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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 327572, 15 pages
http://dx.doi.org/10.1155/2012/327572
Research Article

On the Global Well-Posedness of the Viscous Two-Component Camassa-Holm System

School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China

Received 3 January 2012; Accepted 15 February 2012

Academic Editor: Valery  Covachev

Copyright © 2012 Xiuming Li. 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 establish the local well-posedness for the viscous two-component Camassa-Holm system. Moreover, applying the energy identity, we obtain a global existence result for the system with .

1. Introduction

We are interested in the global well-pose dness of the initial value problesm associated to the viscous version of the two-component Camassa-Holm shallow water system [13], namely, where the variable represents the horizontal velocity of the fluid or the radial stretch related to a prestressed state, and is related to the free surface elevation from equilibrium or scalar density with the boundary assumptions, and as . The parameter characterizes a linear underlying shear flow, so that (1.1) models wave-current interactions [46]. All of those are measured in dimensionless units.

Set , . Then for all , where denotes the spatial convolution. Let , (1.1) can be rewritten as a quasilinear nonlocal evolution system of the type

The system (1.1) without vorticity, that is, , was also rigorously justified by Constantin and Ivanov [1] to approximate the governing equations for shallow water waves. The multipeakon solutions of the same system have been constructed by Popivanov and Slavova [7], and the corresponding integral surface is partially ruled. Chen et al. [8] established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of AKNS hierarchy. More recently, Holm et al. [9] proposed a modified two-component Camassa-Holm system which possesses singular solutions in component . Mathematical properties of (1.1) with have been also studied further in many works. For example, Escher et al. [10] investigated local well-posedness for the two-component Camassa-Holm system with initial data with and derived some precise blow-up scenarios for strong solutions to the system. Constantin and Ivanov [1] provided some conditions of wave breaking and small global solutions. Gui and Liu [11] recently obtained results of local well-posedness in the Besov spaces and wave breaking for certain initial profiles. More recently, Gui and Liu [12] studied global existence and wave-breaking criteria for the system (1.2) with initial data with .

In this paper, we consider the global well-posedness of the viscous two-component Camassa-Holm system

The goal of the present paper is to study global existence of solutions for (1.3) to better understand the properties of the two-component Camassa-Holm system (1.2). We state the main result as follows.

Theorem 1.1. For , there exists a unique global solution of (1.3) such that

To proof Theorem 1.1, we will first establish global well-posedness of the following regularized two-component system with given:

that is, Due to the Duhamel’s principle, we can also rewrite (1.6) as an integral equation where, , , here and in what follows, we denote the Fourier (or inverse Fourier) transform of a function by (or ).

The remainder of the paper is organized as follows. In Section 2, we will set up and introduce some estimates for the nonlinear part of (1.5). In Section 3, we will get the local well-posedness of (1.3) by constructing the global well-posedness of (1.5) using the contraction argument and energy identity. The last section is devoted to the proof of Theorem 1.1.

2. Preliminaries

We will list some lemmas needed in Section 3. First, we state the following lemma which consists of the crucial inequality involving the operator .

Lemma 2.1 (see [13]). For , or more generally for all .

The next two lemmas are regarding the nonlinear part of (1.5).

Lemma 2.2. Consider the following:

Proof. Let us prove (2.3) firstly. Thanks to Lemma 2.1, the Sobolev embedding theorem , and the Hölder’s inequality, we have which yields that Similarly, we can get and then which implies (2.4).

Lemma 2.3. Consider the following:

Proof. We mainly prove (2.9). For this, we have that Therefore, applying Lemma 2.2, we can easily obtain (2.9).
Similarly, we can also obtain (2.10).

Let us state the following lemma, which was obtained in [13] (up to a slight modification).

Lemma 2.4. For any and , there exists such that For any , , and , there exists such that

Next, we consider the nonlinear part of (1.7). When written as , , then we have that is,

First, we have the following basic estimates.

Lemma 2.5. Consider the following:

Proof. First, let us prove (2.16), which implies then we can get that where we applied Lemma 2.1, Sobolev embedding theorem , and Hölder’s inequality. This proves (2.16).
Next, we prove (2.17), which yields that and this ends the proof of (2.17).

Then we have some estimates for and .

Lemma 2.6. Consider the following:

Proof. We mainly prove (2.24). We have that
Multiply to the second equation of (2.14) and integrate with respect to over . After integration by parts, we have for any , which implies By (2.25), together with (2.27), we have that By Lemma 2.5, (2.24) follows.
Similarly, we can also obtain (2.23).

3. Local Well-Posedness

Let , , and define the mapping by

Theorem 3.1. For any , there exist and such that . In addition, is a contraction mapping.

Proof. We first need to show that the map is well defined for some appropriate and . Let , then we have
Considering the terms in (3.3) one by one, from Lemma 2.2, the first term in (3.3) can be estimated as follows: From Lemma 2.3, the second term in (3.3) can be estimated as follows: which implies From Lemma 2.6, the third term in (3.3) can be estimated as follows: Combining (3.4)–(3.7), we have that
With appropriate values of , , and , we are able to have that , that is, is well defined.
Similar to the above argument, we can show that is a contraction mapping, where can be chosen as with appropriate values of and .

Theorem 3.2. For any and , there exist a and a unique solution of (1.5) such that

Proof. Theorem 3.2 is merely Theorem 3.1 with a standard uniqueness argument.

Theorem 3.3. For any and , there exists a unique global solution of (1.5) such that

Proof. To prove Theorem 3.3, we need only to establish the a priori energy identity. Multiplying the first equation in (1.6) by and integrating by parts (with respect to over ), we have that where we used the relation and . Multiplying the first equation in (1.6) by and integrating by parts (with respect to over ), we have that From (3.12) and (3.13), we obtain the energy identity which gives rise to the following inequality independent of and : According to Theorem 3.2 and the energy inequality (3.15), one can extend the local solution to the global one by a standard contradiction argument, which completes the proof of Theorem 3.3.

From Theorem 3.3, one has the local well-posedness of system (1.3).

Theorem 3.4. For , there exist a and a unique local solution of (1.3) such that

Similar to the proof of Theorem  4.1 in [12] (up to a slight modification), we may get the following.

Theorem 3.5. Let , , and let be the corresponding solution to (1.3). Assume that is the maximal time of existence, then

4. Proof of Theorem 1.1

Proof of Theorem 1.1. From Theorem 3.4, we have got the local solution of (1.3). On the other hand, according to the second equation of (1.3), we have An application of Gronwall’s inequality yields Similar to the proof of Theorem 3.3, we may get the energy identity which implies Due to the Sobolev embedding theorem and (4.4), we obtain that for any , Therefore, from Theorem 3.5, we deduce that the maximal existence time . This proves Theorem 1.1.

Acknowledgments

The work of the author is supported in part by the NSF of China under Grant no. 11001111 and no. 11141003. The author would like to thank the referees for constructive suggestions and comments.

References

  1. 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
  2. P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, no. 2, pp. 1900–1906, 1996. View at Publisher · View at Google Scholar
  3. R. Ivanov, “Two-component integrable systems modelling shallow water waves: the constant vorticity case,” Wave Motion, vol. 46, no. 6, pp. 389–396, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Constantin and W. Strauss, “Exact steady periodic water waves with vorticity,” Communications on Pure and Applied Mathematics, vol. 57, no. 4, pp. 481–527, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. R. S. Johnson, “Nonlinear gravity waves on the surface of an arbitrary shear flow with variable depth,” in Nonlinear Instability Analysis, vol. 12 of Adv. Fluid Mech., pp. 221–243, Comput. Mech., Southampton, UK, 1997.
  6. R. S. Johnson, “On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer),” Geophysical and Astrophysical Fluid Dynamics, vol. 57, no. 1–4, pp. 115–133, 1991. View at Publisher · View at Google Scholar
  7. P. Popivanov and A. Slavova, Nonlinear Waves, vol. 4 of Series on Analysis, Applications and Computation, World Scientific, Hackensack, NJ, USA, 2011.
  8. M. Chen, S.-Q. Liu, and Y. Zhang, “A two-component generalization of the Camassa-Holm equation and its solutions,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 1–15, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. D. D. Holm, L. Nraigh, and C. Tronci, “Singular solutions of a modified two-component Camassa-Holm equation,” Physical Review E, vol. 85, no. 1, Article ID 016601, 5 pages, 2012. View at Publisher · View at Google Scholar
  10. 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. Series A, vol. 19, no. 3, pp. 493–513, 2007. View at Zentralblatt MATH
  11. G. Gui and Y. Liu, “On the Cauchy problem for the two-component Camassa-Holm system,” Mathematische Zeitschrift, vol. 268, no. 1-2, pp. 45–66, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. W. K. Lim, “Global well-posedness for the viscous Camassa-Holm equation,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 432–442, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH