International Journal of Partial Differential Equations

Volume 2013 (2013), Article ID 834912, 13 pages

http://dx.doi.org/10.1155/2013/834912

Research Article

On the Local Well-Posedness of the Cauchy Problem for a Modified Two-Component Camassa-Holm System in Besov Spaces

Jiangbo Zhou,¹ Lu Yao,¹ Lixin Tian,¹ and Wenbin Zhang²

¹Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013, China

²Taizhou Institute of Science and Technology, NUST, Taizhou, Jiangsu 225300, China

Received 26 April 2013; Accepted 14 November 2013

Academic Editor: Athanasios N. Yannacopoulos

Copyright © 2013 Jiangbo Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the Cauchy problem for an integrable modified two-component Camassa-Holm system with cubic nonlinearity. By using the Littlewood-Paley decomposition, nonhomogeneous Besov spaces, and a priori estimates for linear transport equation, we prove that the Cauchy problem is locally well-posed in Besov spaces with , and .

1. Introduction

The following modified Camassa-Holm equation with cubic nonlinearity

was proposed as a integrable system by Fuchssteiner [1] and Olver and Rosenau [2] by applying the general method of tri-Hamiltonian duality to the bi-Hamiltonian representation of the modified Korteweg-de Vries equation. Later, it was obtained by Qiao [3] from the two-dimensional Euler equations, where the variables and represent, respectively, the velocity of the fluid and its potential density. Qiao also [3] obtained the cuspons and -shape-peaks solitons of (1). In [4], it was shown that (1) admits a Lax pair and hence can be solved by the method of inverse scattering. Fu et al. [5] showed that the Cauchy problem of (1) is locally well-posed in a range of the Besov spaces. They determined the blow-up scenario and the lower bound of the maximal time of existence. They also described a blow-up mechanism for solutions with certain initial profiles and the nonexistence of the smooth traveling wave solutions was also demonstrated. In addition, they obtained the persistence properties of the strong solutions for (1). Gui et al. [6] investigated the formation of singularities and showed that singularities of the solutions occur only in the form of wave breaking. They obtained a new wave-breaking mechanism for solutions with certain initial profiles. It was proved that the peaked functions of the form

are global weak solutions to (1) [6].

Recently, Song et al. [7] suggested a new integrable two-component vision of (1) as follows:

with , , and . Apparently, it reduces to (1) when . They showed that this system has Lax-pair and is also geometrically integrable. As a consequence of geometric integrability, its conservation laws were constructed by expanding the pseudopotential. Finally, the cuspons and -shape solitons of system (3) were obtained.

In this paper, we are interesting in the local well-posedness of the following Cauchy problem for (3)

with , , and . Although Kato’s theory is a useful method to obtain the local well-posedness of the Cauchy problem in Sobolev space for lots of equations, such as the Camassa-Holm equation [8], the Degasperis-Procesi equation [9], and the Novikov equation [10]. However, it seems to be unapplicable to the Cauchy problem (4). Fortunately, by using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem (4) is locally well-posed in the Besov space with , , and . As a corollary, it is locally well-posed in the Sobolev space with . The Littlewood-Paley decomposition and nonhomogeneous Besov spaces which were introduced in [11] have been used to establish the well-posedness of the Euler equations, the Navier-Stokes equations, the Camassa-Holm equation, the Novikov equation, the two-component Camassa-Holm system, the two-component Degasperis-Procesi system, and the coupled Camassa-Holm equations; see [12–20].

The rest of the paper is organized as follows. In Section 2, we recall some basic results on Besov spaces and linear transport equation. In Section 3, we establish the local well-posedness of the Cauchy problem (4) in a range of Besov spaces. Some open problems are presented in Section 4.

Notation. Since all spaces of functions are over , we drop in our notations of function spaces for simplicity. For a given Banach space , we denote its norm by . We denote or the Fourier transform of the function . Let denote the space of all times continuously differentiable functions defined on with values in space . denotes the space of functions which belong to as a function of for every fixed and and the space of functions which belong to as a function of for every fixed and . denotes the tempered distribution spaces.

2. Preliminaries

In this section, we recall some results on the Littlewood-Paley decomposition, the nonhomogeneous Besov spaces, and the linear transport equation which can be seen in [10–20].

Proposition 1 (Littlewood-Paley decomposition). Let and . There exist two radial functions and such that

Furthermore, let , and define the dyadic operators and low-frequency cut-off operators as follows:

Then, for all , the nonhomogeneous Littlewood-Paley decomposition of a distribution is defined

Definition 2 (Besov space). Let and , . The inhomogenous Besov space ( for short) is defined by

where

If , .

Proposition 3 (properties of Besov space). Let and , ; then the following properties hold.

(1) Topological properties: is a Banach space which is continuously embedded in .

(2) Density: is dense in , .

(3) Sobolev embeddings: if and , then . If , and , then the embedding is locally compact.

(4) Algebraic properties: for , is an algebra. Moreover, is an algebra or ( and ).

(5) Fatou property: if is a bounded sequence of which tends to in , then and

(6) Complex interpolation: if and , , then and

(7) Generalized derivatives: let and be a homogeneous function of degree away from a neighborhood of the origin. There exists a constant depending only on and such that

(8) 1-D Moser-type estimates:

(i)for ,

(ii)for , if and ,

where are constants independent of and .

Lemma 4 (a prior estimates for 1-D linear transport equation). Let , , and . Assume that , and if or otherwise. If solves the following 1-D linear transport equation

then there exists a constant depending only on , , and such that the following statements hold:

(1) If or ,

or

where

(2) If , , and , then

with .

(3) If , then, for all , estimate (17) holds true with .

(4) If , then .

(5) If , then for all .

Lemma 5 (existence and uniqueness). Let , , , , and be as in the statement of Lemma 4. Assume that for some and and if or and and if . Then the problem (15) has a unique solution and the inequalities of Lemma 4 can hold true. Moreover, if , then .

3. Local Well-Posedness

In this section, we discuss the local well-posedness of the Cauchy problem (4).

For , , and , we denote

Our main local well-posedness result is the following theorem.

Theorem 6. Suppose that , , , and . Then, there exists a time such that the Cauchy problem (4) has a unique solution and the map is continuous from a neighborhood of in into for every when and whereas .

Proof. We divide the proof into seven steps. In the following, we denote a generic constant only depending on , , and .

Step 1 (approximate solution). We use a standard iterative process to construct a solution. Starting from , by induction, we define a sequence of smooth functions by solving the following linear transport equations:

with .

Since all the data belong to , Lemma 5 enables us to show by induction that, for all , system (21) has a global solution which belongs to .

Step 2 (uniform bounds). Applying (17) to the first equation in (21) we have

Since is an algebra when , we derive from Proposition 3(7) that

It follows from (22)-(23) that

Similarly, we have

Combining (24) and (25) we arrive at

Choose a such that and suppose that for all ,

Then we have

which along with (26) leads to

Therefore, we conclude that .

If , then is an algebra. Hence, according to Proposition 3(7) we have

If , using the Moser-type estimate (14), we obtain

Thus, using the first equation in (21) and combining (30)-(31) with (29), we have

In a similar way, we can obtain that

Thus, (32) and (33) imply that . Therefore

Step 3 (convergence). Now we are going to show that is a Cauchy sequence in . In fact, according to (21), for all , we obtain that

where

According to (17), for every , the following inequality holds:

is equivalent to

If , using the Moser-type estimate (14), we obtain

If , then is an algebra, and we can verify that (39)–(42) also hold true. Thus, we have

Plugging (39) and (43) into (38) we get

Similarly, we can derive that

Combining (44) and (45), we have

Since

which can be seen on page 2142 of [20], and is uniformly bounded in , then there exists a constant independent of and such that, for all ,

Arguing by induction with respect to the index , we can see that

Since and are bounded independently of , we conclude that there exists some new constant independent of and such that

Hence is a Cauchy sequence in and converges to some function .

Step 4 (existence of solution in ). We will prove that belongs to and satisfies (4). Since in uniformly bounded in , the Fatou property for the Besov spaces (Proposition 3(5)) guarantees that also belongs to .

On the other hand, as converges to in , we conclude that the convergence holds in for any . In fact, if , then

If , then Proposition 3(6) gives us

where .

Passing to the limit in (21) and we can easily conclude that is indeed a solution to the Cauchy problem (4). Thanks to the fact that , the right-hand side of the equations

belongs to . In particular, for the case , Lemma 5 enables us to conclude that for any . Finally, using (53) again, we see that if , and in otherwise. Consequently, belongs to .

Step 5 (uniqueness of solution). Let and be two given solutions of the Cauchy problem (4) with the initial data satisfying .

Denote , , and . It is obvious that , which implies that , and , , and solves the transport equations

where

Applying (17) to the first equation in (54), we obtain that

which is equivalent to

Since is an algebra when , we get

If , using the Moser-type estimate (14), we have

If , then is an algebra. We directly obtain that

It follows from (58), (60), and (62) that

Inserting (63) and (59) (or (61)) into (57), we find that

For the component in (54), we derive the similar result as follows:

Then, it follows from (64) and (65) that

Applying Gronwall’s lemma, we end up with

Step 6 (continuity with respect to the initial data in with ). When , the continuity with respect to the initial data can be obtained from Step 5.

When , by using Proposition 3(6) and (67) we have

where , and .

When , by using Proposition 3(6) and (67) again, we have

where .

Step 7 (continuity with respect to the initial data in ). A standard use of a sequence of viscosity approximate solutions for the Cauchy problem (4) which converges uniformly in gives the continuity of the solution in .

Consequently, we complete the proof of the theorem.

When , the Besov space coincides with the Sobolev space . So we have the following corollary.

Corollary 7. Suppose that and . Then there exists a time such that the Cauchy problem (4) has a unique solution and the map is continuous from a neighborhood of in into

4. Open Problems

In this paper, we use some standard a priori estimates for linear transport equation to prove the local well-posedness of the Cauchy problem for a modified two-component Camassa-Holm system with cubic nonlinearity in Besov spaces with minimal regularity assumptions on the initial data.

There are some interesting but challenging problems which are still open. The first is whether the index is sharp for local well-posedness of the Cauchy problem (4). The second is whether the solution can exist globally in time under certain assumptions on the initial data. The third is whether the solution can blow up in finite time with special initial profiles. We will work on these problems in the near future.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (nos. 11171135, 10420130638), the Nature Science Foundation of Jiangsu (no. BK 2010329), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), and the Taizhou Social Development Project (no. 2011213).

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