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Panpan Zhai, Zhengguang Guo, Weiming Wang, "Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation", Abstract and Applied Analysis, vol. 2013, Article ID 750315, 13 pages, 2013. https://doi.org/10.1155/2013/750315
Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation
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.
In 2001, Dullin et al.  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 , which is also shown in . Another remarkable property is that it is integrable and the solitary waves are nonlinearly stable, and when , there exists a smooth soliton solution .
Instead, taking in (2), it turns out to be the Camassa-Holm equation: which was derived by Camassa and Holm in  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 , global solutions were also discussed. In , global existence of weak solutions was proved, but uniqueness was obtained only under a priori assumption. Furthermore, the authors in  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  as follows, where and are two conserved quantities.
In , 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  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.
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.
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 .
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 ). For all , the following inequality holds: with Moreover, is the optimal constant obtained by the function
Lemma 5 (see ). 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 ). 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 ). 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
with determined in (18). Then we have the following inequality for each and :
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:
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  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 .
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