Attractor for a Viscous Coupled Camassa-Holm Equation
© Lixin Tian and Ying Xu. 2010
Received: 8 February 2010
Accepted: 30 August 2010
Published: 7 September 2010
are all time independent .
Up to now, great efforts have been already devoted to the Camassa-Holm equation. A. Constantin (see [19–21]) considered the Cauchy problem, inverse spectral problem, and inverse scatting transform for Camassa-Holm equation, proving that the corresponding solution to (1.1) does not exist globally for smooth initial data. Rui et al. (see [22, 23]) employed both bifurcation method and numerical simulation to investigate bounded traveling waves of (1.1) in a general compressible hyperelastic rod. Lenells  used the inverse scattering transform to show that a solution of the Camassa-Holm equation is identically zero whenever it vanishes on two horizontal half-lines in the space. In particular, a solution that has compact support at two different times vanishes everywhere, proving that the Camassa-Holm equation has infinite propagation speed. Cohen et al.  presented two new multisymplectic formulations for the Camassa-Holm equation, and the associated local conservation laws were shown to correspond to certain well-known Hamiltonian functionals. The multisymplectic discretisation of each formulation was exemplified by means of the Euler box scheme. Yiping Meng and Lixin Tian  investigated the boundary control of the viscous generalized Camassa-Holm equation on . Long et al.  obtained the loop soliton solution and periodic loop soliton solution , solitary wave solution and solitary cusp wave solution and smooth periodic wave solution and nonsmooth periodic wave solution of (1.1) and also discussed their dynamic characters and relations by the integral bifurcation method. Moreover, Ding and Tian (see [29, 30]) considered the existence of the global solution to dissipative Camassa-Holm equation and the global attractor of semigroup of solutions of dissipative Camassa-Holm equation in . Olson  showed that the Cauchy problem for a higher-order modification of (1.1) is locally well posed for initial data in for , where and the value of depends on the order of equation, proved the existence and uniqueness of solutions of (1.1) by a contraction mapping argument. Moreover, Zhou and Tian  investigated the initial boundary value problem of a generalized Camassa-Holm equation with dissipation and established local well-posedness of this closed-loop system by using Kato's theorem for abstract quasilinear evolution equation of hyperbolic type. Then they obtained a conservation law that enables us to present a blowup result by using multiplier technique. Lixin Tian et. al.  discussed optimal control of the viscous Camassa-Holm equation; they deduce that the norm of solution is related to the control item and initial value in the special Hilbert space according to variational method, optimal control theories and distributed parameter system control theories, The optimal control of the viscous Camassa-Holm equation under boundary condition was given, and the existence of optimal solution of the viscous Camassa-Holm equation was proved. Well-posedness problem and scattering problem for DGH equation were also discussed in .
Equation (1.3) is one of many multicomponent generalizations which are integrable (see [35–37]). It has a Lax pair, and it is bi-Hamiltonian. Constantin and Ivanov  showed how (1.3) arises in shallow water theory, and it was derived from the Green-Naghdi equations by using expansions in terms of physical parameters. Recently, the infinite propagation speed property for (1.3) was proved in . Escher et al.  probed into well-posedness and blowup phenomena of the two-component Camassa-Holm equation in details. Chen et al.  obtained solutions of (1.3) by a reciprocal transformation between (1.3) and the first negative flow of the AKNS hierarchy and stated some examples of peakon and multikink solutions of (1.3). Guan and Yin  presented a new global existence result and several new blowup results of strong solutions to (1.3) as , improving considerably earlier results. Jibin Li and Yishen Li  obtained the existence of solitary wave solutions, kink and antikink wave solutions, uncountable infinite many breaking wave solutions, and smooth and nonsmooth periodic wave solutions with the method of dynamical systems to the two-component generalization of the Camassa-Holm equation. Yujuan Wang et. al. showed that the two-component Camassa-Holm equation possesses a global continuous semigroup of weak conservative solutions for initial data. In  a link between central extensions of superconformal algebra and a supersymmetric two-component generalization of the Camassa-Holm equation was concerned. Deformations of superconformal algebra give rise to two compatible bracket structures. For the system (1.3) particularizes to the Camassa-Holm equation which is a re-expression of geodesic flow on the diffeomorphism group of the circle (see [44, 45]).
Our paper is organized as follows. In Section 2, we give the main definitions and Lemmas. In Section 3, main results are presented, as the core of the paper, and the proofs of the main theorems are completed. Firstly, we prove that (1.9)–(1.13) has a unique solution in infinite time interval then obtain the existence of global solution of (1.9)–(1.13) in by prior estimates. Meanwhile we obtain that the semigroup of the solution operator has an absorbing set. Finally, we demonstrate the long-time behavior of solution of (1.9)–(1.13) that is described by global attractor. In brief, we obtain the existence of the global attractor for (1.9)–(1.13) in .
and where is Laplace operators and is a self-adjoint positive operators with compact inverse. The eigenvalue of is satisfying as , where is the corresponding eigenvector of . For simplicity we will give the following inequalities and only refer to their names wherever necessary.
Lemma 2.2 (consistent Gronwall inequality).
Lemma 2.3 (Sobolev inequality).
Lemma 2.4 (Young inequality).
3. Main Results and the Proof of the Theorems
By means of existence theory of solution to ordinary differential equations, we know that local smooth solution of (3.1)–(3.3) exists. Now we can establish consistent integral estimate on approximate solution with respect to by Galerkin method.
Then we get that are bounded. Considering Aubin's compactness theorem, we conclude that there is a subsequence , , , , so that , , , and Now we replace with . We will prove that satisfy (1.9)–(1.10). That is to say, approximate solution of (3.1)–(3.2) is convergent to solution of (1.9)–(1.10).
The authors really appreciate Professor Yue Liu for his valuable comments and opinions. The paper is supported by the National Nature Science Foundation of China (no. 10771088) and Nature Science Foundation of Jiangsu(no. BK 2010329) and Outstanding Personnel Program in Six Fields of Jiangsu (no. 6-A-029).
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