Attractor for a Viscous Coupled Camassa-Holm Equation

Abstract

The global existence of solution to a viscous coupled Camassa-Holm equation with the periodic boundary condition is investigated. We obtain the compact and bounded absorbing set and the existence of the global attractor for the viscous coupled Camassa-Holm equation in by uniform prior estimate.

1. Introduction

The Camassa-Holm equation

(1.1)

has been paid considerable attention due to its rich phenomenology all the time. Its abstract derivation was first discovered by Fuchssteiner and Fokas [1], while in the physical derivation of Camassa and Holm (see [2, 3]), the equation models unidirectional propagation of shallow water waves, and  represents the fluid velocity in the  direction, equivalently the height of the fluid's free surface above a flat bottom. They also found that the solitary waves interact like solitons. Unlike the Korteweg-de Vries equation (which is an approximation to the equations of motion), this model is obtained by approximating directly in the Hamiltonian for Euler's equations in the shallow water regime (see [3, 4]). Equation (1.1) retains higher-order terms in a small amplitude expansion of incompressible Euler's equations for unidirectional motion of waves at the free surface under the influence of gravity. Dropping these terms leads to the BBM equation, or at the same order, the KdV equation. The Camassa-Holm has quite a few interesting features: it admits solitary waves called "peakons" with the form of . The peakons of (1.1) are orbitally stable [5]—that is, their shape is stable under small perturbations and therefore these waves are recognized physically. For waves that approximate the peakons in a special way, a stability result was proved by a variation method [6]. This is in sharp contrast to the Korteweg-de Vries equation, where solitary waves are generally smooth. The peaked traveling waves of the Camassa-Holm equation replicate a future that is characteristic for waves of great height–waves of the largest amplitude that are exact solutions of the governing equations for water waves (see [7, 8]). A breaking wave is a solution which remains bounded but whose slope becomes unbounded in finite time, and, in contrast to the KdV equation, the Camassa-Holm equation models breaking waves [9], as well as a breaking rod (see [4, 10]), since the equation models the propagation of axisymmetric waves in hyperelastic rods. After breaking, the solution can be continued either as a global conservative weak solution or as a global dissipative solution (see [11–13]). Peakons interact "elastically" in the manner typical of all solitons, and their wave dynamics are now wellunderstood (see [3, 14, 15]). Some authors have even argued recently that the Camassa-Holm equation might be relevant to the modeling of tsunamis (see [16, 17]). Moreover, the equation has a bi-Hamiltonian structure [2]. As the Camassa-Holm is completely integrable, it has many conserved qualities. Especially for smooth solutions, the qualities

(1.2)

are all time independent [18].

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 [24] 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. [25] 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 [26] investigated the boundary control of the viscous generalized Camassa-Holm equation on . Long et al. [27] obtained the loop soliton solution and periodic loop soliton solution [28], 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 [31] 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 [32] 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. [33] 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 [34].

On the basis of deformation of bi-Hamiltonian structure of the hydrodynamic type, Chen et al.[35] obtained the following two-component generalization of (1.1):

(1.3)

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 [36] 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 [38]. Escher et al. [39] probed into well-posedness and blowup phenomena of the two-component Camassa-Holm equation in details. Chen et al. [35] 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 [40] 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 [41] 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.[42] showed that the two-component Camassa-Holm equation possesses a global continuous semigroup of weak conservative solutions for initial data. In [43] 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]).

We know that it is of great use to construct an interacting system of equations [37]:

(1.4)

as and , it, respectively, leads to the Degasperis-Procesi equation and Camassa-Holm equation. Three independent conserved quantities have been obtained as follows:

(1.5)

here is an arbitrary constant.

Ying Fu and Changzheng Qu [46] considered the following coupled Camassa-Holm equation:

(1.6)

which has peakon solitons in the form of a superposition of multipeakons. It has the following conserved qualities:

(1.7)

They investigated local well-posedness and blowup solutions of (1.6) 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 (1.6) by energy estimates; moreover, an existence result for a class of local weak solutions was also given. They also showed that the solution of (1.6) is

(1.8)

for some positive constants .

In the field of infinite-dimensional dynamical systems, one of the most important issues is to obtain the existence of global attractors for the semigroups of solutions associated with some concrete partial differential equations. For instance, Yongsheng Li and Xingyu Yan [47] studied the existence and regularity of the global attractor for a weakly damped forced shallow water equation in . Tian et al.[48] studied the global attractor for the viscous weakly damped forced Korteweg-de Vries equations in . Yanhong Zhang and Chengkui Zhong [49] investigated the existence of global attractors for a nonlinear wave equation. Lixin Tian and Ruihua Tian [50] studied the attractor for the two-dimensional weakly damped KdV equation in belt field. Ying Xu and Lixin Tian [51] investigated attractor for a coupled nonhomogeneous Camassa-Holm equation with periodic boundary condition. Lixin Tian and Jinglin Fan [52] discussed the global attractors for the viscous Degasperis-Procesi equation in . Lixin Tian and Ying Gao [53] obtained global attractors for the viscous Fornberg-Whitham equation [54] in . Here we investigate the existence of global attractor for a viscous coupled Camassa-Holm equation with the periodic boundary condition in as follows:

(1.9)

(1.10)

(1.11)

(1.12)

(1.13)

where . To the authors' knowledge, the problem of global attractor for (1.9)–(1.13) has not been discussed in previous publications.

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 .

2. Preliminaries

Definition 2.1.

Let stand for the inner product and the corresponding norm. one also denotes

(2.1)

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).

Assume that are three positive locally integrable functions defined on is a locally integrable function over , satisfying

(2.2)

where are positive constants. one can get

(2.3)

Lemma 2.3 (Sobolev inequality).

Suppose that and there exists a constant c, such that

(2.4)

Lemma 2.4 (Young inequality).

where As p = q = 2, one has

3. Main Results and the Proof of the Theorems

Based on Galerkin procedure, we will show the existence of global solution of (1.9)–(1.13). Suppose that is an orthonormal basis in the space consisting of eigenfunctions of the operator . , is orthogonal projection from to . By Galerkin procedure [55], (1.9)–(1.13) can be reduced to ordinary differential system

(3.1)

(3.2)

(3.3)

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.

Theorem 3.1.

If , then (1.9)–(1.13) has a global solution in .

Proof.

Taking the inner product of (3.1), and (3.2), respectively, with in and noting that

(3.4)

(3.5)

(3.6)

with the same reason, we obtain that

(3.7)

By integrating by parts we get

(3.8)

From (3.4) and (3.5) we obtain that

(3.9)

Applying Poincaré inequality, we get

(3.10)

Equality (3.9) implies that

(3.11)

(3.12)

where is nonnegative constant.

Integrating (3.9) over yields

(3.13)

Taking the inner product of (3.1) and (3.2), respectively, with in , we get

(3.14)

(3.15)

since

(3.16)

By means of integrating by parts frequently, we obtain that

(3.17)

Associating all the above inequalities leads to

(3.18)

Simplifying the above inequality and employing Young inequality, it follows that

(3.19)

where . By means of Poincaré inequality, we obtain that

(3.20)

Let

(3.21)

from (3.13) we get

(3.22)

Finally we obtain that

(3.23)

Integrating (3.19) over yields

(3.24)

where are positive constants.

Taking the inner product of (3.1) and (3.2), respectively, with , in , we obtain that

(3.25)

By integrating by parts and applying Sobolev inequality, we obtain that

(3.26)

where is a constant depending on As well as

(3.27)

where is a constant depending on Combining all the above inequalities, we have

(3.28)

By employing Young inequality, it follows that

(3.29)

where Based on Poincaré inequality we obtain that

(3.30)

Let

(3.31)

from (3.12) and (3.23), we conclude that is bounded, so we suppose that ; by (3.24), we have

(3.32)

Using Gronwall inequality, we obtain that

(3.33)

Integrating (3.29) over , we obtain that

(3.34)

where are nonnegative constants.

Respectively, taking the inner product of (3.1) and (3.2) with in , we can also get . Connecting (3.12) and (3.23) with (3.33), we can get that

(3.35)

so are bounded.

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).

Let is finite from the above discussion, and by ordinary differential equation (3.1), we have

(3.36)

Now it is clear that

(3.37)

Note that

(3.38)

From the above statements we get

(3.39)

Similarly, we have that

(3.40)

For all , we obtain that

(3.41)

From ordinary differential equation (3.2), we know that

(3.42)

as we know that

(3.43)

From the above discussions we get

(3.44)

Simultaneously, we have that

(3.45)

For all , we obtain

(3.46)

All the above analysis shows that the global solution to (1.9)–(1.13) exists in .

Theorem 3.2.

Denote as the semigroup of the solution operator to (1.9)–(1.13), Then has an absorbing set in

Proof.

Taking the inner product of (1.9) and (1.10),respectively, with in , noting that

(3.47)

and associating with all the above statements, we obtain

(3.48)

According to the Poincaré inequality, we obtain that

(3.49)

where is nonnegative constant.

Integrating (3.48) over , we obtain that

(3.50)

We will obtain the uniform estimate of (1.9)–(1.13) in as follows.

Taking the inner product of (1.9) and (1.10), respectively, with , we have

(3.51)

we also obtain

(3.52)

From all the previous statements we obtain that

(3.53)

where . From Poincaré inequality, we get

(3.54)

Let

(3.55)

From (3.50) we have that

(3.56)

Integrating (3.53) over , it follows that

(3.57)

where are nonnegative constants. Taking the inner product of (1.9) and (1.10), respectively, with , we have that

(3.58)

Integrating by parts and employing Sobolev inequality, we get

(3.59)

where is a constant depending on

(3.60)

where is a constant depending on From all the above inequalities we obtain that

(3.61)

Then it follows that

(3.62)

where By means of Poincaré inequality, we obtain that

(3.63)

Let

(3.64)

from (3.49) and (3.56), we conclude that is bounded; assume that . From (3.57), we have

(3.65)

Then through Gronwall inequality, we obtain

(3.66)

Integrating (3.62) on , we have that

(3.67)

where are nonnegative constants. Then from (3.66), we can get In other words, open ball is the attracting set of in .

Theorem 3.3.

Suppose that then the semigroup of the solution operator to (1.9)–(1.13) has a global attractor in

Proof.

To obtain the existence of the global attractor, we will prove that it is a compact operator. Taking inner product of (1.9) and (1.10) with  in , we obtain

(3.68)

(3.69)

By integrating by parts, we obtain that

(3.70)

Similarly, we obtain

(3.71)

where is a constant depending on Integrating by parts frequently, we obtain

(3.72)

From (3.68) and (3.69), we know that

(3.73)

where Employing Young inequality, we obtain that

(3.74)

where . From Poincaré inequality, we obtain that

(3.75)

Let

(3.76)

Through (3.67), we have

(3.77)

We also get

(3.78)

Through (3.49), (3.56), and (3.66), we assume that

(3.79)

Then we have

(3.80)

and all the analysis indicates that

(3.81)

Finally we obtain that

(3.82)

We know that the injection of into is compact, then we can conclude that is equi-continuity. From Ascoli-Arzela's theorem, we know that has the global attractor in .