International Journal of Partial Differential Equations
Volume 2013 (2013), Article ID 834912, 13 pages
On the Local Well-Posedness of the Cauchy Problem for a Modified Two-Component Camassa-Holm System in Besov Spaces
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.
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 .
The following modified Camassa-Holm equation with cubic nonlinearity was proposed as a integrable system by Fuchssteiner  and Olver and Rosenau  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  from the two-dimensional Euler equations, where the variables and represent, respectively, the velocity of the fluid and its potential density. Qiao also  obtained the cuspons and -shape-peaks solitons of (1). In , it was shown that (1) admits a Lax pair and hence can be solved by the method of inverse scattering. Fu et al.  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.  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) .
Recently, Song et al.  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 , the Degasperis-Procesi equation , and the Novikov equation . 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  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.
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 , 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