SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Blow-up and local weak solution for a modified two-component Camassa-Holm equations

Lixin Tian* and Minyi Zhu

Author affiliations

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, People's Republic of China

For all author emails, please log on.

Citation and License

Boundary Value Problems 2012, 2012:52  doi:10.1186/1687-2770-2012-52

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/52


Received:14 November 2011
Accepted:2 May 2012
Published:2 May 2012

© 2012 Tian and Zhu; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we establish some blow-up results for a modified two-component Camassa-Holm system in Sobolev spaces. We also obtain the existence of the weak solutions of this system in Hs × Hs-1, s > 5/2.

Keywords:
the modified two-component Camassa-Holm equations; blow up; weak solutions.

1. Introduction

The well-known two-component Camassa-Holm equations [1]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M1">View MathML</a>

(1.1)

where m = u - uxx and σ = ± 1. Constantin and Ivanov [2] derived this system in the context of shallow water theory. u can be interpreted as the horizontal fluid velocity and ρ is related to the water elevation in the first approximation [2,3]. They showed that while small initial data develop into global solutions, for some initial data wave breaking occurs. They also discussed the solitary wave solutions. In Vlasov plasma models, system (1.1) describes the closure of the kinetic moments of the single-particle probability distribution for geodesic motion on the simplectomorphisms. While in the large-deformation diffeomorphic approach to image matching, system (1.1) is summoned in a type of matching procedure called metamorphosis (see [4] and the references therein). This system appeared originally in [5]. Based on the deformation of bi-Hamiltonian structure of the hydrodynamic type, Chen et al. [6] obtained system (1.1) when σ = -1. They show that it has the peakon and multilink solitons, and is integrable in the sense that it has Lax-pair. The mathematical properties of system (1.1) have been studied further in many articles, see, e.g., [7-15]. In [4], Holm and Ivanov generalized the Lax-pair formulation of system (1.1) to produce an integrable multi-component family, CH(n, k), of equations with n components and 1 ≤ |k| ≤ n velocities. They determined their Lie-Poisson Hamiltonian structures and gave numerical examples of their soliton solution behavior. Recently, a new global existence result and several new blow-up results of strong solutions for the Cauchy problem of Equation (1.1) with σ = 1 were obtained in [8]. Gui and Liu [14] established the local well posedness for the two-component Camassa-Holm system in a range of the Besov spaces. Chen and Liu [16] discussed the wave-breaking phenomenon of a generalized two-component Camassa-Holm system, and determined the exact blow-up rate of such solutions. The existence and uniqueness of global weak solutions to Equation (1.1) have also been discussed by Guan and Yin [17].

In this article, we consider a two-component generalization of Equation (1.1), that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M2">View MathML</a>

(1.2)

with initial data

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M3">View MathML</a>

(1.3)

where m ≥ 1. It can be reduced to (1.1) as m = 1.

The purpose of this article is to study the well posedness, local weak solution, and blow-up for Cauchy problem (1.2) and (1.3). System (1.2) also conserves conservation laws. Our starting point is to obtain the local well posedness by using Kato's theory, Next, we derive some blow-up results of the solutions by the following transport equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M4">View MathML</a>

which is a crucial ingredient to obtain the blow-up phenomenon. Last, by using the conserves from laws and the contraction mapping theorem, we obtain the existence of weak solutions of Cauchy problem (1.2) and (1.3). These methods are similar to that was used in [18]. However, because of the asymmetry and the high strength of the nonlinearity of Equation (1.3), it is more difficult to estimate the norm of u, ρ, ux, ρx in Sobolev space. In addition, also we get Equation (5.10) which is different with that in [18]. As for the blow-up phenomenon, we get some new results of (1.2) and (1.3).

Guan and Yin [17,19] got the global weak solutions for two-component Camassa-Holm shallow water system; they first obtained approximate solutions for the system, then they prove the compactness of these solutions, and at last they got the global weak solutions. Using the same way, Liu and Yin [20] also got global weak solutions for a periodic two-component μ-Hunter-Saxton system. However, in this article, we add high-order perturbation terms in this system, and by using the conserves laws and the contraction mapping theorem, we obtain the existence of weak solutions.

The remainder of this article is organized as follows. Section 2 is the preliminary. In Section 3, the local well posedness for strong solution of Cauchy problem (1.2) and (1.3) is established by Kato's theory. In Section 4, by transport equation, some blow-up results of the solutions of Cauchy problem (1.2) and (1.3) are obtained. The proof of existence of local weak solution is carried out in Section 5.

2. Blow-up

Lemma 2.1: Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M5">View MathML</a>, s > 5/2, then there exists a maximal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M6">View MathML</a>, and a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M7">View MathML</a> to Cauchy problem (1.2) and (1.3) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M8">View MathML</a>

Moreover, the solution depends continuously on the initial data, i.e., the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M9">View MathML</a>

is continuous.

The proof is similar with Theorem 4.1 in [21].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M10">View MathML</a>, then (1.2) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M11">View MathML</a>

(2.1)

Consider the following initial value problem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M12">View MathML</a>

(2.2)

where u is the first component of the solution z to Equation (1.2).

To prove the blow-up result, we need the following lemma.

Lemma 2.2: Let z0∈ Hs × Hs-1, (s > 5/2), and let T > 0 be the maximal existence time of the corresponding solution z to Equation (2.1), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M13">View MathML</a>

(2.3)

Proof. Differentiating the left-hand side of Equation (2.3) with respect t. It follows from (2.1) and (2.2), that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M14">View MathML</a>

This completes the proof of this lemma.

Theorem 2.1: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M15">View MathML</a>, (s > 5/2), and T be the maximal time of the solution z to Equation (1.2) with the initial data z0. Assume that there exists x0 R such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M16">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M17">View MathML</a>

Then, T is finite and the slope of u tends to negative infinity as t goes to T while u is uniformly bounded on [0, T).

Proof. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M18">View MathML</a> be the solution of Equation (2.1) with the initial data z0, and T be the maximal time of z, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M19">View MathML</a>

From (2.1) and (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M20">View MathML</a>

Differentiating the first equation in (2.1) with respect x, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M21">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M22">View MathML</a>, Then, by Lemma4.1, we have γ(t) = 0, ∀t ∈[0, T).

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M23">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M24">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M26">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M27">View MathML</a>. If let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M28">View MathML</a> then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M29">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M30">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M31">View MathML</a>.

With the inequality above, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M32">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M33">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M34">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M35">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M36">View MathML</a>.

This completes the proof of the theorem.

3. Local weak solution

Definition 3.1: ([22]) Let (u0, ρ0) ∈ H1(R) × H1(R). If (u, ρ) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M37">View MathML</a> and satisfies the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M38">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M39">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M40">View MathML</a>, the set of all the restrictions to ([0,T) × R) × ([0,T) × R) of smooth functions on R2 × R2 with compact support contained in ((-T, T) × R) × ((-T, T) × R). Then, z is called a weak solution to Equation (1.6). If z is a weak solution on [0, T) × [0, T) for every T > 0, then it is called global weak solution to Equation (1.6).

In this section, we discuss the existence of weak solution of Cauchy problem (1.2) and (1.3). To this purpose, we consider the following Cauchy problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M41">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M42">View MathML</a>

(3.2)

where ε is a constant satisfying 0 < ε < 1/4. Note that when ε = 0, system (3.1) and (3.2) is just the system (1.2) and (1.3).

For any 0 < ε < 1/4 and s ≥ 1, the integral operators

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M43">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M44">View MathML</a>

define two bounded linear operator in the indicated Sobolev spaces.

To prove the existence of solutions to the problem (3.1) and (3.2), we apply the two operators above to both sides of (3.1) and then integrate the resulting equations with regard to t. This leads to the following equations.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M45">View MathML</a>

A standard application of the contraction mapping theorem leads to the following existence result.

Theorem 3.1: For each initial data u0 Hs (s ≥ 1), ρ0 Hs-1(s ≥ 2), there exists a T > 0 depending only on the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M47">View MathML</a>, and m such that there exists a unique solution (u, ρ) ∈ C([0,T];Hs) × C([0, T];Hs-1) of system (3.1) and (3.2) in the sense of distribution. If u0 Hs (s ≥ 2), ρ0 Hs-1(s ≥ 3), the solution (u, ρ) ∈ C([0,∞];Hs) × C([0, ∞];Hs-1) exists for all time, in particular, when u0 Hs (s ≥ 4), ρ0 Hs-1(s ≥ 5), the corresponding solution is a classical globally defined solution of (3.1) and (3.2).

The global existence result follows from the conservation law

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M48">View MathML</a>

admitted by (3.1) in its integral form.

Theorem 3.2: Suppose that for some s ≥ 4, the function pair u(x, t) and ρ(x, t) in the solution of Equation (3.1) corresponding to the initial data u0 Hs (s ≥ 4); ρ0 Hs-1(s ≥ 5), then the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M49">View MathML</a>

(3.3)

For any real number q ∈ (1, s] (s ≥ 5), there exists a constant c depending only on q, m, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M50">View MathML</a>

(3.4)

For any q ∈ (1, s-1] (s ≥ 4), there exists a constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M51">View MathML</a>

(3.5)

And for any q ∈ (1, s-2] (s ≥ 5), there exists a constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M52">View MathML</a>

(3.6)

Proof. It is obvious that (3.3) holds. In order to prove (5.4), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M53">View MathML</a>. We rewrite Equation (3.1) in the following equivalent form.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M54">View MathML</a>

(3.7)

For any q ∈ (1, s] (s ≥ 5), applying (Λq uq to the both sides of the first equation of Equation (3.7), respectively, and integrating with regard to x, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M55">View MathML</a>

By using Sobolev embedding theorems, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M56">View MathML</a>

where we have used lemma in [23] with r = q -2 > 0. Also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M57">View MathML</a>

where we have used Lemma in [24] with r = q -1 > 0.

Then, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M58">View MathML</a>

(3.8)

For any q ∈ (1, s-1] (s ≥ 5), applying (Λq-1 ρ) Λq-1 to the both sides of the second equation of Equation (3.7), respectively, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M59">View MathML</a>

(3.9)

Summing up (3.8) and (3.9), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M60">View MathML</a>

For any q ∈ (1, s-1] (s ≥ 4), applying (Λq utq to the both sides of the first equation of Equation (3.7), respectively, and integrating with regard to x, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M61">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M62">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M63">View MathML</a>

where we have used lemma in [24] with r = q -1 > 0. Then, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M64">View MathML</a>

For any q ∈ (1, s-2] (s ≥ 5), applying (Λq-1 ρtq-1 to the both sides of the second equation of Equation (5.7), respectively, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M65">View MathML</a>

This complete the proof of the theorem.

Suppose u0 Hs (s ≥ 1), ρ0 Hs-1(s ≥ 2), and let uε0, ρε0 be the convolution uε0 = φε*u0, ρε0 = φε*ρ0, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M66">View MathML</a> such that the Fourier transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M67">View MathML</a> of φ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M69">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M70">View MathML</a> for any ξ ∈ (-1,1). Then, it follows from Theorem 3.1 that for each ε with 0 < ε <1/4, the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M71">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M72">View MathML</a>

has a unique solution uε (t, x) ∈C([0,∞);Hand ρε (t, x) ∈C([0,∞);H. We first demonstrate the properties of the initial data uε0, ρε0 in the following lemma. The proof is similar to Lemma 5 in [25].

Lemma 3.1: Under the above assumptions, there hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M73">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M74">View MathML</a>

for any ε with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M75">View MathML</a>, where c is a constant independent of ε. The proof is similar to Lemma 5 in [25].

Theorem 3.3: Suppose that u0(x) ∈ Hs(R), s ∈ [1, 3/2]; ρ0(x) ∈ Hs-1(R),

s-1 ∈ [1, 3/2] such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M77">View MathML</a>. Let uε0 = φε*u0, ρε0 = φε*ρ0, be defined the same as above. Then, there exist constants T > 0 and c > 0 independent of ε such that the corresponding solution uε, ρε of (3.10) satisfy the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M79">View MathML</a> for any t ∈ [0,T).

Proof. Use Equation (3.7) with u = uε, ρ = ρε. Differentiating with respect to x on both sides of the first equation in Equation (3.7). Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M80">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M81">View MathML</a>

Let n > 0 be an integer. Then, multiplying the above equation by (ux)2n+1 to integrate with respect to x, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M82">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M83">View MathML</a>.

It follows from Hölder inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M84">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M85">View MathML</a> as p→∞ for any f LL2. Integrating the above inequality over R with respect to t, and taking the limitation as n→∞, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M86">View MathML</a>

It follows from (3.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M87">View MathML</a>

For any given r ∈ (1/2,1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M88">View MathML</a>

Then from (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M89">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M90">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M91">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M92">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M94">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M95">View MathML</a>

Then, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M96">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M97">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M98">View MathML</a> where c is a constant depends on Λ-2 and m.

Also, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M99">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M100">View MathML</a>. From (3.9), we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M101">View MathML</a>

It follows from the contraction mapping theorem that there exists a constant T > 0 such that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M102">View MathML</a>

(3.11)

has a unique solution f(t) ∈ C [0,T]. Theorem II in Section I.1 in [26] shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M103">View MathML</a> for any t ∈ [0,T] which leads to the conclusion of this theorem.

Let u = uε, ρ = ρε, with (3.4) used

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M104">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M106">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M107">View MathML</a>

where q ∈ (0, s], r ∈ (0, s-1], t ∈ [0,T].

Then, it follows from Aubin's compactness theorem [27] that there exist subsequences of {uε}, {ρε} denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M109">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M109">View MathML</a> are weakly convergent to u(t, x)∈ L2([0,T]; Hs), ρ(t, x)∈ L2([0,T]; Hs-1), respectively, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M111">View MathML</a> are weakly convergent to ut(t, x)∈ L2([0,T]; Hs-1), ρt(t, x)∈ L2([0,T]; Hs-2), respectively. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108">View MathML</a> are weakly convergent to u(t, x)∈ L2([0,T]; Hs), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M112">View MathML</a> for any f∈ (L2([0,T]; Hs))* = L2([0,T]; Hs) when n → ∞. Applying Riesz lemma, we conclude that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M113">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M114">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M115">View MathML</a> as n → ∞, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M116">View MathML</a>. Then for any real R > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108">View MathML</a> converges strongly to u L2([0,T]; Hq(-R, R)) for any q ∈ [0, s-1); and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M110">View MathML</a> converges to ut strongly in L2([0,T]; Hr(-R, R)) for any r ∈ [0, s-1]. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M117">View MathML</a> for any g∈ (L2([0,T]; Hs))* = L2([0,T]; Hs) as n → ∞. By Riesz lemma, we conclude that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M118">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M119">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M120">View MathML</a> as n → ∞, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M121">View MathML</a>. Then for any real R > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M108">View MathML</a> converges strongly to ρ L2 ([0,T]; Hq-1(-R, R)) for any q ∈ [0, s-1), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M111">View MathML</a> converge to ut, ρt strongly in L2([0,T]; Hr-1(-R, R)) for any r ∈ [0, s-1]. Hence, the existence of a weak solution to the Cauchy problem (1.2) and (1.3) is established.

Theorem 3.4: Let u0(x) ∈Hs(R)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M122">View MathML</a> and ρ0(x) ∈Hs-1(R) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M123">View MathML</a>, which satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M125">View MathML</a>. Then there exists a constant T > 0 such that the Cauchy problem (1.2) and (1.3) with the initial data has a solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M126">View MathML</a>

in the sense of distribution. And ux, ρx L([0,T] × R).

Proof. It follows from Theorem 3.3 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128">View MathML</a> are bounded in the space L. Hence, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M136">View MathML</a> are also weakly convergent to u2, ρ2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M138">View MathML</a>, uρ, uxρx, uxρ, uρxL2([0,T]; Hr(-R, R)) for any r ∈ [0, s-1] and R > 0, respectively. Therefore, u, ρ satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M139">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M140">View MathML</a>

with u(0,x) = u0(x), ρ(0,x) = ρ0(x), and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M141">View MathML</a>. Moreover, since X = L1([0,T] × R) is a separable Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128">View MathML</a> are bounded sequences in the dual space X* = L([0,T] × R) of X, there are two subsequences of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128">View MathML</a> (still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128">View MathML</a>) weak star convergent to two functions U, P L([0,T] × R), respectively. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/52/mathml/M128">View MathML</a> are also weakly convergent to ux, ρx L([0,T] × R), respectively. It follows that ux = U, ρx = P hold almost everywhere. Hence, ux, ρx L([0,T] × R).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

LT raised the modified two-component Camassa-Holm equations and conceived the local weak solution of the equations. MZ carried out the solution of the weak solution and its blow-up phenomenon. All authors read and approved the final manuscript.

Acknowledgements

The study was supported by the National Nature Science Foundation of China (No. 11171135, 71073072), the Nature Science Foundation of Jiangsu (No. BK 2010329), the Project of Excellent Discipline Construction of Jiangsu Province of China, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 09KJB110003).

References

  1. Ivanov, R: Extended Camassa-Holm hierarchy and conserved quantities. Z Naturforsch A. 61, 133–138 (2006)

  2. Constantin, A, Ivanov, RI: On an integrable two-component Camassa-Holm shallow water system. Phys Lett A. 372, 7129–7132 (2008)

  3. Ivanov, R: Two-component integrable systems modelling shallow water waves. Oberwolfach Rep. 6, 429–462 (2009)

  4. Holm, DD, Ivanov, RI: Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples. J Phys A. 43, 492001 doi:10.1088/1751-8113/43/49/492001 (2010)

  5. Olver, P, Rosenau, P: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E. 53, 1900–1906 (1996)

  6. Chen, M, Liu, SQ, Zhang, Y: A 2-component generalization of the Camassa-Holm equation and its solutions. Lett Math Phys. 75, 1–15 (2006)

  7. Holm, DD, Náraigh, LÓ, Tronci, C: Singular solutions of a modified two-component CH equation. Phys Rev E. 79, 016601 (2009)

  8. Liu, SQ, Zhang, Y: Deformations of semisimple bi-hamiltonian structures of hydrodynamic type. J Geom Phys. 54, 427–453 (2005)

  9. Chen, RM, Liu, Y: Wave breaking and global existence for a generalized two-component Camassa-Holm. Int Math Res Notes. 6, 1381–1416 (2011)

  10. Henry, D: Infinite propagation speed for a two component Camassa-Holm equation. Discr Contin Dyn Syst Ser B. 12, 597–606 (2009)

  11. Zhang, P, Liu, Y: Stability of solitary waves and wave-breaking phenomena for the two-component CH system. Int Math Res Notes. 11, 1981–2021 (2010)

  12. Guan, C, Yin, ZY: Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system. J Diff Equ. 248, 2003–2014 (2010)

  13. Gui, G, Liu, Y: On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. J Funct Anal. 258, 4251–4278 (2010)

  14. Gui, G, Liu, Y: On the Cauchy problem for the two-component Camassa-Holm system. Math Z. 268, 45–66 (2010)

  15. Escher, J: Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discr Contin Dyn Syst. 19, 493–513 (2007)

  16. Chen, RM, Liu, Y: Wave-breaking and global existence for a generalized two-component Camassa-Holm system. Math Phys Sci Int Math Res Notes. 6, 1381–1416 (2011)

  17. Guan, C, Yin, Z: Global weak solutions for a two-component Camassa-Holm shallow water system. J Funct Anal. 260, 1132–1154 (2011)

  18. Fu, Y, Qu, C: Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons. J Math Phys. 50, 012906 doi:10.1063/1.3064810 (2009)

  19. Guan, C, Yin, Z: Global weak solutions for a modified two-component Camassa-Holm shallow water system. Ann I.H. Poincare-AN doi:10.1016/j.anihpc.2011.04.003 (2011)

  20. Liu, J, Yin, Z: Global weak solutions for a periodic two-component μ-Hunter-Saxton system. Math AP. 3, arxXiv 1012.5452v3 (2011)

  21. Fu, Y, Liu, Y, Qu, C: Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons. Math Ann. 348, 415–448 (2000)

  22. Zhang, S, Yin, Z: Global weak solutions for the Dullin-Gottwald-Holm equation. Nonlinear Anal. 72, 1690–1700 (2010)

  23. Kato, T, Ponce, G: Communtator estimates and the Euler and Navier-Stokes equations. Commun Pure Appl Math. 41, 891–907 (1988)

  24. Kato, T: On the Korteweg-de Vries equation. Manuscr Math. 28, 89–99 (1979)

  25. Constantin, A: Existence of permanent and breaking waves for a shallow water equation. A geometric approach Ann Inst Fourier (Grenoble). 50, 321–362 (2000)

  26. Water, W: Differential and Integral Inequalities.p. 362. Springer-Verlag, New York (1970)

  27. Lions, JL: Quelques méthodes de résolution des problemes aux limites non linéaires, Gauthier-Villars, Paris. 574 (1969)