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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 576087, 8 pages
http://dx.doi.org/10.1155/2014/576087
Research Article

A Stochastic Weakly Damped, Forced KdV-BO Equation

Department of Mathematics, Tongji University, Shanghai 200092, China

Received 14 January 2014; Revised 13 April 2014; Accepted 13 April 2014; Published 5 May 2014

Academic Editor: Tonghua Zhang

Copyright © 2014 Guolian Wang. 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.

Abstract

We investigate the long time behavior of the damped, forced KdV-BO equation driven by white noise. We first show that the global solution generates a random dynamical system. By energy type estimates and dispersive properties, we then prove that this system possesses a weak random attractor in the space .

1. Introduction

The deterministic Korteweg-de Vries-Benjamin-Ono (KdV-BO) equation describes a large class of internal waves in the ocean and stratified fluid. The well-posedness for this equation was studied in [1, 2]. When the surface of the fluid is submitted to a nonconstant pressure, or when the bottom of the layer is not flat, a forcing term has to be added to the equation [3, 4]. The long time behavior of the forced generalized KdV-BO was studied in [5]. In this paper we are interested in the case when the forcing term is random. The well-posedness for the stochastic KdV-BO driven by the additive noise was studied in [6]. We have found no studies on the long time behavior of the stochastic KdV-BO equation.

In this paper, we consider the long time behavior of the following stochastic damped, forced KdV-BO equation: where , , and are real constants and . denotes the Hilbert transform

The forcing term is time independent, and is a random process defined on . is a linear operator. Also, is a two-parameter Brownian motion on , that is, a zero mean Gaussian process whose correlation function is given by for , .

Alternatively we consider a cylindrical Wiener process by setting where is an orthonormal basis of and is a sequence of mutually independent real Brownian motions in a fixed probability space adapted to a filtration .

Let us write the Itô form of (1) as follows: with initial condition

The purpose of this paper is to study the long time behavior of the problem (5) with initial data. Before describing our works, we recall some facts related to this paper. The Cauchy problem for the deterministic KdV-BO equation, that is, in (5), was considered in [1]. There the authors obtained well-posedness results by using Fourier restriction norm method in Bourgain’s type spaces with . Based on the global existence results given in [1], the attractor of the damped forced KdV-BO equation was obtained in [5]. The stochastic KdV-BO equation (i.e., in (5)) was studied in space with in [6]. By introducing some useful inequalities to deal with the irregularity caused by stochastic term, well-posedness results were obtained in [6]. Following the work in [6], we will construct the attractor to the Cauchy problem for the stochastic damped, forced KdV-BO equation.

Before stating our main result precisely, we introduce some notations.

Denote by and the inner product and the norm in , respectively. And is the norm of the Banach space .

Given two separable Hilbert spaces and , we denote by the space of Hilbert-Schmidt operators from into . Its norm is given by where is any orthonormal basis of . When , , is simply denoted by .

The proof of the global well-posedness of the solution (5)-(6) is similar to [6]. Here, we only give the following global existence results.

Assume that (or ; let (or be -measurable. Then, the solution is global and belongs to (or ) for any .

We now give our main result about the long time behavior of the KdV-BO equation based on its global existence results.

Theorem 1. Under the assumption that , , , and is -measurable, then the random dynamical system associated with the stochastic equation (5) with initial value possesses a universal weak random attractor in .

The paper is organized as follows. Section 2 contains some concepts about the random dynamical system, and Lemma 2 which gives the existence conditions and the structure of the attractor. Then we show that the unique solution of problem (5)-(6) generates a random dynamical system in Section 3. In Section 4, we prove that there exists a compact random absorbing set, which leads to the existence of a random attractor, that is, Theorem 1.

2. Preliminaries on Random Dynamical System

We now recall some concepts and results from [79].

Let be a probability space and a family of measure preserving transformations such that is measurable, , and for all . The flow together with the corresponding probability space is called a metric dynamical system.

A continuous random dynamical system (RDS) on a polish space with Borel sigma-algebra over on is by definition a measurable map such that -a.s.(i) on ;(ii) for all (cocycle property);(iii) is continuous.

A set-valued map , the set of all subsets of , is called a random compact set, if is a compact -a.s. and if is measurable with respect to for each , where .

Let be a random set and ; one says attracts if

A random set is said to be a random attractor for the RDS if (i) is a random compact set;(ii) is invariant, that is, , for all ;(iii) attracts all deterministic bounded sets .

Similar to the deterministic theory, the existence result of random attractors can be stated as follows (see [8, 9]).

Lemma 2. If there exists a random compact set absorbing every bounded set , then the RDS possesses a random attractor , where is the omega-limit set of .

3. Solve the Equations and Generate an RDS

We consider the following linear problem to (5)-(6): whose solution is given by the stochastic integral (see [10])

From now on we turn our attention to study the well-posedness of a weakly damped, forced KdV-BO equation with random parameter by change of variable.

Let us study (5) by means of the change of variable and then satisfies (5) if and only if is a solution of This is a weakly damped, forced KdV-BO equation with random parameter with the following initial condition:

The estimation about the random parameter and the bilinear term in (14) can be obtained by using the method in [6]. Then by the fixed point argument the global existence results to the random parameter Cauchy problem (14)-(15) can be obtained.

Theorem 3. Assume that ; let (or . Then, the solution of problem (14) and (15) is global and belongs to for any .

We summarize the above existence results for of (14) with initial condition , , as follows.(i)Under the assumption of Theorem 3, for , and any and , there exists a unique solution .(ii)Under the assumption of Theorem 3, for , and any , and , there exists a unique solution .(iii)Denoting such a solution , the mapping is continuous for all .

Now we construct an RDS modeling the stochastic weakly damped, forced KdV-BO equation. For example, consider a set of continuous functions with value 0 at Let be the Borel sigma-algebra induced by the compact open topology of , and let be a Wiener measure on . We write . The time shift is simply defined by and then is an ergodic metric dynamical system which models white noise.

Having the mapping , we define where is a solution to (14) with and satisfies

Obviously, for , we have Thanks to (17), for any , , we have Therefore, the process defined by is cocycle. It is continuous RDS on over and models the dynamical system associated with the stochastic equation (5) with initial value .

4. Compact Random Absorbing Set

In the following computations is fixed. We usually denote by ;    by ; and by for any in this part.

First note that, for the Hilbert transform, we have for any

Before we prove the existence of a compact random absorbing set, we first give some estimates about the solution of problem .

Let us introduce the following space to study the solution of problem : for some .

Lemma 4 (see [11]). Assume that for some ; then is almost surely in for any and any such that . More precisely,
let , and then one also has where and depend on , respectively.

Remark 5. We have to impose a stronger condition on in the present paper than that on in [11]. Thus, the solution of the linear problem is more regular, which will be used in proving the boundedness of in . More precisely, we can get

4.1. Absorption in at Time

Let and be given, and let be the solution of (14) with initial condition .

Multiplying (14) in by , we obtain It follows that Using Gronwall Lemma for and noticing Lemma 4, we get Noticing we get that .

Then we have the following proposition.

Proposition 6. There exists a random radius , such that for all there exists such that the following holds -a.s. For all and all with , the solution of problem (5) with initial condition satisfies the inequality

Proof. Given , there exists such that for all . Put Then the proof of the proposition is completed.

We can also get an auxiliary estimate from (30) by the Gronwall Lemma with .

Consider the following: This inequality will be useful in the following proof.

4.2. Absorption in at Time

To obtain the estimate, we multiply (14) by and integrate by part to get implies Moreover Combining (37), (38), and (39) , we have Denote And noticing that we deduce that Now we stop to estimate the right hand side of (43) term by term:

Then combining the above estimates, we get Applying the Gronwall Lemma for , we find

We can read the boundedness of the right hand side of (46) from the following estimates. Using (36), we have We also have The boundedness of other terms about the right hand side of (46) is obvious. Noticing (25)–(28), we get where Then we get the following proposition.

Proposition 7. There exists a random variable , such that for all there exists such that the following holds -a.s. For all and all with , satisfies

Proof. Given , there exists such that for all . Put Then the proof of the proposition is completed.

Corollary 8. There exists a random variable , such that the solution of problem (5) with the initial condition satisfies

Proof. We can deduce from the presentation of that Then By , Then the proof of the corollary is completed.

4.3. Construct the Attractor

Before we construct the random attractor, it is worthwhile to point out that the domain about the space variable is unbounded which makes the embedding to be noncompact. Fortunately, we can get weak compactness in the space by Lemma 9 given below. Motivated by the definition of weak attractor for the deterministic system in [12], we extend this definition to the RDS. Then we get weak random global attractor for (5) with initial condition .

Lemma 9 (see [13]). The solution operator is weakly continuous in in the sense that if converges weakly in to some as , then converges to weakly in for all .

Having Proposition 7, Corollary 8, and Lemma 9 in hand, one begins to construct the weak random attractor in space .

From Section 4.2, one gets an RDS for the stochastic weakly damped, forced KdV-BO equation with additive noise; that is, One recalls in [8] that if and are random sets such that for -almost all there exists a time such that for all then is said to absorb . Of course, if absorbs , then attracts .

Denote It is easy to see that is an absorbing set for RDS in . Furthermore it is a weak compactness according to Lemma 9.

Proposition 10. Let where is the omega-limit set of . Here the closures are taken with respect to the weak topology of . Then is included in and is nonempty. It is invariant by ; that is,

The proof of Proposition 10 is easy using the following facts.(a) is weakly continuous on , which is an obvious corollary of Lemma 9.(b)A point if and only if there exist two sequences , such that and .Then has the following properties:(i) is random compact set;(ii) for all ;(iii) attracts all deterministic bounded sets under the sense that where denotes the distance in the weak topology of .

Then we get Theorem 1 in this paper.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11101309).

References

  1. B. Guo and Z. Huo, “The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 444–458, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. Linares, “L2 global well-posedness of the initial value problem associated to the Benjamin equation,” Journal of Differential Equations, vol. 152, no. 2, pp. 377–393, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. L. Herman, “The stochastic, damped KdV equation,” Journal of Physics A: Mathematical and General, vol. 23, no. 7, pp. 1063–1084, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Scalerandi, A. Romano, and C. A. Condat, “Korteweg-de Vries solitons under additive stochastic perturbations,” Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 58, no. 4, pp. 4166–4173, 1998. View at Google Scholar · View at Scopus
  5. B. Guo and Z. Huo, “The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in L2,” Discrete and Continuous Dynamical Systems A, vol. 16, no. 1, pp. 121–136, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. Wang and B. Guo, “Well-posedness of stochastic Korteweg-de Vries-Benjamin-Ono equation,” Frontiers of Mathematics in China, vol. 5, no. 1, pp. 161–177, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. L. Arnold, Random Dynamical Systems, Springer, Berlin, Germany, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  8. H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probability Theory and Related Fields, vol. 100, no. 3, pp. 365–393, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. Crauel, A. Debussche, and F. Flandoli, “Random attractors,” Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp. 307–341, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  11. A. de Bouard and A. Debussche, “On the stochastic Korteweg-de Vries equation,” Journal of Functional Analysis, vol. 154, no. 1, pp. 215–251, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J.-M. Ghidaglia, “Finite-dimensional behavior for weakly damped driven Schrödinger equations,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 5, no. 4, pp. 365–405, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. Rosa, “The global attractor of a weakly damped, forced Korteweg-de Vries equation in H1(),” Matemática Contemporânea, vol. 19, pp. 129–152, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet