Random attractor for a stochastic viscous coupled Camassa-Holm equation
© Huang et al.; licensee Springer 2013
Received: 10 October 2012
Accepted: 10 April 2013
Published: 23 April 2013
In this paper, we study the asymptotic dynamics of the stochastic viscous coupled Camassa-Holm equation with periodic boundary condition. We investigate the existence of a random attractor for the dynamical system associated with the equation. The random attractor is invariant and attracts every pull-back tempered random set under the forward flow. We establish the asymptotic compactness of the random dynamical system by compactness of embedding of Sobolev space.
models the unidirectional propagation of shallow water waves over a flat bottom [1–4]. It has been paid considerable attention due to its rich nonlinear phenomenology. It is completely integrable  and it has stable solitons . It admits the peakons which is also stable [6, 7]. In [8, 9], it has been shown that (1.1) is locally well-posed for initial data (). More interestingly, there are a rich variety of global strong solutions and blow-up solutions [8–10]. Xin and Zhang obtained the global existence of weak solution . Bressan and Constantin obtained the existence of global conservative and dissipative solutions [6, 12].
which has peakon solitons in the form of a superposition of multipeakons. They investigated local well-posedness and blow-up solutions of (1.2) by means of Kato’s semigroup approach to nonlinear hyperbolic evolution equation and obtained a criterion and condition on the initial data guaranteeing the development of singularities in finite time for strong solutions of (1.2) by energy estimates. Moreover, an existence result for a class of local weak solutions was also given.
The authors proved that (1.3) has a unique solution in infinite time interval by prior estimates and demonstrated its long time behavior which was described by a global attractor.
They considered the well-posedness and discussed the existence of global solution of (1.4) in by using prior estimates. Finally, they showed the long time behavior of solution and obtained the existence of global attractor of (1.4).
where , , , . is a given function defined on , which described in the following. is a two-side real-valued Wiener process on a probability space which will be specified later.
A system is usually uncertain in reality 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 also rather essential phenomena [16–20].
Attractor is an important concept to describe long time behavior of solutions for a given system. It is known that the long time behavior of random systems is captured by a pullback random attractor, which was introduced in [17, 21, 22] as an extension of attractor theory of deterministic system in [23–27]. The existence of random attractors for stochastic differential equations has been studied extensively by many authors [28–35]. To our best knowledge, the problem of random attractor for (1.5) has not been discussed. We think it is a significant work to obtain a random attractor for the system.
The paper is organized as follows. In the next section, we review the pullback random attractor theory for random dynamical systems and some lemmas. In Section 3, we define a random dynamical system for the stochastic viscous coupled Camassa-Holm equation. Then we derive the uniform estimates of solutions in Section 4. These estimates are necessary for proving the existence of bounded random absorbing sets and the asymptotic compactness of the random dynamical system and prove the existence of a pullback random attractor in . We conclude that the global attractor persists under a white noise.
Let be a separable Hilbert space with Borel σ-algebra , and be a probability space.
Definition 2.1 is called a metric dynamical system if is -measurable, is the identity on Ω, for all and for all .
is the identity on X;
for all ;
is continuous for all .
Hereafter, we assume that Φ is a continuous RDS on X over .
Definition 2.4 Let be a collection of random subsets of X. Then is called inclusion-closed if and with for all imply that .
Definition 2.6 The RDS Φ is said to be -pullback asymptotically compact in X if for P-a.e. , has a convergent subsequence in X whenever , and with .
is compact, and is measurable for every ;
- (ii)is invariant, that is,
- (iii)attracts every set in , that is, for every ,
where d is the Hausdorff semi-metric given by for any and .
Some basic inequalities which will be used frequently in the following consideration are presented as follows.
Lemma 2.1 (Gagliardo-Nirenberg inequality)
Lemma 2.2 (Poincaré inequality)
Lemma 2.3 (Young inequality)
, where , . As , one has .
3 Stochastic viscous coupled Camassa-Holm equation
For convenience, we introduce some marks. We denote and the norms in and respectively. We denote and . The Laplace operator Δ is an isomorphism from to H. The eigenvalues of Δ has the form where and . Then the Poincaré inequality is simplified down to , where and λ denotes the first eigenvalue of Δ.
where . W is a two-sided real-valued Wiener process on a probability space, which will be determined below.
It’s known that there exists a -invariant set of full P measure such that is continuous in t for every , and the random variable is tempered [16, 17, 21, 28]. We give some properties of the process as follows.
We will consider (3.13)-(3.17) for and write as Ω from now on.
generates a continuous random dynamical system, where .
generates a random dynamical system associated with (3.1)-(3.5). Note that the two random dynamical systems are equivalent by (3.19). It is easy to check that has a random attractor provided possesses a random attractor. So, we only need to consider the random dynamical system .
4 Uniform estimates of solution
In this section, we derive uniform estimates on the solution of the stochastic viscous coupled Camassa-Hlom equation when . These estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system. From now on, we always assume that is the collection of tempered random subsets of E.
where c is the constant in Lemma 2.1.
which completes the proof. □
which completes the proof. □
which completes the proof. □
The next lemma will illustrate the existence of the random absorbing set for Φ in .
Then . Further, (4.67) indicates that is a random absorbing set for Φ in , which completes the proof. □
Next we derive uniform estimates for in .
which completes the proof. □
which completes the proof. □
5 Random attractors
In this section, we prove the existence of a -random attractor for the random dynamical system Ψ associated with the stochastic viscous coupled Camassa-Holm equation (3.1)-(3.5) with periodic boundary condition. It follows from Lemma 4.4 that Φ has a closed random absorbing set in , which along with the -pullback asymptotic compactness will imply the existence of a unique -random attractor. The -pullback asymptotic compactness of Φ is given below and will be proved by the compactness of embedding of Sobolev space.
Lemma 5.1 The random dynamical system Φ is -pullback asymptotically compact in : that is, for P-a.e. , the sequence has a convergent subsequence in provided , and .
strongly in as desired. □
We are now in a position to present our main result: the existence of a -random attractor for Ψ in .
Theorem 5.1 The random dynamical system Ψ has a unique -random attractor in .
Proof Notice that Φ has a closed random absorbing set in by Lemma 4.4, and is -pullback asymptotically compact in by Lemma 5.1. Hence, the existence of a unique -random attractor for Φ follows from Theorem 2.1 immediately. Then (3.19) implies that Ψ has a -random attractor in . □
We have checked the persistence of global attractor of viscous coupled Camassa-Holm equation with periodic boundary condition under a white noise.
The authors would like to thank reviewers for the valuable suggestions and comments. The paper is supported by the National Natural Science Foundation (No. 11171115).
- Camassa R, Holm DD: An integrable shallow water equation with peaked solutions. Phys. Rev. Lett. 1993, 71: 1661–1664. 10.1103/PhysRevLett.71.1661MathSciNetView ArticleGoogle Scholar
- Camassa R, Holm DD, Hyman JM: A new integrable shallow water equation. Adv. Appl. Mech. 1994, 31: 1–33.View ArticleGoogle Scholar
- Dullin HR, Gottwald GA, Holm DD: On asymptotically equivalent shallow water wave equations. Phys. D 2004, 190: 1–14. 10.1016/j.physd.2003.11.004MathSciNetView ArticleGoogle Scholar
- Dullin HR, Gottwald GA, Holm DD: CH, Korteweg-de Vries-J-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 2003, 33: 73–95. 10.1016/S0169-5983(03)00046-7MathSciNetView ArticleGoogle Scholar
- Constantin A, Strauss W: A stability of the Camassa-Holm solitons. J. Nonlinear Sci. 2002, 12: 415–422. 10.1007/s00332-002-0517-xMathSciNetView ArticleGoogle Scholar
- Bressan A, Constantin A: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 2007, 183: 215–239. 10.1007/s00205-006-0010-zMathSciNetView ArticleGoogle Scholar
- Constantin A, Strauss W: Stability of peakons. Commun. Pure Appl. Math. 2000, 53: 603–610. 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-LMathSciNetView ArticleGoogle Scholar
- Constantin A, Escher J: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1998, 26: 303–328.MathSciNetGoogle Scholar
- Rodriguez-Blanco G: On the cauchy problem for the Camassa-Holm equation. Nonlinear Anal. 2001, 46: 309–327. 10.1016/S0362-546X(01)00791-XMathSciNetView ArticleGoogle Scholar
- Constantin A, Escher J: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998, 181: 229–243. 10.1007/BF02392586MathSciNetView ArticleGoogle Scholar
- Xin ZP, Zhang P: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 2000, 53: 1411–1433. 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5MathSciNetView ArticleGoogle Scholar
- Bressan A, Constantin A: Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 2007, 5: 1–27. 10.1142/S0219530507000857MathSciNetView ArticleGoogle Scholar
- Fu Y, Qu C: Well posedness and blow-up solution for a new coupled Cammassa-Holm equations with peakons. J. Math. Phys. 2009., 50: Article ID 012906Google Scholar
- Tian L, Xu Y: Attractor for a viscous coupled Camassa-Holm equation. Adv. Differ. Equ. 2010., 2010: Article ID 512812Google Scholar
- Tian L, Xu Y, Zhou J: Attractor for the viscous two-component Camassa-Holm equation. Nonlinear Anal. 2012, 13: 1115–1129. 10.1016/j.nonrwa.2011.09.005MathSciNetView ArticleGoogle Scholar
- Arnold L: Random Dynamical Systems. Springer, Berlin; 1998.View ArticleGoogle Scholar
- Crauel H, Flandoli F: Attractors for random dynamical systems. Probab. Theory Relat. Fields 1994, 100: 365–393. 10.1007/BF01193705MathSciNetView ArticleGoogle Scholar
- Chueshov I: Monotone Random Systems Theory and Applications. Springer, Berlin; 2002.View ArticleGoogle Scholar
- Da Prato G, Zabczyk J: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge; 1992.View ArticleGoogle Scholar
- Øksendal B: Stochastic Differential Equations. Springer, Berlin; 2000.Google Scholar
- Crauel H, Debussche A, Flandoli F: Random attractor. J. Dyn. Differ. Equ. 1997, 9: 307–341. 10.1007/BF02219225MathSciNetView ArticleGoogle Scholar
- Flandoli F, Schmalfuß B: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 1996, 59: 21–45.View ArticleGoogle Scholar
- Babin AV, Vishik MI: Attractors of Evolution Equations. North-Holland, Amsterdam; 1992.Google Scholar
- Hale JK: Asymptotic Behaviour of Dissipative Systems. Am. Math. Soc., Providence; 1988.Google Scholar
- Robinson JC: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge; 2001.View ArticleGoogle Scholar
- Sell R, You Y: Dynamics of Evolutionary Equations. Springer, New York; 2002.View ArticleGoogle Scholar
- Teman R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York; 1998.Google Scholar
- Bates PW, Lisei H, Lu K: Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 2006, 6: 1–21. 10.1142/S0219493706001621MathSciNetView ArticleGoogle Scholar
- Bates PW, Lu K, Wang B: Random attractors for stochastic reaction-diffusion equations on unbounded domains. J. Differ. Equ. 2009, 246: 845–869. 10.1016/j.jde.2008.05.017MathSciNetView ArticleGoogle Scholar
- Caraballo T, Langa JA, Robinson JC: A stochastic pitchfork bifurcation in a reaction-diffusion equation. Proc. R. Soc. Lond. Ser. A 2001, 457: 2041–2061. 10.1098/rspa.2001.0819MathSciNetView ArticleGoogle Scholar
- Han X, Shen W, Zhou S: Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differ. Equ. 2011, 250: 1235–1266. 10.1016/j.jde.2010.10.018MathSciNetView ArticleGoogle Scholar
- Wang B: Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domians. J. Differ. Equ. 2009, 246: 2506–2537. 10.1016/j.jde.2008.10.012View ArticleGoogle Scholar
- Wang Z, Zhou S, Gu A: Random attractor of the stochastic strongly damped wave equation. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 1649–1658. 10.1016/j.cnsns.2011.09.001MathSciNetView ArticleGoogle Scholar
- Wang Z, Zhou S, Gu A: Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains. Nonlinear Anal. 2011, 12: 3468–3482. 10.1016/j.nonrwa.2011.06.008MathSciNetView ArticleGoogle Scholar
- Wang X, Li S, Xu D: Random attractors for second-order stochastic lattice dynamical systems. Nonlinear Anal. 2010, 72: 483–494. 10.1016/j.na.2009.06.094MathSciNetView ArticleGoogle Scholar
- Crauel H: Random point attractors versus random set attractors. J. Lond. Math. Soc. 2002, 63: 413–427.MathSciNetView ArticleGoogle Scholar
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