- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 548310, 8 pages

http://dx.doi.org/10.1155/2013/548310

## 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.

#### Abstract

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.

#### 1. Introduction

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 [1] (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, [2–5] 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 [6]. Our modification breaks the structure of DGH2 system derived by following Ivanov’s approach [7] by the authors in [8]. 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’ 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 [9–11] 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’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 [12–19] and its two-component generalizations [20–27]. 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.

#### 2. Preliminaries

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 .

We can apply Kato’s theory [28] to establish the following local well-posedness theorem for (3).

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 .*

The proof of Theorem 1 is similar to the one in [11]. Moreover, using the techniques in [11], one can get the criterion for finite time wave breaking to (3) as follows.

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 3 (see [29]). *Assume that a differentiable function satisfies
**
with constants . If the initial datum , then the solution to (9) goes to before tends to .*

Lemma 4 (see [19]). *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 [30] that
In order to arrive at our result, we need the following three claims.*Claim 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 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 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 [30]
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)* on and on or on and on ,*(ii)

*,*

*for some point . Then the solution to system (3) with the initial value blowsup in finite time.*

*Proof. *As shown in Theorem 6, condition (i) guarantees that for all . Then,
where is a constant. By setting , we obtain
Applying Lemma 3, we have
when
This completes the proof.

#### Acknowledgments

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.

#### References

- H. R. Dullin, G. A. Gottwald, and D. D. Holm, “An integrable shallow water equation with linear and nonlinear dispersion,”
*Physical Review Letters*, vol. 87, no. 19, Article ID 194501, pp. 4501–4504, 2001. View at Scopus - Y. Liu, “Global existence and blow-up solutions for a nonlinear shallow water equation,”
*Mathematische Annalen*, vol. 335, no. 3, pp. 717–735, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Tian, G. Gui, and Y. Liu, “On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation,”
*Communications in Mathematical Physics*, vol. 257, no. 3, pp. 667–701, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhou, “Blow-up of solutions to the DGH equation,”
*Journal of Functional Analysis*, vol. 250, no. 1, pp. 227–248, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhou and Z. Guo, “Blow up and propagation speed of solutions to the DGH equation,”
*Discrete and Continuous Dynamical Systems B*, vol. 12, no. 3, pp. 657–670, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. D. Holm, L. Ó Náraigh, and C. Tronci, “Singular solutions of a modified two-component Camassa-Holm equation,”
*Physical Review E*, vol. 79, no. 1, Article ID 016601, 13 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - A. Constantin and R. I. Ivanov, “On an integrable two-component Camassa-Holm shallow water system,”
*Physics Letters A*, vol. 372, no. 48, pp. 7129–7132, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Guo, H. Gao, and Y. Liu, “On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system,”
*Journal of the London Mathematical Society*, vol. 86, no. 3, pp. 810–834, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - R. M. Chen and Y. Liu, “Wave breaking and global existence for a generalized two-component Camassa-Holm system,”
*International Mathematics Research Notices*, no. 6, pp. 1381–1416, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Guan and Z. Yin, “Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,”
*Journal of Differential Equations*, vol. 248, no. 8, pp. 2003–2014, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Guan, K. H. Karlsen, and Z. Yin, “Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation,” in
*Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena*, vol. 526 of*Contemporary Mathematics*, pp. 199–220, American Mathematical Society, Providence, RI, USA, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,”
*Physical Review Letters*, vol. 71, no. 11, pp. 1661–1664, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,”
*Acta Mathematica*, vol. 181, no. 2, pp. 229–243, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,”
*Communications on Pure and Applied Mathematics*, vol. 51, no. 5, pp. 475–504, 1998. View at Zentralblatt MATH · View at MathSciNet - A. A. Himonas, G. Misiołek, G. Ponce, and Y. Zhou, “Persistence properties and unique continuation of solutions of the Camassa-Holm equation,”
*Communications in Mathematical Physics*, vol. 271, no. 2, pp. 511–522, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Jiang and S. Hakkaev, “Wave breaking and propagation speed for a class of one-dimensional shallow water equations,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 647368, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus - H. P. McKean, “Breakdown of a shallow water equation,”
*The Asian Journal of Mathematics*, vol. 2, no. 4, pp. 867–874, 1998. View at Zentralblatt MATH · View at MathSciNet - Y. Zhou, “Wave breaking for a periodic shallow water equation,”
*Journal of Mathematical Analysis and Applications*, vol. 290, no. 2, pp. 591–604, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhou, “On solutions to the Holm-Staley $b$-family of equations,”
*Nonlinearity*, vol. 23, no. 2, pp. 369–381, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - J. Escher, O. Lechtenfeld, and Z. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,”
*Discrete and Continuous Dynamical Systems A*, vol. 19, no. 3, pp. 493–513, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Guo, “Blow-up and global solutions to a new integrable model with two components,”
*Journal of Mathematical Analysis and Applications*, vol. 372, no. 1, pp. 316–327, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Guo and L. Ni, “Persistence properties and unique continuation of solutions to a two-component Camassa-Holm equation,”
*Mathematical Physics, Analysis and Geometry*, vol. 14, no. 2, pp. 101–114, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Guo and Y. Zhou, “On solutions to a two-component generalized Camassa-Holm equation,”
*Studies in Applied Mathematics*, vol. 124, no. 3, pp. 307–322, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Guo, M. Zhu, and L. Ni, “Blow-up criteria of solutions to a modified two-component Camassa-Holm system,”
*Nonlinear Analysis. Real World Applications*, vol. 12, no. 6, pp. 3531–3540, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Zhang and Y. Liu, “Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,”
*International Mathematics Research Notices*, no. 11, pp. 1981–2021, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Zhu, “Blow-up, global existence and persistence properties for the coupled Camassa-Holm equations,”
*Mathematical Physics, Analysis and Geometry*, vol. 14, no. 3, pp. 197–209, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Zhu and J. Xu, “On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system,”
*Journal of Mathematical Analysis and Applications*, vol. 391, no. 2, pp. 415–428, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kato, “On the Cauchy problem for the (generalized) Korteweg-de Vries equation,” in
*Studies in Applied Mathematics*, vol. 8 of*Advances in Mathematics, Supplementary Studies*, pp. 93–128, Academic Press, New York, NY, USA, 1983. View at Zentralblatt MATH · View at MathSciNet - Y. Zhou, “Blow-up of solutions to a nonlinear dispersive rod equation,”
*Calculus of Variations and Partial Differential Equations*, vol. 25, no. 1, pp. 63–77, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Zhu, L. Jin, and Z. Jiang, “A new blow-up criterion for the DGH equation,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 515948, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet