A nonlinear generalization of the famous Camassa-Holm model is investigated. Provided that initial value u₀ ∈ H^s(R)(1 ≤ s ≤ 3/2) and satisfies an associated sign condition, it is shown that there exists a unique global weak solution to the equation in space u(t, x) ∈ L²([0, +∞), H^s(R)) in the sense of distribution, and u_x ∈ L^∞([0, +∞) × R).

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 k > 0 [3, 8] or peakons if k = 0 [3, 9]. Equation (1) is also regarded as a model of the geodesic flow for the H¹ right invariant metric on the Bott-Virasoro group if k > 0 and on the diffeomorphism group if k = 0 (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 m ≥ 0 is a natural number. Obviously, (2) reduces to (1) if m = 0. The authors applied the pseudoparabolic regularization technique to build the local well-posedness for (2) in Sobolev space H^s(R) with s > 3/2 via a limiting procedure. Provided that the initial value u₀ satisfies a sign condition and u₀ ∈ H^s(R)(s > 3/2), it is shown that there exists a unique global strong solution for (2) in space C([0, ∞); H^s(R))⋂ C¹([0, ∞); H^s−1(R)). 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 H^q(R) with 0 ≤ q ≤ 1/2, which are derived from the equation itself, we prove that there exists a unique global weak solution to (2) in space H^s(R) with 1 ≤ s ≤ 3/2 if u₀ ∈ H^s(R), 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 ϕ(t, x) with compact support in [0, +∞) × R is denoted by

. L^p = L^p(R) (1 ≤ p < +∞) is the space of all measurable functions h such that

. We define L^∞ = L^∞(R) with the standard norm

. For any real number s, we let H^s = H^s(R) denote the Sobolev space with the norm defined by

()

where

For T > 0 and nonnegative number s, let C([0, T); H^s(R)) denote the Frechet space of all continuous H^s-valued functions on [0, T). We set .

Defining

()

and letting ϕ_ε(x) = ε^−(1/4)ϕ(ε^−(1/4)x) with 0 < ε < 1/4 and u_ε0 = ϕ_ε⋆u₀ (convolution of ϕ_ε and u₀), we know that u_ε0 ∈ C^∞ for any u₀ ∈ H^s with s > 0. Notation

(or equivalently

) means that

(or equivalently

) for an arbitrary sufficiently small ε > 0.

For the equivalent form of (2), we consider its Cauchy problem

()

Definition 1. A function u(t, x) ∈ L²([0, +∞), H^s(R)) is called a global weak solution to problem (5) if for every T > 0, u(t, x) ∈ H^s(R), u_t(t, x) ∈ H^s−1(R), and all , it holds that

()

with u(0, x) = u₀(x).

Now, we give the main result of this work.

Theorem 2. Let u₀(x) ∈ H^s(R), 1 ≤ s ≤ 3/2, , and k ≥ 0 (or equivalently , k ≤ 0). Then, problem (5) has a unique global weak solution u(t, x) ∈ L²([0, +∞), H^s(R)) in the sense of distribution, and u_x ∈ L^∞([0, +∞) × R).

3. Several Lemmas

Lemma 3 (see [20].)Let u₀(x) ∈ H^s(R) with s > 3/2. Then, the Cauchy problem (5) has a unique solution

()

where T > 0 depends on

Lemma 4 (see [20].)Let u₀(x) ∈ H^s, s > 3/2, and (or equivalently k ≤ 0, . 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 s > 3/2, and the function u(t, x) is a solution of problem (5) and the initial data u₀(x) ∈ H^s. Then, the following inequality holds:

()

For q ∈ (0, s − 1], there is a constant c such that

()

For q ∈ [0, s − 1], there is a constant c such that

()

For (2), consider the problem

()

Lemma 6 (see [20].)Let u₀ ∈ H^s, s ≥ 3, and let T > 0 be the maximal existence time of the solution to problem (5). Then, problem (14) has a unique solution p ∈ C¹([0, T) × R). Moreover, the map p(t, ·) is an increasing diffeomorphism of R with p_x(t, x) > 0 for (t, x)∈[0, T) × R.

Differentiating (14) with respect to x 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 u₀ ∈ H^s with s ≥ 3, and let T > 0 be the maximal existence time of the problem (5). It holds that

()

where (t, x)∈[0, T) × R and y : = u − u_xx + k/2(m + 1).

Lemma 8. If u₀ ∈ H^s, s ≥ 3, such that , k ≥ 0 (or equivalently, ), then the solution of problem (5) satisfies

()

Proof. Using u₀ − u_0xx + k/2(m + 1) ≥ 0, it follows from Lemma 7 that u − u_xx + k/2(m + 1) ≥ 0. Letting Y₁ = u − u_xx, 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

, k ≤ 0, we still know that (18) is valid.

Lemma 9. For s > 0, u₀ ∈ H^s, it holds that

()

where c is a constant independent of ε.

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 u depending on the parameter ε. We write u_ε(t, x) to represent the solution of problem (23). Using Lemma 3 derives that u_ε(t, x) ∈ C^∞([0, T), H^∞(R)) since

Lemma 10. Provided that u₀ ∈ H^s, 1 ≤ s ≤ 3/2, k ≥ 0, and (or equivalently , k ≤ 0), then there exists a constant c₀ > 0 independent of ε such that the solution of problem (23) satisfies

()

Proof. Using identity (10) and Lemma 9, if u₀ ∈ H^s(R) with 1 ≤ s ≤ 3/2, we have

()

where c is independent of ε.

From Lemma 8, we have

()

which completes the proof.

Lemma 11. For any f₁ ∈ L^∞, f₂ ∈ H^z with z ≤ 0, it holds that

()

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

4. Existence and Uniqueness of Global Weak Solution

Provided that 1 ≤ s ≤ 3/2, for problem (23), applying Lemmas 5, 9, and 10, and the Gronwall’s inequality, we obtain the inequalities

()

where q ∈ (0, s], r ∈ [0, s − 1], and c is a constant independent of ε. It follows from the Aubin’s compactness theorem that there is a subsequence of {u_ε}, denoted by

, such that

and their temporal derivatives

are weakly convergent to a function u(t, x) and its derivative u_t in L²([0, T], H^s) and L²([0, T], H^s−1), respectively, where T is an arbitrary fixed positive number. Moreover, for any real number R₁ > 0,

is convergent to the function u strongly in the space L²([0, T], H^q(−R₁, R₁)) for q ∈ (0, s] and

converges to u_t strongly in the space L²([0, T], H^r(−R₁, R₁)) for r ∈ [0, s − 1].

4.1. The Proof of Existence for Global Weak Solution

For an arbitrary fixed T > 0, from Lemma 10, we know that

is bounded in the space L^∞. Thus, the sequences

, and

are weakly convergent to u, u_x,

, and

in L²([0, T], H^r(−R₁, R₁)) for any r ∈ [0, s − 1), separately. Using

, we know that u satisfies the equation

()

with u(0, x) = u₀(x) and

. Since X = L¹([0, T] × R) is a separable Banach space and

is a bounded sequence in the dual space X^* = L^∞([0, T] × R) of X, there exists a subsequence of

, still denoted by

, weakly star convergent to a function v in L^∞([0, T] × R). As

weakly converges to u_x in L²([0, T] × R), it results that u_x = v almost everywhere. Thus, we obtain u_x ∈ L^∞([0, T] × R). Since T > 0 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 u(t, x) and v(t, x) to problem (5) with the same initial value u(0, x) ∈ H^s(R), 1 ≤ s ≤ 3/2, we consider its associated regularized problem (23). Letting w_ε = u_ε(t, x) − v_ε(t, x), from Lemma 10, we get and which is independent of ε. Still denoting u = u_ε, v = v_ε, and w = w_ε, it holds that

()

Multiplying both sides of (30) by w, we get

()

Using

, we have

()

Applying Lemma 11 repeatedly, we have

()

For I₅, using Lemma 11 derives

()

Using (32)–(34), we get

()

Applying w(0) = 0 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).

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The Global Weak Solution for a Generalized Camassa-Holm Equation

Abstract

1. Introduction

2. Main Results

3. Several Lemmas

4. Existence and Uniqueness of Global Weak Solution

4.1. The Proof of Existence for Global Weak Solution

Acknowledgment

References

Citing Literature

References

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The Global Weak Solution for a Generalized Camassa-Holm Equation

Abstract

1. Introduction

2. Main Results

3. Several Lemmas

4. Existence and Uniqueness of Global Weak Solution

4.1. The Proof of Existence for Global Weak Solution

Acknowledgment

References

Citing Literature

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