Journal of Applied Mathematics

Volume 2012 (2012), Article ID 197672, 20 pages

doi:10.1155/2012/197672

Research Article

On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

Yongsheng Mi,1,2 Chunlai Mu,¹ and Weian Tao²

¹College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

²College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling, Chongqing 408100, China

Received 16 May 2012; Revised 24 July 2012; Accepted 30 July 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Yongsheng Mi 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 Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

1. Introduction

In this paper, we consider the Cauchy problem of the following weakly dissipative modified two-component Camassa-Holm system:

where , , and is a nonnegative dissipative parameter.

The Camassa-Holm equation [1] has been recently extended to a two-component integrable system (CH2)

with , which is a model for wave motion on shallow water, where describes the horizontal velocity of the fluid, and is in connection with the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. Moreover, and satisfy the boundary conditions: and as . The system can be identified with the first negative flow of the AKNS hierarchy and possesses the interesting peakon and multikink solutions [2]. Moreover, it is connected with the time-dependent Schrödinger spectral problem [2]. Popowicz [3] observes that the system is related to the bosonic sector of an supersymmetric extension of the classical Camassa-Holm equation. Equation (1.2) with becomes the Camassa-Holm equation, which has global conservative solutions [4] and dissipative solutions [5].

Since the system was derived physically by Constantin and Ivanov [6] in the context of shallow water theory (also by Chen et al. in [2] and Falqui et al. in [7]), many researchers have paid extensive attention to it. In [8], Escher et al. establish the local well-posedness and present the precise blow-up scenarios and several blow-up results of strong solutions to (1.2) on the line. In [6], Constantin and Ivanov investigate the global existence and blow-up phenomena of strong solutions of (1.2) on the line. Later, Guan and Yin [9] obtain a new global existence result for strong solutions to (1.2) and get several blow-up results, which improve the recent results in [6]. Recently, they study the global existence of weak solutions to (1.2) [10]. In [11], Henry studies the infinite propagation speed for (1.2). Gui and Liu [12] establish the local well-posedness for (1.2) in a range of the Besov spaces, they also derive a wave breaking mechanism for strong solutions. Mustafa [13] gives a simple proof of existence for the smooth travelling waves for (1.2). Hu and Yin [14, 15] study the blow-up phenomena and the global existence of (1.2) on the circle.

Recently, the CH2 system was generalized into the following modified two-component Camassa-Holm (MCH2) system:

where , , denotes the velocity field, is taken to be a constant, and is the downward constant acceleration of gravity in applications to shallow water waves. This MCH2 system admits peaked solutions in the velocity and average density, we refer this to [16] for details. There, the authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply singularity in the pointwise density at the point of vertical slope. Some other recent work can be found in [17–25]. We find that the MCH2 system is expressed in terms of an averaged or filtered density in analogy to the relation between momentum and velocity by setting , but it may not be integrable unlike the CH2 system. The important point here is that MCH2 has the following conservation law:

which play a crucial role in the study of (1.3). Noting that for the CH2 system, we cannot obtain the conservation of norm.

In general, it is quite difficult to avoid energy dissipation mechanisms in a real world. Ghidaglia [26] studies the long time behaviour of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system. Recently, Hu and Yin [27] study the blow-up and blow-up rate of solutions to a weakly dissipative periodic rod equation. In [28, 29], Hu considered global existence and blow-up phenomena for a weakly dissipative two-component Camassa-Holm system on the circle and on the line. However, (1.1) on the circle (periodic case) has not been studied yet. The aim of this paper is to study the blow-up phenomena of the strong solutions to (1.1). We find that the behavior of solutions to the weakly dissipative modified two-component periodic Camassa-Holm system (1.1) is similar to that of the modified two-component Camassa- Holm system (1.3), such as the local well-posedness and the blow-up scenario. In addition, we also find that the blow-up rate of (1.1) is not affected by the weakly dissipative term, but the occurrence of blow-up of (1.1) is affected by the dissipative parameter.

This paper is organized as follows: In Section 2, we establish local well-posedness of the Cauchy problem associated with (1.1). In Section 3, we derive precise the blow-up scenario of strong solution and the blow-up rate. In Section 4, we discuss the blow-up phenomena of (1.1).

2. Local Well-Posedness

In this section, by applying Kato’s semigroup theory [30], we can obtain the local well-posedness for the Cauchy problem of (1.1) in , , with with (the circle of unit length).

First, we introduce some notations. All spaces of functions are assumed to be over ; for simplicity, we drop in our notation for function spaces if there is no ambiguity. If is an unbounded operator, we denote by the domain of . denotes the commutator of two linear operators and . denotes the norm of Banach space . We denote the norm and the inner product of ; , by and , respectively.

For convenience, we state here Kato’s theorem in the form suitable for our purpose.

Consider the following abstract quasilinear evolution equation:

Let and be Hilbert spaces such that is continuously and densely embedded in and let be a topological isomorphism. denotes the space of all bounded linear operator from to (and we write , if ).

Theorem 2.1 (see [30]). Assume that

(i) for with

 and uniformly on bounded sets in .(ii), where is bounded, uniformly on bounded sets in . Moreover,

(iii) and extends also to a map from into is bounded on bounded sets in and

 where, , , and depend only on and depends only on . If the above conditions (i), (ii), and (iii) hold, given , there is a maximal depending only on and a unique solution u to (2.1) such that

Moreover, the map is continuous from to .

We now provide the framework in which we will reformulate system (1.1). With , , and , we can rewrite (1.1) as follows:

Note that if is the kernel of , where stands for the integer part of , then for all , and . Here we denote by the convolution. Using this identity, we can rewrite (2.6) as follows:

or we can write it in the following equivalent form:

Theorem 2.2. Given , then there exist a maximal and a unique solution to (1.1) or (2.7) such that

Moreover, the solution depends continuously on the initial data, that is, the mapping is continuous and the maximal time of existence can be chosen to be independent of .

The remainder of this section is devoted to the proof of Theorem 2.2.

Let , and

Set , , and . Obviously, is an isomorphism of onto . In order to prove Theorem 2.2 by applying Theorem 2.1, we only need to verify and which satisfy the conditions (i)–(iii).

We break the argument into several lemmas.

Lemma 2.3. The operator , with , , belongs to .

Lemma 2.4. The operator , with , , belongs to .

Lemma 2.5. , with , . The operator . Moreover,

Lemma 2.6. The operator with , . Then and

for and .

The proof of the above five lemmas can be done similarly as in [8], therefore we omit it here.

Hence, according to Kato’s theorem (Theorem 2.1), in order to prove Theorem 2.2, we only need to verify condition (iii), that is, we need to prove the following lemma.

Lemma 2.7. Let , and

Then is bounded on bounded sets in and satisfies

(a), (b).

Proof. Let , . Since is a Banach algebra, it follows that

This proves (a). Taking in the above inequality, we obtain that is bounded on bounded set in .

Next, we prove (b). Note that is a Banach algebra. Then, we have

This proves (b) and completes the proof of the Lemma 2.7.

Proof of Theorem 2.2. Combining Theorem 2.1 and Lemmas 2.3–2.7, we can get the statement of Theorem 2.2.

3. The Precise Blow-Up Scenario and Blow-Up Rate

In this section, we present the precise blow-up scenario and the blow-up rate for strong solutions to (2.7).

Lemma 3.1. Let , , and let be the maximal existence time of the solution to (2.7) with the initial data . Then for all , we have

Proof. Denote

In view of the identity , we can obtain from (2.7),

Therefore, an integration by parts yields

Thus, the statement of the conservation law follows.

Lemma 3.2 (see [31]). For every , we have

where the constant is sharp.

(ii) For every , we have

with the best possible constant lying within the range . Moreover, the best constant is .

So, if , then by Lemmas 3.1 and 3.2, we have

for all .

Theorem 3.3. Let , be given and assume that is the maximal existence time of the corresponding solution to (2.7) with initial data , if there exists such that

then the norm of does not blow-up on .

The proof of the theorem is similar to the proof of Theorem  3.1 in [20], we omit it here.

Consider the following differential equation equation:

where denotes the first component of the solution to (2.7). Applying classical results in the theory of ordinary differential equations, one can obtain the following result on which is crucial in the proof of blow-up scenario.

Lemma 3.4 (see [8]). Let , , and be the maximal existence time of the corresponding solution to (3.7). Then (3.7) has a unique solution . Moreover, the map is an increasing diffeomorphism of with

The following result is proved only with regard to , since we can obtain the same conclusion for the general case by using Theorem 2.1 and a simple density argument.

We now present a precise blow-up scenario for strong solutions to (2.6).

Theorem 3.5. Let , , and let be the maximal existence of the corresponding solution to (2.7). Then the solution blows up in finite time if and only if

Proof. Multiplying the first equation in (2.6) by and integrating by parts, we obtain

Repeating the same procedure to the second equation in (2.6) we get

A combination of (3.7) and (3.9) yields

Differentiating the first equation in (2.6) with respect to , multiplying by , then integrating over , we obtain

Similarly,

A combination of (3.12)–(3.16) yields

Assume that there exists and such that and for all , then it follows from Lemma 2.4 that

Therefore,

The above discussion shows that if there exist and such that and for all , then there exist two positive constants and such that the following estimate holds

This inequality, Sobolev’s embedding theorem and Theorem 3.3 guarantee that the solution does not blow-up in finite time.

On the other hand, we see that if

then by Sobolev’s embedding theorem, the solution will blow-up in finite time. This completes the proof of the theorem.

Lemma 3.6 (see [32]). Let and . Then for every , there exists at least one point with

The function is absolutely continuous on with

Theorem 3.7. Let , be the corresponding solution to (2.7) with initial data and satisfies , for all , be the maximal existence time of the solution. Then we have

Proof. Applying Theorems 2.1 and a simple density argument, we only need to show that the above theorem holds for some . Here, we assume to prove the above theorem.

Define now

and let be a point where this minimum is attained. Clearly, since . Differentiating the first equation of (2.7) with respect to , in view of , we have

Evaluating (3.26) at and using Lemma 3.6, we obtain

where . By Lemma 3.1 and Young’s inequality, we have for all that

This relation together with (3.7) and implies that there is a constant such that

where depends only on and . It follows that

Choose . Since by Theorem 3.5, there is some with and . Let us first prove that

Since is locally Lipschitz, there is some such that

Note that is locally Lipschitz (it belongs to by Lemma 3.6) and therefore absolutely continuous. Integrating the previous relation on yields that

It follows from the above inequality that

The obtained contradiction completes the proof of the relation (3.31). By (3.30)-(3.31), we infer

For , integrating (3.35) on to get

Since on , it follows that

By the arbitrariness of , the statement of the theorem follows.

4. Blow-Up

In this section, we discuss the blow-up phenomena of (2.7) and prove that there exist strong solutions to (2.7) which do not exist globally in time.

Theorem 4.1. Let , and be the maximal existence time of the solution to (2.7) with the initial data . If there exists some such that

then the existence time is finite and the slope of tends to negative infinity as goes to while remains uniformly bounded on .

Proof. As mentioned earlier, here we only need to show that the above theorem holds for . Differentiating the first equation of (2.7) with respect to , in view of , we have

Define now

and let be a point where this minimum is attained. It follows that

Clearly since . Evaluating (4.2) at , we obtain

here, we used Lemma 3.2 and

Inequality (4.5) and Lemma 3.4 imply

that is,

Take

It then follows that

Note that if , then , for all . Therefore, we can solve the above inequality to obtain

Due to , then there exists , and , such that . This completes the proof of the theorem.

Theorem 4.2. Let , be the corresponding solution to (2.7) with initial data and satisfies , for all , be the maximal existence time of the solution. If satisfies the following condition:

where . Then the corresponding solution to (2.7) blows up in finite time.

Proof. In view of (4.2), we obtain

Note that

Thus,

Using the following inequality:

and letting

we obtain

Taking

we get

Note that if

then

for all . From the above inequality, we obtain

Since then there exists

such that . On the other hand,

Applying Theorem 3.5, the solution blows up in finite time.

Acknowledgments

This work was partially supported by NSF of China (11071266), Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, and the Educational Science Foundation of Chongqing China (KJ121302).

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