The local well-posedness, blow-up criteria and Gevrey regularity of solutions for a two-component high-order Camassa-Holm system

This paper studies the Cauchy problem for a two-component high-order Camassa Holm system proposed in Escher and Lyons (2015). First, we investigate the local well-posedness of the system in the Besov spaces B-p,r(s) x B-p,r(s-2) with s > max{3+1/p, 7/2, 4-1/p} and p, r is an element of [1, infinity]. Second, by means of the Littlewood-Paley decomposition technique and the conservative property at hand, we derive a blow-up criteria for the strong solution. Finally, we study the Gevrey regularity and analyticity of the solutions to the system in the Gevrey-Sobolev spaces. In particular, we get a lower bound of the lifespan and the continuity of the data-to-solution mapping. (C) 2016 Elsevier Ltd. All rights reserved.