The Uniqueness of Strong Solutions for the Camassa-Holm Equation

has been investigated by many scholars. Equation (1) has peaked solitary wave solutions, which takes the form ce, c ∈ R. The existence and uniqueness of the global weak solutions for (1) have been given byConstantin andEscher [2] and Constantin and Molinet [3] in which the m = u 0 − u 0xx is a positive (or negative) Radon measure. The local wellposedness of strong solutions for the Camassa-Holm model and its various generalized forms are provided in [4–8]. For the initial value u 0 satisfying u 0 − u 0xx ≥ 0 or u 0 − u 0xx ≤


Introduction
The integrable Camassa-Holm model [1]   −   + 3  = 2    +   (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  =  0 −  0 is a positive (or negative) Radon measure.The local wellposedness of strong solutions for the Camassa-Holm model and its various generalized forms are provided in [4][5][6][7][8].For the initial value  0 satisfying  0 −  0 ≥ 0 or  0 −  0 ≤ 0, it is shown in [9] that the Camassa-Holm equation has unique global strong solutions in the Sobolev space   () with  > 3/2.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][13][14][15][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  ∈ [0,  0 ), functions  and V are two local or global strong solutions of problem (2) with initial data (0, ⋅) =  0 ∈  1 () and V(0, ⋅) = V 0 ∈  1 (), respectively.Constant  depends on ‖ 0 ‖  1 () , ‖V 0 ‖  1 () , and the maximum existence time  0 .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.
In fact, the proof of ( 11) can also be found in [17].
From Theorem 6, we immediately obtain the uniqueness result.
Theorem 7. Let (, ) be a strong solution of (1) with  0 ∈  1 (), and let  0 be the maximum existence time of solution .Then any strong solution of (1) is unique.