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

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 ^{s }^{s-1},

1. Introduction

The well-known two-component Camassa-Holm equations

where _{xx }

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

with initial data

where

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,

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 _{x}, ρ_{x }

Guan and Yin

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

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

is continuous.

The proof is similar with Theorem 4.1 in

Let

Consider the following initial value problem,

where

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

**Lemma 2.2**: Let _{0}^{s }^{s-1}, (

** Proof**. Differentiating the left-hand side of Equation (2.3) with respect

This completes the proof of this lemma.

**Theorem 2.1**: Let _{0}. Assume that there exists _{0 }∈

Then,

** Proof**. Let

From (2.1) and (2.2), we have

Differentiating the first equation in (2.1) with respect

Note that

Thus

Since

Note that

Since

With the inequality above, we get

Since

This completes the proof of the theorem.

3. Local weak solution

**Definition 3.1**: (_{0}, _{0}) ∈ ^{1}(^{1}(

for all ^{2 }× ^{2 }with compact support contained in ((-

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:

where

For any 0 <

and

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

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

**Theorem 3.1: **For each initial data _{0 }∈ ^{s }_{0 }∈ ^{s-1}(^{s}^{s-1}) of system (3.1) and (3.2) in the sense of distribution. If _{0 }∈ ^{s }_{0 }∈ ^{s-1}(^{s}^{s-1}) exists for all time, in particular, when _{0 }∈ ^{s }_{0 }∈ ^{s-1}(

The global existence result follows from the conservation law

admitted by (3.1) in its integral form.

**Theorem 3.2: **Suppose that for some _{0 }∈ ^{s }_{0 }∈ ^{s-1}(

For any real number

For any

And for any

** Proof**. It is obvious that (3.3) holds. In order to prove (5.4), let

For any ^{q }u^{q }

By using Sobolev embedding theorems, we have

where we have used lemma in

where we have used Lemma in

Then, we get

For any ^{q-1 }ρ^{q-1 }

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

For any ^{q }u_{t}^{q }

and

where we have used lemma in

For any ^{q-1 }_{t}^{q-1 }to the both sides of the second equation of Equation (5.7), respectively, then we obtain

This complete the proof of the theorem.

Suppose _{0 }∈ ^{s }_{0 }∈ ^{s-1}(_{ε0}, _{ε0 }be the convolution _{ε0 }= _{ε}_{0}, _{ε0 }= _{ε}_{0}, where

has a unique solution _{ε }^{∞}([0,∞);^{∞ }and _{ε }^{∞}([0,∞);^{∞}. We first demonstrate the properties of the initial data _{ε0}, _{ε0 }in the following lemma. The proof is similar to Lemma 5 in

**Lemma 3.1: **Under the above assumptions, there hold

for any

**Theorem 3.3**: Suppose that _{0}(^{s}_{0}(^{s-1}(

_{ε0 }= _{ε}_{0}, _{ε0 }= _{ε}_{0}, be defined the same as above. Then, there exist constants _{ε}, ρ_{ε }

** Proof**. Use Equation (3.7) with

Let _{x}^{2n+1 }to integrate with respect to

where

It follows from

Note that ^{∞}∩^{2}. Integrating the above inequality over

It follows from (3.3) that

For any given

Then from (3.4), we have

and

Thus,

and

Then, we get

It follows that

^{-2 }and

Also, we can obtain

where

It follows from the contraction mapping theorem that there exists a constant

has a unique solution

Let _{ε}, ρ _{ε}

where

where

Then, it follows from Aubin's compactness theorem _{ε}_{ε}^{2}([0,^{s}^{2}([0,^{s-1}), respectively, and _{t}^{2}([0,^{s-1}), _{t}^{2}([0,^{s-2}), respectively. Because ^{2}([0,^{s}^{2}([0,^{s}^{2}([0,^{s}

Since ^{2}([0,^{q}_{t }^{2}([0,^{r}^{2}([0,^{s}^{2}([0,^{s}

Since ^{2 }([0,^{q-1}(-_{t}, ρ_{t }^{2}([0,^{r-1}(-

**Theorem 3.4: **Let _{0}(^{s}_{0}(^{s-1}(

in the sense of distribution. And _{x}, ρ_{x }^{∞}([0,

** Proof. **It follows from Theorem 3.3 that

and

with _{0}(_{0}(^{1}([0,^{∞}([0,^{∞}([0,_{x}, ρ_{x }^{∞}([0,_{x }_{x }_{x}, ρ_{x }^{∞}([0,

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).