Abstract and Applied Analysis

Volume 2013 (2013), Article ID 750315, 13 pages

http://dx.doi.org/10.1155/2013/750315

Research Article

Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation

Panpan Zhai, Zhengguang Guo, and Weiming Wang

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China

Received 11 April 2013; Accepted 22 May 2013

Academic Editor: Carlo Bianca

Copyright © 2013 Panpan Zhai 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.

This paper is concerned with blow-up phenomena and persistence properties for an integrable two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data which guarantee blow-up phenomena of solutions in finite time for both periodic and nonperiodic cases, respectively. Furthermore, the persistence properties of solutions to the system are investigated.

1. Introduction

In 2001, Dullin et al. [1] derived a new equation by using the method of asymptotic analysis and a near-identity normal form transformation in water wave theory to describe the unidirectional propagation of surface waves in a shallow water regime, it reads as follows:

Since (1) was derived by Dullin et al., we call it DGH equation. Let ; then we can rewrite (1) as the following form:

where and are squares of length scales, the constant is the critical shallow water speed for undisturbed water at rest at spatial infinity, is the mean fluid depth, and is the gravitational constant. This equation is connected with two separately integrable soliton equations for shallow water waves.

When , (2) becomes the KdV equation as follows:

Bourgain proved that solutions to the KdV equation are global as long as the initial data is square integrable [2], which is also shown in [3]. Another remarkable property is that it is integrable and the solitary waves are nonlinearly stable, and when , there exists a smooth soliton solution [4].

Instead, taking in (2), it turns out to be the Camassa-Holm equation:

which was derived by Camassa and Holm in [5] by approximating directly the Hamiltonian for Euler’s equations in the shallow water regime in 1993. Many researches have been carried out on the Camassa-Holm equation in recent years. Local well-posedness has been proven by several authors; see [6–9]. Blow-up phenomena have been investigated in [10–15]. Besides, in [11], global solutions were also discussed. In [16], global existence of weak solutions was proved, but uniqueness was obtained only under a priori assumption. Furthermore, the authors in [17] studied the persistence properties of solution to the Camassa-Holm equation. In [18–20], the soliton solution was studied.

It is very interesting that (2) still preserves the bi-Hamiltonian structure and complete integrability. Indeed, (2) can be rewritten in two compatible Hamiltonian forms in terms of [1] as follows,

where

and are two conserved quantities.

In [21], Tian et al. studied the well-posedness of the Cauchy problem and the scattering problem of the DGH equation. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation was investigated, and the convergence of the solutions to DGH equation as was studied. Besides, data of the scattering problem for the equation could be explicitly expressed. The blow-up phenomenon of solutions to the DGH equation was the subject of [21–24].

In this paper, we are interested in the following Cauchy problem for an integrable two-component DGH equation. It reads as

The variable describes the horizontal velocity of the fluid, and denotes the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. This model can be derived by Constantin and Ivanov’s approach [25] from shallow water theory, which includes the two-component Camassa-Holm system [26–32] as its special case with and in (7). From a geometric point of view, (7) is the model for geodesic motion on the semidirect product Lie group of diffeomorphisms acting on densities, with respect to the -norm of velocity and the -norm on the density. Mathematically, (7) admits not only breaking-wave solutions but also global in time solutions. In view of the context of hydrodynamics, we have the assumptions , as , at any instant . For convenience of later discussion, letting , , and , we have as . Then it follows that

Define ; then the operator can be expressed by

where is the Green function for and denotes the convolution. It is easy to get . For the periodic case, is the Green function for in the unit circle. While, for the nonperiodic case, the Green function for in is . Using this identity, we can rewrite (1) as the following nonlocal form:

We note that less deep results exist yet for this model except for the recent ones [33, 34]. Our motivation here is to explore some new blow-up conditions especially for the initial data were added on some different quantities. This paper is organized as follows. In Section 2, we recall some preliminary results on local well-posedness and blow-up scenario and present some useful conversation laws. In Section 3, we study the blow-up phenomenon of strong solutions in both periodic and nonperiodic cases. In Section 4, persistence properties of solutions to the system are studied.

2. Preliminaries

We can use Kato’s theory [35] to establish the following local well-posedness theorem for (10); its proof is referred to in the discussions of [33, 34], so we omit it here.

Theorem 1. Assume an initial data . Then there exists a maximal and a unique solution

of system (10). Moreover, the solution depends continuously on the initial value , and the maximal time of existence is independent of .

Next, we will give some useful conserved quantities, which are important to discuss the blow-up criteria.

Proposition 2. Suppose ; then the solution of the system (10) guaranteed by Theorem 1 satisfies

Proof. Consider

From the first equation of (10), we have

Similarly, from the second equation of (10), we can get

So we can obtain

This completes the proof.

We also note that the quantities and are also conserved for , . These conservation laws will play important roles in considering the behavior of solutions. In what follows, we denote by .

Moreover, using the techniques in [36], one can get the following criterion for finite time wave breaking to (10).

Theorem 3 (see [34]). Let with , and let be the maximal time of existence of the solution to the model (10) with initial data . Then the corresponding solution blows up in finite time if and only if

We will then need to introduce the standard particle trajectory method for later use. Now consider the following initial value problem:

where is the first component of the solution to system (10) with initial data , and is the maximal time of existence. By direct calculation, we have

Then,

which means that is a diffeomorphism of the line for every . Consequently, the -norm of any function is preserved under the family of the diffeomorphism ; that is,

Similarly,

3. Blow-Up Criteria

In this section, we establish sufficient conditions to guarantee the formation of singularities for the corresponding solutions to (10) in periodic and nonperiodic cases. These sufficient conditions are different from each other.

Hereinafter, we investigate the blow-up phenomenon with periodic setting; that is, . First, we introduce the following lemmas for later use.

Lemma 4 (see [23]). For all , the following inequality holds:

with

Moreover, is the optimal constant obtained by the function

Lemma 5 (see [23]). For all , the following inequality holds:

where

Moreover, is the minimum value, so in this sense, is the optimal constant which is obtained by the associated Green function

Lemma 6 (see [37]). Assume that a differentiable function satisfies

with the constants . If the initial datum , then the solution to the previous equation goes to in finite time.

Lemma 7 (see [24]). Assume , . If , then

Now we give some sufficient conditions on the initial data to guarantee finite time wave breaking for the periodic case.

Theorem 8. Assume that the initial data , satisfies

where is defined as in Lemma 5, and constant is determined later. Moreover, for convenience, we suppose that . Then the corresponding strong solution to (10) blows up in finite time.

Proof. By assumption and Theorem 1, there exists a solution to system (10) corresponding to the initial value . Suppose that the statement is not true; that is, the corresponding solution does not blow up in finite time, and then due to Theorem 3, there exists , such that

Let

with determined in (18). Then we have the following inequality for each and :

Since

we have, by solving it,

So we can get

then we have

Therefore, the following inequality holds:

With the notation of , differentiating both sides of the first equation of (10) with respect to variable , we obtain

We note that ; then (40) can be rewritten as follows:

Multiplying on both sides and integrating by parts with respect to , one obtains

where we have used the following identity:

Since

it follows that

We can also get

So we obtain

Due to the theory of Riccati type differential equations in Lemma 6, if (31) holds, then we have

as tends to some .

On the other hand, the majorization

implies that

It contradicts the assumption

By Theorem 3, we know that the solution must blow up in finite time.

Next, we find if the initial velocity has zero mean and initial energy is sufficiently large, wave breaking can occur.

Theorem 9. Suppose , and satisfies . If the initial energy satisfies for some positive constant , and then holds for nontrival . Besides, assume that just for convenience. Then the solution to the system (10) with initial value blows up in finite time.

Proof. From the first equation of (10), we know that

On the other hand, we know that in view of the hypothesis, and the following inequality holds:

Using Lemma 7 and (53), we obtain

Similar to Theorem 8, assume that the solution of system (10) does not blow up; that is, there is , such that

then

So we can get

If , then . Since for some constant , due to Lemma 7, there is some such that

Moreover, Lemma 7 also implies that

Hence,

Note that is bounded. In view of Hölder’s inequality, there exists

For simplicity of notations, we denote by and the following quantities:

respectively. Therefore we have

First, we can easily get , and it is not difficult to find that there exists a time such that . Then for all , we have

Solving this inequality yields

which goes to as tends to ; that is, there exists a time such that

Since

it shows that

Then it contradicts the assumption

By Theorem 3, we know that the solution must blow up in finite time. This finishes the proof.

From now on, we give some blow-up criteria for the nonperiodic case.

Theorem 10. Assume that the initial data , satisfy

where is a constant determined later. And assume that just for convenience. Then is finite and the slope of tends to negative infinity as goes to while is uniformly bounded on .

Proof. Let be the solution to system (10) with the initial data and let be the maximal existence time of solution. Defining

one can get that is almost everywhere differentiable on with

Note that , and with (40), we have

and with the relation , we can get

Since and are both bounded, then there exists a , so that , and thus we have

Note that if , then for all . So there exists a such that

This completes the proof of the theorem.

The authors in [25] claimed that the corresponding solution to the two-component Camassa-Holm equation exists globally in time provided that some initial data satisfy smallness conditions. However, we find surprisingly the smallness of initial energy can also lead to breaking wave solutions. This is stated as follows.

Theorem 11. Let with and let be the maximal time of existence of the solution to (10) with initial data . Assume there is some such that , and there holds for suitable and . Moreover,

Then the corresponding solution to (10) blows up in finite time in the following sense, and there exists a with

such that

Proof. As in Theorem 8, we define

so that

We recall that defined by (18) is a diffeomorphism of the line for any , so there exists an such that

Differentiating the first equation of (10), we can get the expression of as follows:

Besides, along the trajectory of , we have

Then choose and , which implies that

Therefore we can obtain

where we use the fact

We define

For any , then

We also have

Thus, we can get

Therefore, when holds, we always have . In this case, we can obtain

so is decreasing strictly in such interval. Assume that the solution of (10) does not blow up; that is, it exists globally in time; that is, . Then integrating (92) over with yields

We can also find that , which leads to

Integrating both sides, we have

where we use the result obtained previously. Solve the inequality

That implies that , which leads to a contradiction as . This completes the proof.

4. Persistence Properties

Attention now is turned to determining the persistence properties of solutions to the two-component DGH equation. We show that certain decay properties of the initial data persist in corresponding solution in later time as long as it exists. Precisely, we prove that the corresponding solution and its first order spatial derivatives preserve the exponential decay as their initial values do. This is a very important investigation since we can obtain the detailed asymptotic behavior of solution from the initial values. The main idea comes from the recent works of Zhou et al. [17, 38]. Now, we state our result as follows.

Theorem 12. Assume that for some and , is a strong solution of the initial value problem to the system (10) and just for convenience. If , satisfy

for some . Then

uniformly in the time interval .

Notation. Consider

Proof. First, we figure out the estimates on , , , and . Then we use the weight function to obtain the desired result.

Step 1. Estimate for .

Multiplying the first equation of (10) by with and then integrating both sides with respect to variable, we can get

The first term of the previous identity is

and the estimate of the second term is

In view of Hölder’s inequality, we can obtain the following estimate for the third and the last terms in (100):

Putting all the previous inequalities into (100) yields

Thus, according to the Sobolev embedding theorem, there exists a constant

such that applying Gronwall’s inequality gives us

For any , we know that

Thus, taking the limits in (106), we get

Similarly, for the second equation of (10), we have

and using the same method, one can get

Using the Gronwall’s inequality, one gets

Taking the limits, one gets

Step 2. Estimate for , .

We will establish an estimate on using the same method. Differentiating the first equation (10) with respect to variable produces the following equation:

Then, multiplying (113) by with , integrating the result in the variable, and considering the second term in the previous identity with integration by parts, one can get

so we have

Similarly, we can get the following inequality:

Using Gronwall’s inequality, we can similarly get

Taking the limits, one gets

Differentiating the second equation of (10), one has

and using the same method, we can obtain

In a similar way, we should deal with the second term as follows:

thus we can get the estimates of as follows:

In what follows, we use the weight function to get the desired result. We shall now introduce the weight function with , which is independent of as follows:

Note that . Multiplying the first equation of (10) and (106) by , we have

In order to get the estimate of , we need to eliminate the second derivatives as follows:

where we use the fact . Hence, as in the weightless case, we get the following inequality in view of the estimates of and :

Similarly, we get estimates for and as follows:

On the other hand, a simple calculation shows that there exists , depending only on such that for any ,

Therefore for any appropriate function , one can get

and using the same method,

Besides,

In a similar way, we have

Then we can get the following estimate:

Using Gronwall’s inequality of integral form, one can get

Taking the limit as goes to infinity, we can obtain

This completes the proof.

Acknowledgments

This work was partially supported by Zhejiang Provincial Natural Science Foundation of China under LQ12A01009 and LY12A01014 and NSFC under Grant no. 11226172.

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