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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 548310, 8 pages
Blowup Phenomena for a Modified Dullin-Gottwald-Holm Shallow Water System
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
Received 6 April 2013; Accepted 11 June 2013
Academic Editor: Sergey Piskarev
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.
We discuss blowup phenomena for a modified two-component Dullin-Gottwald-Holm shallow water system. In this paper, some new blowup criteria of strong solutions involving the density and suitable integral form of the momentum are established.
We consider the following two-component DGH type system: where , denotes the velocity field, is the downward constant acceleration of gravity in applications to shallow water waves, and , where is taken to be a constant. It is obvious that if , then (1) reduces to the well-known Dullin-Gottwald-Holm equation  (DGH equation for short). There are some contributions to DGH equation concerning the well-posedness, scattering problem, blowup phenomenon, and so forth; see, for example, [2####^~^~^~^~^~^####x2013;5] and references therein. We find that (1) is expressed in terms of an averaged filtered density component in analogy to the relation between momentum and velocity by setting and the velocity component . The idea is actually from the recent work . Our modification breaks the structure of DGH2 system derived by following Ivanov####^~^~^~^~^~^####x2019;s approach  by the authors in . The motivation of current research is stated as follows. From geometric point of view, (1) 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 filtered density. From a physical point of view, (1) admits wave breaking phenomena in finite time which attracts researchers####^~^~^~^~^~^####x2019; interest. We also find that the -norm of is conserved with respect to time variable. This makes further different discussions on the singularities, unlike those for the DGH2 system or two-component Camassa-Holm system, possible. In the previous works [9####^~^~^~^~^~^####x2013;11] on the two-component Camassa-Holm equation and its modified version, blowup conditions were established in view of the negativity of initial velocity slope at some point; basically, the initial integral form of momentum is never involved. That is why we consider this kind of blowup condition in this paper. Precisely, we show the solutions blowup in finite time provided that the initial density and momentum satisfy certain sign conditions. To our knowledge, less results exist yet for the formation of singularities of (1) although the approaches we applied here are standard. The methods in previous works cannot be moved to this model parallelly. For convenience, let and ; then the operator can be expressed by its associated Green####^~^~^~^~^~^####x2019;s function with Using this identity, system (1) takes an equivalent form of a quasilinear evolution equation of hyperbolic type as follows:
The current paper is based on some results on the Camassa-Holm equation [12####^~^~^~^~^~^####x2013;19] and its two-component generalizations [20####^~^~^~^~^~^####x2013;27]. We investigate further formation of singularities of solutions to (3) with the case of and , just for simplicity mathematically. This paper is organized as follows. In Section 2, we recall some preliminary results on the well-posedness and blowup scenario. In Section 3, the detailed blowup conditions are presented.
In this section, for completeness, we recall some elementary results and skip their proofs since they are not the main concern of this work. For convenience, in what follows, we let and .
Theorem 1. Assume an initial data . Then there exists a maximal and a unique solution of system (3). Moreover, the solution depends continuously on the initial value , and the maximal time of existence is independent of .
Theorem 2. Let with , and let be the maximal time of existence of the solution to (3) with initial data . Then the corresponding solution blowsup in finite time if and only if
Lemma 4 (see ). Suppose that is twice continuously differential satisfying Then blowsup in finite time. Moreover the blowup time can be estimated in terms of the initial datum as
We also need to introduce the standard particle trajectory method for later use. Consider now the following two initial value problems: where is the first component of the solution to system (3) with initial data , and is the maximal time of existence. By direct computation, 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 diffeomorphisms ; that is, Similarly,
3. Blowup Phenomenon
In this section, we show that blowup phenomenon is the only one way that singularity arises in smooth solutions. We start this section with the following useful lemma.
Lemma 5. Let , . is assumed to be the maximal existence time of the solution to system (3) corresponding to the initial data . Then for all , one has the following conservation law:
Proof. We will prove that is a conserved quantity with respect to time variable. Here we use the classical energy method. Multiplying the first equation in (3) by and integrating by parts, we obtain Similarly, we have the following inequality for the second equation (3): This implies that Thus, we have This completes the proof.
Using this conservation law, we obtain where
Theorem 6. Suppose that , , , and the initial data satisfies the following conditions:(i) and on ,(ii) and ,for some point . Then the solution to system (3) with the initial value blowsup in finite time.
Proof. Differentiating the first equation of (3) with respect to , we obtain
Applying the relation yields
From (23) we have
where we used the fact proved in  that
In order to arrive at our result, we need the following three claims.
Claim####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;1. for all in its lifespan; is defined in (9).
It is worth noting the equivalent form of the first equation in (3) in what follows: From the previous equation, we can get Since defined by (10) is a diffeomorphism of the line for any , so there exists an such that When , we have Now we prove that . It is easy to get Since integrating the previous equation, we can obtain thus we have So we can get
then we have Our claim is proved.
Claim####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;2. For any fixed , for all . For any fixed , if , then where the condition (i) is used. Similarly, for , we also have So Claim 2 is proved. Consequently, we can obtain Thus, one can get
Claim####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;3. for all . Furthermore, is strictly decreasing.
Suppose that there exists a such that on and . From the expression of in terms of , we can rewrite and as follows: Letting then Integrating by parts, the first term of (42) yields For the second term of (42), we have the following equation in the view of Claim 1: Here we have used Combining the previous equations together, and with the help of (38), (42) reads as where Claim 2 and the inequality  have been used. From the continuity property, we have Similarly, Thus, by continuity property, Summarizing (48) and (50), we obtain That is a contradiction. On the other hand, from the expression of in terms of , we can easily get that . So we complete the proof of Claim 3.
Furthermore, due to (46) and (49), we can obtain Integrating (39) and then substituting it into the previous inequality, we have Let ; then we can complete the proof with the help of Lemma 4.
Remark 7. We note that if the condition (i) is replaced by the following one: and ,then Claim 2 also holds; that is, the theorem always holds with anyone of (i) and (i').
As a corollary of Theorem 6, we have the following.
Theorem 8. Suppose that , and the initial data satisfies the following conditions:(i)####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;on and ####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;on or ####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;on and ####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;on ,(ii),for some point . Then the solution to system (3) with the initial value blowsup in finite time.
This work was partially supported by Natural Science Foundation of China under Grant no. 11226172 and Zhejiang Provincial Natural Science Foundation of China under Grant nos. LQ12A01009 and LY12A01014.
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