- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- 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

Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 409760, 7 pages

http://dx.doi.org/10.1155/2013/409760

## The Uniqueness of Strong Solutions for the Camassa-Holm Equation

^{1}The School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China^{2}The School of Finance, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 28 February 2013; Accepted 10 May 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 Meng Wu and Chong Lai. 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

Assume that there exists a strong solution of the Camassa-Holm equation and the initial value of the solution belongs to the Sobolev space . We provide a new proof of the uniqueness of the strong solution for the equation.

#### 1. Introduction

The integrable Camassa-Holm model [1] has been investigated by many scholars. Equation (1) has peaked solitary wave solutions, which takes the form . The existence and uniqueness of the global weak solutions for (1) have been given by Constantin and Escher [2] and Constantin and Molinet [3] in which the is a positive (or negative) Radon measure. The local well-posedness of strong solutions for the Camassa-Holm model and its various generalized forms are provided in [4–8]. For the initial value satisfying or , it is shown in [9] that the Camassa-Holm equation has unique global strong solutions in the Sobolev space with . If the initial data satisfy certain conditions, we know that the local strong solutions blow up in finite time [10, 11]. It means that the slope of the solution becomes unbounded while the solution itself remains bounded. For other techniques to obtain the dynamic properties for the Camassa-Holm equation and other related shallow water equations, the reader is referred to [12–16] and the references therein.

We consider the equivalent form of the Cauchy problem for (1) where . If (1) has a suitable smooth strong solution, we have the conservation law which derives

The objective of this work is to give a new proof of the uniqueness for the solutions of the Camassa-Holm equation (1). Firstly, we establish the following inequality: where , functions and are two local or global strong solutions of problem (2) with initial data and , respectively. Constant depends on , , and the maximum existence time . Secondly, from (5), we immediately arrive at the goal of the uniqueness. Here we state that the approach to establish (5) is the device of doubling variables which was presented in Kruzkov's paper [17].

This paper is organized as follows. Several lemmas are given in Section 2, while the proofs of the main results are established in Section 3.

#### 2. Notations and Several Lemmas

Set for an arbitrary . The space of all infinitely differentiable functions with compact support in is denoted by . We define as a function which is infinitely differentiable on such that , for and . For any number , we let . Then we have that is a function in and Assume that the function is locally integrable in . We define the approximation of function as

We call a Lebesgue point of function if At any Lebesgue point of the function , we have . Since the set of points which are not Lebesgue points of has measured zero, we get as almost everywhere.

We introduce notation connected with the concept of a characteristic cone. For any , we define . Let designate the cone . We let designate the cross section of the cone by the plane .

Let , where .

Lemma 1 (see [17]). *Let the function be bounded and measurable in cylinder . If and any number , then the function
**
satisfies .*

Lemma 2 (see [17]). *If the function is bounded, then the function satisfies the Lipschitz condition in and , respectively.*

Lemma 3. *Let . It holds that the function satisfies
*

The proof of Lemma 3 can be found in [13, 15] (see [13, Lemma 5.1]).

Lemma 4. *Let be the strong solution of problem (2), , and . Then
**
where is an arbitrary constant.*

*Proof. *Let be an arbitrary twice smooth function on the line . We multiply (2) by the function , where . Integrating over and transferring the derivatives with respect to and to the test function , for any constant , we obtain
in which we have used .

We have
Let be an approximation of the function and set . Using the properties of the , (12), and (13) and sending , we have
which completes the proof.

In fact, the proof of (11) can also be found in [17].

Lemma 5. *Assume and are two strong solutions of problem (2). It has
*

*Proof. *We have
in which we have used the Fubini theorem, and . The proof is completed.

#### 3. Main Results

Theorem 6. *Let and be two local or global strong solutions of problem (2) with initial data and , respectively. Let be the maximum existence time of solutions and . For any , it holds that
**
where depends on and . *

From Theorem 6, we immediately obtain the uniqueness result.

Theorem 7. *Let be a strong solution of (1) with , and let be the maximum existence time of solution . Then any strong solution of (1) is unique.*

*Proof of Theorem 6. *For an arbitrary , set . Let . We assume that outside some cylinder
We define
where and . The function is defined in (6). Note that

Taking and assuming outside the cylinder , from Lemma 4, we have
Similarly, it has
from which we obtain

We will show that
We note that the first two terms in the integrand of (23) can be represented in the form
Since and , from Lemma 2, we know satisfies the Lipschitz condition in and , respectively. By the choice of , we have outside the region
Considering the estimate and the expression of function , we have
where the constant does not depend on . Using Lemma 1, we obtain as . The integral does not depend on . In fact, substituting , , , and noting that
we have
Hence
Since
we obtain
By Lemmas 1 and 3, we have as . Using (28), we have
From (30) and (33), we prove that inequality (24) holds.

Let
We define
and choose two numbers and , . In (24), we choose
where
We note that function outside the cone and outside the set . For , we have the relations
Applying (24) and (35)–(38), we have the inequality
where .

From (39), we obtain
Using Lemma 5, we have
where and is defined in Lemma 5.

Letting in (41) and sending , we have

By the properties of the function for , we have
where is independent of .

Set
Using the similar proof of (43), we get
from which we obtain
Similarly, we have
Then, we get
Furthermore, if , we have

Let and , and note that

Thus, from (42), (43), (48), (49), and (50), for any , we have
from which we complete the proof of Theorem 6 by using the Gronwall inequality.

#### Acknowledgments

Thanks are given to referees whose suggestions are very helpful to the paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

#### References

- 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, “Global weak solutions for a shallow water equation,”
*Indiana University Mathematics Journal*, vol. 47, no. 4, pp. 1527–1545, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Constantin and L. Molinet, “Global weak solutions for a shallow water equation,”
*Communications in Mathematical Physics*, vol. 211, no. 1, pp. 45–61, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Constantin and J. Escher, “Global existence and blow-up for a shallow water equation,”
*Annali della Scuola Normale Superiore di Pisa. Classe di Scienze*, vol. 26, no. 2, pp. 303–328, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. A. Himonas and G. Misiołek, “The Cauchy problem for an integrable shallow-water equation,”
*Differential and Integral Equations*, vol. 14, no. 7, pp. 821–831, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,”
*Journal of Differential Equations*, vol. 248, no. 8, pp. 2038–2063, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. A. Li and P. J. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,”
*Journal of Differential Equations*, vol. 162, no. 1, pp. 27–63, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Rodríguez-Blanco, “On the Cauchy problem for the Camassa-Holm equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 46, no. 3, pp. 309–327, 2001. 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 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, “On the Cauchy problem for a family of quasilinear hyperbolic equations,”
*Communications in Partial Differential Equations*, vol. 23, no. 7-8, pp. 1449–1458, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Bressan and A. Constantin, “Global conservative solutions of the Camassa-Holm equation,”
*Archive for Rational Mechanics and Analysis*, vol. 183, no. 2, pp. 215–239, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. M. Coclite, H. Holden, and K. H. Karlsen, “Global weak solutions to a generalized hyperelastic-rod wave equation,”
*SIAM Journal on Mathematical Analysis*, vol. 37, no. 4, pp. 1044–1069, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation—a Lagrangian point of view,”
*Communications in Partial Differential Equations*, vol. 32, no. 10–12, pp. 1511–1549, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,”
*Communications on Pure and Applied Mathematics*, vol. 53, no. 11, pp. 1411–1433, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Yin, “On the Cauchy problem for an integrable equation with peakon solutions,”
*Illinois Journal of Mathematics*, vol. 47, no. 3, pp. 649–666, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Kruzkov, “First order quasi-linear equations in several independent variables,”
*Mathematics of the USSR-Sbornik*, vol. 10, no. 2, pp. 217–243, 1970. View at Google Scholar