  to 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 attractor of the system has the same finite Hausdorff dimension for different parameter values . One can also show that is continuous as varies in . See  for details and  for 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-Weber  proved the existence of solution as well as its density for a class of stochastic Cahn-Hilliard equations with additive noise using an appropriate convolution semigroup (in the sense of that in ) posed on cubic domains. The authors in  derived the existence for a generalized stochastic Cahn-Hilliard equation in general convex or Lipschitz domains. The main novelty was the derivation of space-time Hölder estimates for the Greens kernel of the stochastic problem, by using the domains geometry, which can be very useful in many other circumstances. In , 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 Flandoli  (see also Schmalfuss ) introduced 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 case , (1.1) reduces to the stochastic Cahn-Hilliard equation which was studied in , 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 Section 2, we first make some preliminary works, then we state our main results. In Section 3, 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. Section 4 consists 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 Sections 5 and 6, respectively. Finally, the last section stands as an appendix for some basic knowledge of random dynamical system(RDS).
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 of . We define the linear operator with domain . is positive and selfadjoint. By spectral theory, we can define the powers and spaces with norms for real . Note that . It is well known that is a subspace of and is on a norm equivalent to the usual one. Moreover, we have the following Poincare inequality and interpolation inequality: where is the first eigenvalue of .
We can define to be the Green’s operator for . Thus, By Rellich’s Theorem, we know that is compact, and is a linear and bounded operator. Finally, we introduce the invertible operator , defined by For each and , we know that is bounded and has a bounded inverse (see [10, 11]). We also define the operator with domain By definition, it is clear that in case .
Lemma 2.1. For , there exist , and such that where
Proof. Here, we only verify (2.8) is valid; the proofs of (2.6) and (2.7) can be found in . Since is bounded, there exists , such that . Then, for any , we have
which completes the right part of (2.8).
Now, we proof the left part of (2.8) let denote the eigenvalues of , repeated with the respective multiplicity, and the corresponding unit eigenvector is denoted by , which forms an orthonormal basis for . We have Since , there exist , such that . Consequently, which finishes the proof.
2.2. Assumptions on the Noise
The stochastic process , defined on a probability space , is a two-side in time Wiener process on which is given by the expansions where is a basis of consisting of unit eigenvectors of , is a bounded sequence of nonnegative numbers, and is a sequence of mutually independent real valued standard Brownian motions in a fixed probability space adapted to a filtration .
For convenience, we will define the covariance operator on as follows: The process will be called as the -Wiener process. We need to impose on one of the following assumptions: (Q1), (Q1*)and ,(Q2)for some and ,(Q2*) (for some and , and ,
where is given in Section 4. It is obvious that
2.3. Main Results
We will assume throughout the paper that the space dimension and the integer in (1.3) satisfy the following growth condition:
Under the above assumptions on the noise, we can now put the original problem (1.1)-(1.2) in an abstract form with which we will also associate the following initial condition: Note that since is bounded from into itself for each , (2.19) is qualitatively of second order in space for although it also has a nonlocal character. In contrast, for the equation is of fourth-order in space and local in character. Thus, is a singular limit for the equation.
Definition 2.2. Let be an interval in . We say that a stochastic process is a mild solution of the system (2.19)-(2.20) in , if moreover, it satisfies in the following integral equation: where is called stochastic convolution (see Section 3 for details), is a positive constant chosen in Section 3 and .
The main results of the paper are contained in the following two theorems.
Theorem 2.3. Let , and, the hypothesis be satisfied. Then for every , there is a unique maximally defined mild solution of (2.19)-(2.20) in for all .
Let , and, the hypothesis be satisfied. Then for every , there is a unique maximally defined mild solution of (2.19)-(2.20) in for all .
Theorem 2.4. (i) Let , and, the hypothesis be satisfied. Then the stochastic flow associated with (2.19)-(2.20) has a compact stochastic attractor at time 0, which pullback attracts every bounded deterministic set .
(ii) Let , and, the hypothesis be satisfied. Then the stochastic flow associated with (2.19)-(2.20) has a compact stochastic attractor at time 0, which pullback attracts every bounded deterministic set .
3. Stochastic Convolution
Some regularity properties satisfied by are given below.
Lemma 3.1. Assume that holds. Then, has a version which is Hölder continuous with respect to for any .
Proof. We only consider the case . For the sake of simplicity, we also assume that . The eigenvectors of can be given explicitly as follows:
with corresponding eigenvalues
where varies in . Using (2.14), we find that
where , and hence,
For any , one trivially verifies that there is a constant independent of such that for any and Thus, we have
Now, let . We may assume that . Then, Let , and let Since the function g is a Lipschitzoneon , we always have . Observe that By , we know that for some . Therefore, by (3.8) and (3.11), one deduces that there exists a constant such that As is a Gaussian process, we find that for each , there is a constant such that Now, thanks to the well-known Kolmogorov test, one concludes that is -Hölder continuous in . Because and are arbitrary, we see that the conclusion of the lemma holds true. The proof is complete.
Lemma 3.2. Assume holds. Then, for any , there exists a such that for all ,
Proof. Since , one can now easily choose a large enough so that , and the proof is complete.
Similarly, we can verify the following basic fact.
Lemma 3.3. Assume holds. Then, has a version which is Hölder continuous with respect to for any .
Lemma 3.4. Assume that holds. Then, for any , there exists such that for all ,
4. Stochastic dissipativeness in
It is well known that in the deterministic case without forcing terms, is a Lyapunov functional of the system (i.e. ), where is the primitive function of which vanishes at zero. In this section, we will prove a similar property for the stochastic equation by adapting some argument in .
Assume that satisfies (2.19)-(2.20). As usual, we may assume in advance that is sufficiently regular so that all the computations can be performed rigorously. Applying the Itô formula to , we obtain where denote, respectively, the first and second derivative of . Since there exists such that for , where . And for , where we have used (2.8). Simple computations show that and hence, where is the orthonormal basis of as in (2.14), and .
We infer from (3.3) that where depends only on . Therefore, Set such that then where depends on , , and . Let satisfy then Finally, Since we have from (4.2) that Further, by (4.4), (4.5) and (4.14), it holds that where . This is precisely what we promised.
Now, by directly applying the classical Gronwall Lemma, we have the following lemma.
As a consequence, we immediately obtain the following basic result.
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:
Definition 5.1. Let be an interval in . We say that a stochastic process is a mild solution of the system (5.4) in , if and satisfies in the following integral equation:
Theorem 5.2. Let . Suppose that the Hypothesis (Q2) is satisfied.
Then, for every , there is a unique globally defined mild solution of (5.4) in with for all .
Proof. We only consider the case where . First, it is easy to verify that .
Indeed, by Lemma 3.3, we see that is -Hölder continuous with respect to . Recall that is a polynomial of degree with (in case ). One deduces that there exist such that
It then follows from [11, Lemma 47.4] that there is a unique maximally defined mild solution of (5.4) in on satisfying .
for all . Furthermore, we also know that is a strong solution in . Hence, it satisfies in the strong sense that
In what follows, we show , thus proving the theorem.
Simple computations yields Since is a polynomial of degree 3, there exist and such that Therefore, By the Nirenberg-Gagiardo inequality, there exist such that Hence, By (Q2) and Lemma 3.2, we know that for . , there exists an such that (for all ). On the other hand, by Lemma 3.2 and Corollary 4.2, we find that . is bounded in . Thus, for . , there exist such that where the continuous imbedding is used. Consequently, we have
Similarly for . , one easily deduces that there exists such that It then follows from (5.12) that for . ,
Now, taking the inner-product of equation (5.11) with , one obtains By (5.20), we deduce that . Furthermore, by Young's inequality and , we know that there exists such that . Applying the gronwall lemma on (5.23), one gets This implies that the weak solution solution does not blow up in finite time in the space . Hence, , for all .
Theorem 5.3. Let , and let Hypothesis (Q1) be satisfied. Then, for every , there is a unique maximally defined mild solution of (5.4) in for all with for .
Proof. As noted above, is a positive selfadjoint linear operator on with compact resolvent. The negative operator generate an analytic semigroup . It is easy to verify by Lemma 3.1 that .
It then follows from [11, Lemma 47.4] that there is a unique maximally defined mild solution of (5.4) in on with
where , and . Furthermore, is a strong solution in and hence solves (5.4) in the strong sense. To complete the proof of the theorem, there remains to check that .
Equation (5.4) is equivalent to Multiplying (5.28) by , one gets We observe that We take and such that for all . Then, where we have used Hölder's inequality, Young's inequality, and the appropriate imbeddings in dimension and 3. We also know by (2.6) that there exists such that Combining the last two inequalities together, we deduce that there exists constants such that In view of (2.7), there exists such that Using the gronwall lemma on (5.35), the following inequality holds: Lemma 3.1 guarantees that . This and (5.36) implies that the mild solution does not blow up in finite time in the space . It follows that . The proof is complete.
Remark 5.4. The conclusions in Theorem 2.3 are 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 Theorem 2.3, the mapping defines a stochastic flow , Notice that . (i),(ii) is continuous in , and is continuous in for .
6.2. Compactness Properties of Stochastic Flow
Lemma 6.1. (i) Under Assumption , the stochastic flow is uniformly compact at time 0. More precisely, for all bounded and each , is v relatively compact in .
(ii) Under Assumption , the flow , , is uniformly compact at time 0. More precisely, for all bounded and each , is . relatively compact in .
Proof. (i) Let be a given bounded deterministic set. By Lemma 3.4, we know that for . , there exists , such that , . Define , where denotes the open ball centered at 0 with radius in . Then, is . bounded, and ., where and is an arbitrary constant satisfying .
Since for fixed the operator is compact, we see that , , and are relatively compact sets in . Now, we show that . is relatively compact. To this end, we first give an estimate on the Kuratowski measure of .
For , one has Since is a positive sectorial operator on , there exists a constant such that Recall that . So there is a such that . Therefore It follows that where denotes the Kuratowski measure of noncompactness on . Now since , , and are relatively compact sets in ., we have Letting , one immediately concludes that . , hence is relatively compact.
(ii) The proof of the compactness result for () is fully analogous, and is thus omitted.
6.3. The Random Attractors
Now, we show that the system possesses a random attractor for every .
Proof of Theorem 2.4. We infer from the proofs of Theorem 5.2 and Lemma 6.1 that there exists such that for any , we can define an absorbing set for at time 0 by and for (), for any we can define an absorbing set for at time 0 by where denotes the open ball centered at 0 with radius in . Now the conclusions of the theorem immediately follows from Proposition A.6
Basic knowledge of RDS
Let be a complete metric space, and let be a probability space. We consider a family of mappings satisfying . (i),(ii) is continuous in , for all .
Definition A.1. We say that is an absorbing set at time , if . (i) is bounded,(ii)for all there exists such that , for all .
Definition A.2. Given and , we say that is uniformly compact at time if for all bounded set , there exist , such that . is relatively compact in .
Definition A.3. Given and , for any set , we define the random omega limit set of a bounded set at time as
Definition A.4. Let be a metric space, and let a family of operators that maps into itself. We say that is a stochastic attractor if . (i) is not empty and compact,(ii) for all ,(iii)for every bounded set , .
Remark A.5. (i) In the stochastic case, it is not possible to construct the