Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 691731, 15 pages

http://dx.doi.org/10.1155/2013/691731

Research Article

Initial Boundary Value Problem of the General Three-Component Camassa-Holm Shallow Water System on an Interval

Lixin Tian, Qingwen Yuan, and Lizhen Wang

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 19 July 2012; Accepted 6 November 2012

Academic Editor: Feliz Minhós

Copyright © 2013 Lixin Tian et al. 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 study the initial boundary value problem of the general three-component Camassa-Holm shallow water system on an interval subject to inhomogeneous boundary conditions. First we prove a local in time existence theorem and present a weak-strong uniqueness result. Then, we establish a asymptotic stabilization of this system by a boundary feedback. Finally, we obtain a result of blow-up solution with certain initial data and boundary profiles.

1. Introduction

It is well known that the Camassa-Holm equation has attracted much attention in the past decade. It is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed, as well as water waves moving over an underlying shear flow. It was first introduced by Fo kas and Fuchssteiner as a bi-Hamiltonian model. Cauchy problem and initial boundary value problem for Camassa-Holm equation have been studied extensively in a number of papers (see [1–15] and the references within).

Fu and Qu in [16] proposed a coupled Camassa-Holm equation,

with , which has peakon solitons in the form of a superposition of multipeakons and may as well be integrable. They investigated the local well-posedness and blow-up solutions of system (1) 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 system (1) by energy estimates. Recently the initial boundary value problem for the system (1) has been established in [17]; moreover, the local well-posedness and blow-up phenomena for the coupled Camassa-Holm equation were also established in [16, 18–32]. In [33], Tian and Xu obtained the compact and bounded absorbing set and the existence of the global attractor for viscous system (1) with the periodic boundary condition in by uniform prior estimate.

Recently, Fu and Qu in [34] introduced a general three-component Camassa-Holm equation as follows:

where , , and . Equation (2) also has peakon solitons in the form of a superposition of multipeakons. Such system also conserves the -norm conservation law. Moreover, the well-posedness and blow-up phenomena for system (2) with peakons have been established in [35]. To our knowledge, the initial boundary value problem of (2) has not been studied yet. The first aim of this paper is to consider an initial boundary value problem of the following

where .

Then, we will consider the asymptotic stabilization of (3) by means of a stationary feedback law acting on the inhomogeneous boundary condition. Following the step in [11], we convert the initial boundary value problem of (3) on the interval into an ODE system and two PDE systems. Then, we can consider the system (3) easily. Consequently, we obtain a local in time existence theorem, a weak-strong uniqueness result, asymptotic stabilization result on the interval, and a result of blow-up solution, respectively.

Our paper is organized as follows. In Section 2, we will consider an initial boundary value problem and the uniqueness of the solution to (3). By using the feedback law enjoyed by (3), the asymptotic stabilization on an interval is considered in Section 3. Finally, in Section 4, a result of blow-up solution with certain initial data and boundary profiles will be established.

First, we begin with a general remark that will be used many times later.

Remark 1. Let be a positive number and . Changing in, in,   in, and in , and it will be more convenient for us to analysis the system, if we define the following sets

Let , then the operator can be expressed as

where . Now, let ,  , where is an auxiliary function which lifts the boundary values , , , , and defined by

where .

Setting , , and , we can further rewrite the system (3) as

where functions , and in are the boundary values and , and in are the initial datum.

Lemma 2. We have and , . Moreover, we also have the bounds

Proof. can be expressed, respectively, as

Estimates (9) and (10) can be easily obtained from the above expressions.

2. Initial Boundary Value Problem

First, we define what we mean by a weak solution to (8). Our test functions will be in the space:

Definition 3. When , the function is the weak solution to (8) if for all :

It is obvious that ; therefore, a weak solution to system (8) is also a solution to (8) in the distribution sense. And it is clear that a regular weak solution is a classical solution.

Definition 4. For , we consider the maximal solution satisfying

We consider that is the flow of . For , is defined on a set . Here is basically the entrance time in of the characteristic curve going through .

Remark 5. Obviously implies that .

In the following, we consider a partition of , which allows us to distinguish the different influence zones in .

Definition 6. Let such that and ,

Those points of the set are tangent to the boundary, which are precisely the singular points of and . It's obviously that the sets , and constitute a partition of . Furthermore, if , then , and if , then .

Definition 7. Here, we consider the case of data ; . We define the functions , and in the following way.

When ,  ,  , and  , when ,

when ,

when ,

Lemma 8. Since and satisfies (8), we immediately get that is the weak solution of (8) and . However, the functions , and satisfy the following estimates:

Definition 9. We can define operator and a domain for the system (8) by:  ,

where

Obviously is convex and is compact with respect to the norm . We will endow with the norm . There exist positive numbers , and such that maps into itself.

Theorem 10. There exists , and is a weak solution of (8) with and . Moreover, any such solution is in fact in . Furthermore, the existence time of a maximal solution , with

Proof. For , we consider , and in such that the sets and have only a finite number of connected components.

Let , where and

Now, if (see (21)), we have

For all , we have

We also define that . If , then from Lemmas 2 and 8, we derive that

Finally, to obtain , it is sufficient to show that

if we have chosen and ; it is easy to choose to satisfy the second inequality. For the above two inequalities, we just choose and sufficiently large and then close to 0. More precisely:

Maximizing the bound of , we can get minimum existence. Then, we get the result announced, where , .

Lemma 11. The operator is continuous with respect to .

Proof. The proof is omitted here; one can see a similar proof in [8, Proposition 2.4].

Now, we can apply Shauder's fixed point theorem to the operator , and we get the result that there exist fixed points such that , , and , so we know that there exists a wake solution of (9).

2.1. Uniqueness

We will prove the weak-strong uniqueness of weak solution of (8) in the following.

Theorem 12. Let be the weak solution of (7)-(8), then it is unique in .

Proof. Define ,  ,   and ,  ,  , then we have

where , and is the unique weak solution of

Let

with ,  , and  , where .

For , we have , and .

Then, we get the uniqueness result.

For , we have

For , we have

For , we have

Now since , , and bounded, we see that for some , ,  ,

since , and are bounded, we get that for some , , ,

We can obtain that

As a result, we get the result of the uniqueness by Gronwall's inequality when , , . Then, we complete the proof of the uniqueness results.

3. Asymptotic Stabilization

3.1. Preliminary Results

The equilibrium state that we want to stabilize is , , and . A natural idea is using Lyapunov indirection method to investigate whether the linearized system around the equilibrium state is stabilizable or not. Its stabilization would provide a local stabilization result on the nonlinear system. However, there is a difficulty in the stabilization problem. We have to prescribe , and we just need to make a continuous transition at , and that asymptotically converge in time. For convenience, the system (6)–(8) can rewrite in the following

where

Our feedback law for (3) reads

where , , , , and ,, a symmetric matrix, is the unique matrix solution to the matrix function:

for some symmetric positive-definite matrices and . Indeed, let be the Lyapunov candidate, and that asymptotically converges in time is equivalent to that the time derivative of the , is strictly negative. A fixed-point strategy will be used again to prove the existence of a solution to the closed-loop system, we begin by defining the domain of the operator.

Definition 13. Let be the space of satisfying

(1),  ,  ,(2),  ,(3) is nonincreasing, and .

Lemma 14. The domain is nonempty, convex, bounded, and closed with respect to the uniform topology.

The proof is elementary and one notices that , so is nonempty.

Now for , we define and as the solutions of

One has the following exact formulas:

Therefore, we have the following inequalities:

Let , where , and in turn those provide

Now, if is the flow of , is , and since , is nondecreasing. This allows us to define the entrance time and then the operator as follows. Let .

Now, for , with the following:

(1)if ,

(2)if ,

(3).

Lemma 15. (1) The operator maps to .

(2) The family is uniformly bounded and equicontinuous.

(3) is continuous w.r.t. the uniform topology.

The proof is very similar to [10], except for the state here is a three-component vector and the proof is omitted.

Now, we can apply Schauder’s fixed point theorem to and get fixed point of .

3.2. Stabilization and Global Existence

Theorem 16. For any , there exists a weak solution of (39) satisfying

Furthermore, any maximal solution of (39) and (41) is global, and if we let

then we have

To finish the proof of Theorem (39), we have to prove the global existence of a maximal solution and the estimate (51).

Proof . First, we rewrite (46) as the following:

where .

For is the solution of the transport (39) and it satisfies

Combining those facts, we get for the following:

We have also imposed and thanks to the existence theorem that a maximal solution of the closed loop system is global. To get a more precise statement, we consider all the between time and , and we obtain.

For ,

We define

and we set , where

Then, we have

as long as the quantity is not equal to zero, it strictly decreases, so if , for small enough , and we have the following.

If we define       , we get that

This provides , the inequality (which was clear when

4. Blow-Up Phenomena

In this section, we present a result with the initial data and boundary profiles under a special condition that ensure strong solutions to following system blow-up in finite time as follows:

where ,  ,  . This imply that

From Definition 3, we can also define

where , and denote the solution to (62). Applying classical results in the theory of ordinary differential equations, one can obtain a result on which is crucial in studying blow-up phenomena.

From (62), we obtain that

Lemma 17. Let , then (64) has a unique solution . Moreover, the map is an increasing diffeomorphism with

Proof. The proof is omitted here, one can see a similar proof in [12].

Now, we have the following lemma that the potential ,  ,   with compactly supported initial datum ,  ,   also has compact support as long as it exists.

Lemma 18. Assume that with , is the corresponding solution, if has compact support, then also has compact support, moreover, we can obtain that

Proof. Since

Similarly,

So it follows that

We obtain

From Lemma 17, we have

If there exist four constants , and such that , , , and , we can get that

Similarly,

Theorem 19. Let , and , and be the solution in (62) with time . Then, is finite if and only if

or

or

Proof. Let , and , and be the solution (62) with time . We know that , and . By the definition of , and , we have

Hence, .

Multiplying the first equation by , the second one by , and the third one by , after integration by parts and adding up the results, we see that

So, we have

By Gronwall's inequality, we get

The above inequality, Soblev’s embedding theorem, ensure that the solution cannot blow up in finite time.

On the other hand, if

or

or

then the solution will blow up in finite time.

Acknowledgments

The paper is supported by the National Nature Science Foundation of China (nos. 71073072 and 11171135) and the National Nature Science Foundation of Jiangsu (no. BK 2010329) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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