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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 409760, 7 pages
The Uniqueness of Strong Solutions for the Camassa-Holm Equation
1The School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
2The 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.
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.
The integrable Camassa-Holm model  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  and Constantin and Molinet  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  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.
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 .
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 ). Let the function be bounded and measurable in cylinder . If and any number , then the function satisfies .
Lemma 2 (see ). If the function is bounded, then the function satisfies the Lipschitz condition in and , respectively.
Lemma 3. Let . It holds that the function satisfies
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.
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.
Proof of Theorem 6. For an arbitrary , set . Let . We assume that outside some cylinder
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.
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).
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