Existence of Random Attractor for Stochastic Fractional Long-Short Wave Equations with Periodic Boundary Condition

Liu, Na; Xin, Jie

doi:https://doi.org/10.1155/2017/3642548

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2017 | Article ID 3642548 | https://doi.org/10.1155/2017/3642548

Existence of Random Attractor for Stochastic Fractional Long-Short Wave Equations with Periodic Boundary Condition

Na Liu¹and Jie Xin^1,2

Academic Editor: Ciprian G. Gal

Received30 Mar 2017

Accepted08 Jun 2017

Published26 Jul 2017

Abstract

We consider the asymptotic behaviors of stochastic fractional long-short equations driven by a random force. Under a priori estimates in the sense of expectation, using Galerkin approximation by the stopping time and the Borel-Cantelli lemma, we prove the existence and uniqueness of solutions. Then a global random attractor and the existence of a stationary measure are obtained via the Birkhoff ergodic theorem and the Chebyshev inequality.

1. Introduction

Long-short wave resonance equations arise in the study of the interaction of the surface waves with both gravity and capillary modes presence and also in the analysis of internal waves, as well as Rossby wave in [1]. In the plasma physics, they describe the resonance of the high-frequency electron plasma oscillation and associated low-frequency ion density perturbation in [2]. Long wave and short wave equations with periodic boundary condition have attracted considerable attention as a result of their rich physical and mathematical properties. Guo [3, 4] proved the existence of global solutions for long-short wave equations and generalized long-short wave equations. The existence of global attractor was studied in [5–9].

The stochastic partial differential equation (SPDE) is a kind of partial differential equation with random term and random coefficients, which come from the random environmental effects and the errors of measurement. SPDE is used to describe better complex phenomenon, for example, quantum field theory, statistical mechanics, and financial mathematics; see [10–12] and so on. In [13–19], the authors obtained the existence and uniqueness of the solution and of attractors for SPDEs.

In this paper, we consider that the random environmental effects and the errors of measurement are included into the model of fractional long-short wave equations. More specifically, we study the following equations: with the initial conditionand periodic boundary condition where in ; ; . The unknown complex valued function is short surface wave packet and the unknown real valued function is a long interfacial wave. and are independent value Wiener processes which are from the errors of measurement or the random environmental effects and can be seen in detail in the next section.

Since the solution , if it exists, is a -periodic function, we have the Fourier expansion where .

Hence, and is defined by Sincethe following definitions make sense. Letand let be a complete distance space of the set under the norm It is easy to get it to be a Banach space, and that is space-periodic with the period and its order derivatives are in . And , when . is a Hilbert space with the inner product

The rest of this paper is arranged as follows. In Section 2, we present some preliminaries results. In Section 3, we give a series of time uniform a priori estimates in different energy spaces which will be used to prove our main results; see [20]. In Section 4, we show the existence and uniqueness of solutions for (1)-(2). In Section 5, the random weak attractor and the stationary measure are constructed.

2. Preliminary

In this paper, we outline the variational framework for studying problems (1)-(2) and list some preliminary lemmas which will be used later.

We define a complete probability space . denotes the expectation operator with respect to . Stochastic terms and are defined on bywhere is a standard complex valued Wiener process, is a standard real valued Winer process independent of , and are sufficiently smooth functions in a manner. The different inner product spaces for the solution of (1)-(2) are defined as Endow each with the usual norm, and satisfy with compact embedding.

Let be three Banach reflective spaces which satisfy with compact and dense embedding. The Banach space can be defined as follows: endowed with the natural norm We obtain the following lemma with regard to compactness result by [21].

Lemma 1. If is bounded in , then is precompact in .

In order to get maximal estimates on stochastic integrals, we need another lemma. and are Hilbert spaces which are separable and is a -Wiener process on with . Let be the space of Hilbert-Schmidt operators from to . For such operators, we obtain the following lemma by [22].

Lemma 2. For any and any -valued predictable process , we get where and are some positive constants dependent on .

3. Uniform a Priori Estimates in Time

3.1. A Priori Estimates in

Lemma 3. Provided that , then for any and , we obtain .

Proof. Taking the inner product of (1) with , we get Taking the imaginary part of (19) and applying the Itô formula to , one gets By (20), using Hölder’s and Young’s inequalities, we can obtain On the one hand, multiplying by and integrating from to on both sides of (21), we get where is independent of .
On the other hand, integrating from to and taking the supremum and the expectation on both sides of (21), we can obtain By Lemma 2, for any positive constant , we obtain From (24), for any , there exists a positive constant such that On the basis of the above estimates, we can further give an estimate of for any . Applying the Itô formula and Hölder’s inequality, we obtain On the one hand, multiplying by , integrating from to , and taking the expectation on both sides of (26), we can have where is independent of .
On the other hand, by (26), we have Integrating from to and taking the supremum and the expectation on both sides of (28), we get By Lemma 2, for any positive constant , we have where Then inserting (30) into (29), and by (27), for any , we obtain where is a positive constant depending on , and .

Lemma 4. Provided that , then for any and , we have

Proof. Taking the inner product of (2) with , we have Applying the Itô formula to , since , we get From (1), we get that Notice that From (33), (34), and (35), we can have Applying the Itô formula to , and taking the inner product of (1) with , we obtain Taking the real part of (37), we get It is easy to check that Otherwise, by (2), we can have .
So From (2), we can obtain Using Hölder’s inequality, the Gagliardo-Nirenberg inequality, and Young’s inequality, we can estimate each term. Now, let and take . Putting (36) and (41) into the result, we get Multiplying by , integrating from to , and taking expectation on both sides of (42) yield From (27), we can estimate (43) and obtain where is independent of .
Since for any , we obtain where is independent of .
Besides, we estimate for . Firstly, applying the Itô formula to , we have Multiplying by , integrating from to , and taking expectation on both sides of (47) yield From (27) and (48), we obtain where is independent of .
Consequently, from (45), we obtain On the one side, integrating from to on both sides of (42), we can deduce Taking the supremum and expectation on both sides of (51) yields And then by estimating each term of the right hand side of (52), we obtain On the other side, for , integrating from to and taking the supremum and the expectation on both sides of (47) yield Now we estimate each term of (54). For the third term on the right hand of (54), For the fifth and sixth terms on the right hand of (54), we have For the seventh and eight terms on the right hand of (54), using a similar method, we estimate For the last term on the right hand of (54), we estimate By the above estimates, we clearly obtain In addition, by (46), we can get Then we obtain Then the proof is complete.

3.2. A Priori Estimates in

Lemma 5. Provided that , then for any and , we have

Proof. Taking the inner product of (1) with , we get Applying the Itô formula to , we have Since and we obtain According to the above inequality, we can estimate each term on the right hand side of (65) using Höder’s inequality, the Gagliardo-Nirenberg, and Young’s inequality. So, by (65), we get Otherwise, applying the Itô formula to , we have Since and we obtain Thus, by (66) and (69), taking , and letting we get Multiplying by , integrating from to , and taking expecting on both sides of (71), by (46), we obtain where is independent of . Since for any , we obtain where is independent of .
Next step, we estimate for . On the one hand, applying the Itô formula to , we obtainBy (45), we obtain where is independent of . Thus, from (73), for any , we get Integrating from to and taking the supremum and expectation on both sides of (71), as with the estimates in Lemma 4 for each term, we deduce On the other hand, for , integrating from to and taking the supremum and the expectation on both sides of (71), and estimating each term, we get Therefore, from (73), it is inferred that for So then, we have and the proof is complete.

3.3. A Priori Estimates in

By using the similar method and idea as Sections 3.1 and 3.2, we can achieve a priori estimates in . For simplicity, we only provide the idea of the proof. Using (1)-(2) and applying the Itô formula to and , respectively, we can get some inequalities by Höder’s inequality, the Gagliardo-Nirenberg, and Young’s inequality. After that, taking the supremum and expectation for inequalities and estimating on and by Gronwall-type, we can deduce the following lemma.

Lemma 6. Provided that , and , then for any and , we have .

4. Proofs of Theorems 7 and 8

Based on the prior estimate, we acquire the existence and uniqueness of a solution for the stochastic fractional long-short wave equations (1)-(2) in spaces .

Theorem 7. If , and , then there exists a unique solution , almost surely satisfying (1)-(2). In addition, is continuous from to .

Proof. First, we know that . Let be an orthonormal basis of eigenvectors of the Laplace operator on , which is an orthonormal basis of Consider as the projection from onto the space spanned by . Then the approximation solution solves the approximation problem where is the projector onto the first vectors , and commutes with the operator . We will treat the above equations pathwise by introducing the following random processes solving with periodic boundary conditions and initial conditions In accordance with the same method as in Section 3, for any and almost all , we have Therefore, we can get the following estimate: where a positive constant is independent of . Moreover, for any , holds for a positive constant dependent on . Set as the solution of the following equations: Let , and satisfy for and for . It is easy to see that (88)-(89) are random differential equations with Lipschitz nonlinearity in finite dimension. Afterwards, for almost all , we have a unique solution for (88)-(89). We define the stopping time as follows: if the set is nonempty, or else . Because is increasing in , let almost surely. For , we obtain satisfying (82). On the basis of the estimates given in Section 3.2 and (86)-(87), for any , we obtain where the positive constant , which is dependent on and . And for , we obtain with the positive constant , which is dependent on but independent of . Let . On the other side, we get where for and for . Then, by (93), we get In the light of the above estimate and the Borel-Cantelli lemma, for any , we obtain So we know that satisfies the following random differential equations: with initial conditions Then satisfies the estimates (93) and (94), and for any , we find that is the unique global solution of (82).
And then, we will investigate (99)-(100) for fixed . Firstly, by (94), for any , we can know that We let Then . Thus, for any fixed , there exists with such that Then we can extract a subsequence of , which are still denoted by , such that These convergences are sufficient to pass the limit in linear terms; however, in fact, for nonlinear terms, we need a strong convergence of . From (100) and the estimate (104), it is easy to get . Further, we can extract a subsequence of which is still denoted by such that For the nonlinear term, we can pass the limit by a standard procedure. Thus, we prove that is a weak solution of with initial conditions Then is a solution of (1)-(2) and satisfies the estimates given in Section 3.2. The continuity of the solution can be proved in the following. For any , it is obvious that . Then, we obtain that there is for almost all by Lemma in [23]. Noticing almost surely and applying similar methods, we can get according to [21]. So we get almost surely by definition of and . Thus, the solution is continuous from to almost surely.
Because the noise is additive, we can use the same approach as [7]. Therefore, the solution is unique almost surely in .

Theorem 8. If , and , then there exists a unique solution almost surely satisfy (1)-(2). Moreover, is continuous from to .

As a matter of fact, from Theorems 7 and 8, a continuous random dynamical system can be defined in and , respectively. Then we can, respectively, construct a random attractor endowed with the weak topology for the continuous random dynamical system in and .

The proof of Theorem 8 is similar to that of Theorem 7. Here we omit the detail of the proof.

5. Proofs of Theorems 10 and 11

In this section, the asymptotic behavior of solution for the corresponding problem is studied. We will construct a random attractor for stochastic fractional long-short wave equations in equipped with the weak topology. In [13–16], we can obtain some basic concepts related to random attractors for random dynamical systems.

Motivated by [13, 14], we can find the following existence result for a random attractor for a continuous RDS. It is a sufficient condition for the existence of random attractors.

Theorem 9 (see [13, 14]). Assume is a RDS on a Polish space and there exists a random compact set absorbing every bounded deterministic set . Then we define a global random attractor for RDS by

Next step, according to Theorem 9 and the priori estimates in Section 3, we research the random attractors for the stochastic long-short wave equations in and .

Theorem 10. If , and , then (1)-(2) have a global random weak attractor which is a random tempered compact set in endowed with the weak topology.

Proof. On the basis of the former analysis, we can consider the properties of solution a of the system and it (107) has a unique solution for almost all . Noticing that system (107) has coefficients driven by , a random dynamical system can be defined by on . Thus, also defines a continuous random dynamical system on , which is denoted by , and is weakly continuous almost surely on . denotes the ball center at with radius in . Using estimates made in accordance with Section 3, there is a random variable such that, for any , . Therefore, there exists a random time , such that, for all and almost all , We define the random attractor as follows: where the closure is taken with respect to the weak topology of . Next we prove that is tempered. According to the estimates obtained in Section 3, we obtain Then, from Birkhoff’s ergodic Theorem [24], on a -invariant subset of with full probability measure; that is, is tempered. So we get that is tempered. The proof is completed.

Theorem 11. If , and , then (1)-(2) have a global random weak attractor which is a random tempered compact set in endowed with the weak topology.

Based on Theorem 10, we can prove Theorem 11 using the same methods and ideas of proof. More precisely, there exists a random attractor for the stochastic fractional long-short wave equations in .

Theorem 12. If , then (1)-(2) have one stationary measure on and .

Proof. If , (1)-(2) has a unique solution with by the results given in Sections 3 and 4, which, for any , satisfies for a positive constant which is independent of .
Let be the distribution of . According to the classical Bogolyubov-Krylov argument [22], we define as for any Borel set of : namely, . From (114), we obtain According to Chebyshev’s inequality and the fact that has a compact embedding into is tight in . Then there exists a sequence with as and a probability measure on such that weakly as . Thus, using the standard argument as in [24], is a stationary measure for stochastic fractional long-short wave equations on . In addition, from (114), is in fact supported on ; that is to say, is a stationary measure for stochastic fractional long-short wave equations on . The proof is completed.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was supported by the This work was supported by the NSF of China (no. 11371183) and the NSF of Shandong Province (no. ZR2013AM004).

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Copyright © 2017 Na Liu and Jie Xin. 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.

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