#### Abstract

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 .

#### 1. Introduction

In recent years, a lot of works have been carried out to investigate the Camassa-Holm equation [1], 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 [4]). Johnson [2], Constantin and Lannes [5] 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. [20] 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 [20].

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.

The structure of this paper is as follows. The main result is given in Section 2. Several lemmas are given in Section 3. Section 4 establishes the proof of the main result.

#### 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

Lemma 3 (see [20]). *Let with . Then, the Cauchy problem (5) has a unique solution
**
where depends on .*

Lemma 4 (see [20]). *Let , , and (or equivalently , . Then, problem (5) has a unique solution satisfying
*

Using the first equation of system (5) derives from which one has the conservation law

Lemma 5 (see [20]). *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 [20]). *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 [21]).

Lemma 7 (see [20]). *Let with , and let be the maximal existence time of the problem (5). It holds that
**
where and .*

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 [15].

From Lemma 3, it derives that the Cauchy problem has a unique solution depending on the parameter . We write to represent the solution of problem (23). Using Lemma 3 derives that since .

Lemma 10. *Provided that , , , and (or equivalently , ), then there exists a constant independent of such that the solution of problem (23) satisfies
*

*Proof. *Using identity (10) and Lemma 9, if with , we have
where is independent of .

From Lemma 8, we have
which completes the proof.

Lemma 11. *For any , with , it holds that
*

The proof of this lemma can be found in [15].

#### 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.

#### Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).