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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 492847, 17 pages
Uniformly Random Attractor for the Three-Dimensional Stochastic Nonautonomous Camassa-Holm Equations
Department of Mathematics, South China University of Technology, Guangzhou 510640, China
Received 25 November 2012; Revised 3 July 2013; Accepted 17 July 2013
Academic Editor: Ziemowit Popowicz
Copyright © 2013 Zhehao Huang and Zhengrong Liu. 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.
We consider the uniformly random attractor for the three-dimensional stochastic nonautonomous Camassa-Holm equations in the periodic box 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 and for the equations.
The Camassa-Holm equation 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 , and it has stable solutions . It possesses the peakons which has been proved stable [6, 7]. It has been shown that (1) is locally well-posed for initial data [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  considered the three-dimensional Camassa-Holm equations subject to periodic boundary conditions: 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  the authors analyzed the effects produced by stochastic perturbations in the deterministic version of the three-dimensional Lagrangian averaged Navier-Stokes- model: that is, the persistence of exponential stability as well as possible stabilization effects produced by the noise.
In  the authors proved the existence of the pullback and forward attractors for three-dimensional Lagrangian averaged Navier-Stokes- model with delay:
In  the author investigated the existence of finite dimensional uniform attractor for three-dimensional nonautonomous Camassa-Holm equations with periodic boundary conditions:
In [23, 24] the author studied the existence of uniform attractor and convergence of the attractor as for a nonautonomous three-dimensional Lagrangian averaged Navier-Stokes- model with singularly oscillating external force: where
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 : where is the modified pressure, while is the pressure, is the constant viscosity, and is a constant density. The function is a given body forcing, and , are scale parameters. are two-sided real-valued Wiener processes on a probability space which will be specified later. is a bounded linear operator. Also observing that at the limiting case , , 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 . In the approach presented in , to construct the uniform attractor, instead of the associated process , one should consider a family of processes , , in some Banach space where the functional parameter , , is called the symbol and is the symbol space including . 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 which can be constructed by a closed random absorbing set for an asymptotically compact stochastic mapping is given by where 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 ) random attractor which all solutions approach as . 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 ) random attractor for the system.
The paper is organized as follows. In Section 2, 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 ) 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 ) 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 be a separable Hilbert space with the Borel -algebra , and let be a probability space.
Definition 1. is called a measurable flow on probability space if is -measurable, is the identity on , for all , , and for all .
Definition 2. A random bounded set of , is called tempered with respect to if for a.e. , where .
Definition 3. A random set is called an absorbing set of a stochastic mapping in if for every random bounded set and a.e. , there exists such that
Let be a three-parameter family of mappings acting on :
Definition 4. A three-parameter family of random mappings is said to be a process in if it is -measurable and for a.e. it satisfies
Definition 5. A family of processes , , acting in is said to be -continuous, if, for a.e. and fixed , , , the random mapping is continuous from into .
Definition 6. A random curve , , is said to be a complete trajectory of the process if for a.e.
Definition 7. The random kernel of the process consists of all bounded complete trajectories of the process :
Definition 8. The random set is said to be the random kernel section at time , .
Let , where is a random set. Denote that is closure of the set and .
Definition 9. A random set is called the uniformly (with respect to ) random (pullback) omega-limit set of which can be characterized as follows, analogously to that for semigroups,
Let , and its Kuratowski measure of noncompactness is defined by
Definition 10. A family of processes , , is said to be uniformly (with respect to ) random (pullback) omega-limit compact if, for a.e. and any , the set is bounded for every and .
We now present a method to verify the uniformly (with respect to ) random (pullback) omega-limit compactness.
Definition 11. A family of processes , , is said to satisfy uniformly (with respect to ) condition (C) if, for a.e. and any fixed , tempered set , , there exist and a finite dimensional subspace of such that(i) is bounded, and(ii), ,where is a bounded projector.
Therefore, we have the following results.
Theorem 12. Let be a metric space, and let be a continuous invariant semigroup on . A family of processes , , acting in is -continuous (weakly) and possesses the compact uniformly (with respect to ) random attractor satisfying if it(i)has a bounded uniformly (with respect to ) random absorbing set , and(ii)satisfies uniformly (with respect to ) condition (C).
Moreover, if is a uniformly convex Banach space, then the converse is true.
3. Functional Setting
We consider the probability space where is the Borel -algebra induced by the compact-open topology of and the corresponding Wiener measure on . Then we will identify with Define the time shift by Then is a family of measure preserving transformations on probability space in Definition 1.
Next we define a symbol space for (8). We assume that the function is translation bounded. That is, for , we have
Definition 13 (cf. ). A function is said to be normal if, for any , there exists such that
We denote by the set of all normal functions in . Obviously, , and it is proved in  that is a closed subset of . Let a fixed symbol be normal functions in . That is, the family of translation forms a normal function set in , where is an arbitrary interval of the time axis . Therefore,
After integrating (8), one can easily see that On the other hand, because of the spatial periodicity of solution, we have . Then we have 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 , , and .
Next we introduce some essential functional spaces.(i)We denote by a vector-valued trigonometric polynomial defined on , such that and , and we let and be the closure of in and in , respectively. We can observe that , the orthogonal complement of in , is (cf. [38, 41]).(ii)We denote by the orthogonal projection, usually referred to as Helmholtz-Leray projector, and by the Stokes operator with domain . Note that, in the case of periodic boundary condition, is a self-adjoint positive operator with compact inverse. Hence, the space has an orthogonal basis of eigenfunctions of , that is, , with In fact, these eigenvalues have the form with .(iii)We denote by the inner product and by the corresponding norm. By virtue of Poincaré inequality one can show that there is a constant , such that Moreover, one can show that (cf. [38, 42]). We denote by the dual of . Hereafter 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 yhat , and we set for every , . That is, for every fixed , is a linear operator acting on . Note that We also denote that for every . Using the identity one can easily show that for every , , , where denotes the adjoint operator of the linear operator defined above. As a result we have
The next lemma will present some properties of the bilinear operator .
Lemma 14 (cf. ). The operator can be extended continuously from with values in , and it satisfies for every , , . Moreover, and in particular Furthermore, one has for every , , and and by symmetry one has for every , , and . Also for every , , and . In addition, for every , , and .
4. Stochastic Nonautonomous Camassa-Holm Equations
We now apply the result in Section 2 to the stochastic nonautonomous Camassa-Holm equations. To associate a family of processes with the stochastic equations over and , 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 operator in (8) as follows: where , .
Given , consider the Ornstein-Uhlenbeck equation: One can easily check that a solution to (42) is given by It is known that the random variable is tempered and that is a.e is continuous. Now we put . By (42) we have Employing Cauchy-Schwarz’s inequality, we get
To show that (8) corresponds to a process , we let , where is a solution of (8). Then for , we have defined in the periodic box , satisfying We apply to (46) and use the notation in Section 3 to obtain the equivalent system of equations satisfying the initial condition By a Galerkin method as in , it can be proved that if and , for a.e. , (48) has a unique solution satisfying for any and such that, for almost all and for any ,
Now for any , (48) with instead of possesses a corresponding process acting on . It is analogous to the proof in  to prove that, for a.e , the family of processes is -continuous. Let be the so-called kernel of the process .
5. Uniformly Random Attractor for Stochastic Nonautonomous Camassa-Holm Equation
In , 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 random attractor for the stochastic nonautonomous Camassa-Holm equations in and .
Lemma 15. Let be tempered and . Then the process corresponding to (48) possesses a uniformly (with respect to ) random absorbing set in .
Proof. Letting in (51), we have
Now we estimate the second term of (53) on the right-hand side. Applying Lemma 14 we have
By Poincaré’s inequality, we have
Associating with the above inequalities and employing Poincaré’s inequality, we have
Applying Gronwall’s lemma, we have
Replacing by in (57) and (58), we have
Note that are stationary and ergodic (cf. ), then it follows from ergodic theorem that
On the other hand, we have
Employing (61), we have
So for a.e. , there are and such that, for and ,
Since are tempered, the integral
is convergent. For any , there exists such that , and we have
Then by piecewise integration, we have
By differential mean value theorem, for each , there exists , such that
By the property of normal function, there exists , which just depends on , such that
It is easy to check that the series is convergent. Associating with (65)–(68), for any , we have
Therefore (64) and (69) imply that, for , the second term on the right-hand side of (59) can be bounded by
where . For being tempered, there exists such that, when ,
Letting and , when , we have
We define In conclusion, is a uniformly random absorbing set for in , which complete the proof.
Lemma 16. Let be tempered and . Then the process corresponding to (48) possesses a uniformly with respect to random absorbing set in .
Proof. Integrating (56) into where , we have
Replacing by in (75), we have
As the previous consideration in Lemma 15, for where , we have
Associating (76) with (77), we have where
Now we let in (51), and we have Employing Lemma 14, the bilinear term in (80) can be bounded by The other terms are bounded by Associating with all the above inequalities, we have