Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and ApplicationsView this Special Issue
The Global Weak Solution for a Generalized Camassa-Holm Equation
A nonlinear generalization of the famous Camassa-Holm model is investigated. Provided that initial value and satisfies an associated sign condition, it is shown that there exists a unique global weak solution to the equation in space in the sense of distribution, and .
In recent years, a lot of works have been carried out to investigate the Camassa-Holm equation , which is a completely integrable equation. In fact, the Camassa-Holm equation arises as a model describing the unidirectional propagation of shallow water waves over a flat bottom [1–3]. The equation was originally derived much earlier as a bi-Hamiltonian generalization of the Korteweg-de Vries equation (see ). Johnson , Constantin and Lannes  derived models which include the Camassa-Holm equation (1). It has been found that (1) conforms with many conservation laws (see [6, 7]) and possesses smooth solitary wave solutions if [3, 8] or peakons if [3, 9]. Equation (1) is also regarded as a model of the geodesic flow for the right invariant metric on the Bott-Virasoro group if and on the diffeomorphism group if (see [10–14]). The well-posedness of local strong solutions for generalized forms of (1) has been given in [15–17]. The sharpest results for the global existence and blow-up solutions are found in Bressan and Constantin [18, 19].
Recently, Li et al.  studied the following generalized Camassa-Holm equation: where is a natural number. Obviously, (2) reduces to (1) if . The authors applied the pseudoparabolic regularization technique to build the local well-posedness for (2) in Sobolev space with via a limiting procedure. Provided that the initial value satisfies a sign condition and , it is shown that there exists a unique global strong solution for (2) in space . However, the existence and uniqueness of the global weak solution for (2) is not investigated in .
The objective of this paper is to establish the well-posedness of global weak solutions for (2). Using the estimates in with , which are derived from the equation itself, we prove that there exists a unique global weak solution to (2) in space with if , and satisfies an associated sign condition.
2. Main Results
Firstly, we give some notations.
The space of all infinitely differentiable functions with compact support in is denoted by . is the space of all measurable functions such that . We define with the standard norm . For any real number , we let denote the Sobolev space with the norm defined by where .
For and nonnegative number , let denote the Frechet space of all continuous -valued functions on . We set .
Defining and letting with and (convolution of and ), we know that for any with . Notation (or equivalently ) means that (or equivalently ) for an arbitrary sufficiently small .
For the equivalent form of (2), we consider its Cauchy problem
Definition 1. A function is called a global weak solution to problem (5) if for every , , and all , it holds that with .
Now, we give the main result of this work.
Theorem 2. Let , , , and (or equivalently , ). Then, problem (5) has a unique global weak solution in the sense of distribution, and .
3. Several Lemmas
Using the first equation of system (5) derives from which one has the conservation law
Lemma 5 (see ). Let , and the function is a solution of problem (5) and the initial data . Then, the following inequality holds:
For , there is a constant such that
For , there is a constant such that
For (2), consider the problem
Lemma 6 (see ). Let , , and let be the maximal existence time of the solution to problem (5). Then, problem (14) has a unique solution . Moreover, the map is an increasing diffeomorphism of with for .
Differentiating (14) with respect to yields which leads to
The next lemma is reminiscent of a strong invariance property of the Camassa-Holm equation (the conservation of momentum ).
Lemma 8. If , , such that , (or equivalently, ), then the solution of problem (5) satisfies
Proof. Using , it follows from Lemma 7 that . Letting , we have from which we obtain On the other hand, we have The inequalities (19), (20), and (21) derive that inequality (18) is valid. Similarly, if , , we still know that (18) is valid.
Lemma 9. For , , it holds that where is a constant independent of .
The proof of this lemma can be found in Lai and Wu .
Lemma 10. Provided that , , , and (or equivalently , ), then there exists a constant independent of such that the solution of problem (23) satisfies
Lemma 11. For any , with , it holds that
The proof of this lemma can be found in .
4. Existence and Uniqueness of Global Weak Solution
Provided that , for problem (23), applying Lemmas 5, 9, and 10, and the Gronwall’s inequality, we obtain the inequalities where , and is a constant independent of . It follows from the Aubin’s compactness theorem that there is a subsequence of , denoted by , such that and their temporal derivatives are weakly convergent to a function and its derivative in and , respectively, where is an arbitrary fixed positive number. Moreover, for any real number , is convergent to the function strongly in the space for and converges to strongly in the space for .
4.1. The Proof of Existence for Global Weak Solution
For an arbitrary fixed , from Lemma 10, we know that is bounded in the space . Thus, the sequences , , , and are weakly convergent to , , , and in for any , separately. Using , we know that satisfies the equation with and . Since is a separable Banach space and is a bounded sequence in the dual space of , there exists a subsequence of , still denoted by , weakly star convergent to a function in . As weakly converges to in , it results that almost everywhere. Thus, we obtain . Since is an arbitrary number, we complete the global existence of weak solutions to problem (5).
Proof of Uniqueness. Suppose that there exist two global weak solutions and to problem (5) with the same initial value , , we consider its associated regularized problem (23). Letting , from Lemma 10, we get and which is independent of . Still denoting , and , it holds that
Multiplying both sides of (30) by , we get Using , , , , we have Applying Lemma 11 repeatedly, we have For , using Lemma 11 derives Using (32)–(34), we get Applying results in . Consequently, we know that the global weak solution is unique.
This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).