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

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