Abstract and Applied Analysis

Volume 2014 (2014), Article ID 520218, 4 pages

http://dx.doi.org/10.1155/2014/520218

Research Article

Wave Breaking for the Modified Two-Component Camassa-Holm System

Wujun Lv and Weiyi Zhu

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 8 April 2014; Accepted 21 April 2014; Published 5 May 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Wujun Lv and Weiyi Zhu. 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.

Some new sufficient conditions to guarantee wave breaking for the modified two-component Camassa-Holm system are established.

1. Introduction

This paper concerns the following modified two-component Camassa-Holm system (MCH2, for simplicity):

where , expresses the velocity field, and is the downward constant acceleration of gravity in applications to shallow water waves. In this paper, we let .

Let ; then the operator can be denoted by its associated Green’s function as

Let and . So system (1) is equivalent to the following one:

The MCH2 system admits peaked solutions in the velocity and average density and we refer it to reference [1]. The local posedness, precise blow-up scenarios, and the existence of strong solutions which blow up in finite time can be found in [2–5]. Note that the MCH2 system is a modified version of the 2-component Camassa-Holm (CH2, for simplicity) system to allow a dependence on the average density (or depth, in the shallow water interpretation) as well as the pointwise density . Meanwhile, the MCH2 may not be integrable unlike the CH2 system. The characteristic is that it will amount to strengthening the norm for from to in the potential energy term [5]. Also, the MCH2 admits the following conserved quantity:

This paper mainly studies wave breaking phenomenon, and we aim at improving previous results which were proved in [3, 6]. Our method is partially motivated by [7]. The remaining of this paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, we establish a new blow-up criterion for the MCH2. Finally, we establish a similar criterion for the CH2 system in Section 4.

2. Preliminaries

In this section, we recall some results without the proofs for conciseness. The first one is concerning local well-posedness and blow-up scenario.

Lemma 1 (see [2]). Given to system (3), , there exists a maximal , and a unique solution to system (3). Then the corresponding solutions blow up in finite time if and only if

We also need to introduce the standard particle trajectory [8]. Let be the particle line evolved by the solution; that is, it satisfies

Taking the derivative with respect to , we get

Hence

Thus, the map is a diffeomorphism of the real line.

3. Blowup for the MCH2 System

In this section, we establish a new sufficient condition to guarantee blowup for system (3), which is an improvement of that in [3].

Theorem 2. Suppose to system (3), and . And the initial data satisfies the following two conditions:

for some point . Then the solution to our system (3) with initial value blows up in finite time.

Remark 3. In [17] conditions and are needed to guarantee blowup, which implies condition (10). In addition, is required. So obviously Theorem 2 is an improvement of that in [3]. On the other hand, our condition is a local version and is easy to check. For nonlocal conditions, we refer to [5, 9].

Now we give a proof for Theorem 2.

Proof. Let us first consider the case . As in [10], we will look for . Applying to differentiate (3) with respect to yields

Let . Recalling that , we show that and are continuous on and is Lipschitz, uniformly with respect to in any compact time interval in .

We get

where we used , , and .

As

we get

it is easy to get in (8), so .

Consider ; we can refer to [3].

The obvious factorization ; this leads us to study the functions of the form:

Computing the derivatives with respect to using the definition of the flow map (6) gives

In fact, the next lemma will be used.

Lemma 4. Consider

Proof. Consider

So we get

The same computations also obtain that

We have

taking the linear combination in the two last inequalities implies estimate (17).

Similarly,

It is convenient to establish the following fundamental proposition.

Proposition 5. as in Theorem 2. Set

Then, for all , the function is monotonically increasing and is monotonically decreasing.

It is easy to factorize

from inequality (12) we get

Now let be such that . Proposition 5 yields, for all ,

where we used , then we get and .

Assume, by contradiction, ; set ; thus we get

Set ; then ; we can find such that . For , then ; we obtain

This implies that, for ,

From above, must blow up in finite time, and , so the condition of the blowup scenario (5) is fulfilled.

4. Blowup for the CH2 System

In this section, we consider the following two-component Camassa-Holm system:

The CH2 system appears initially in [11]. Wave breaking mechanism was discussed in [3, 12–14]. The existence of global solutions was analyzed in [6, 15, 16]. This system also has the following conservation laws [17]:

In [6], a blow-up condition is established as , and ; here . Similar to Theorem 2, we can do the following improvement.

Theorem 6. Suppose to system (30), , and ; furthermore

for some point . Then the solution to our system (30) with initial value blows up in finite time.

The proof is similar to Theorem 2 and we omit it.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank anonymous reviewers for their valuable comments and careful reading.

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