Global existence and well-posedness of the 2D viscous shallow water system in Sobolev spaces with low regularity

In this paper we consider the Cauchy problem for the 2D viscous shallow water system in Hs(R2), s>1. We first prove the local well-posedness of this problem by using the Littlewood-Paley theory, the Bony decomposition, and the theories of transport equations and transport diffusion equations. Then, we get the global existence of the system with small initial data in Hs(R2), s>1. Our obtained result improves considerably the recent result in [14].