Global existence and well-posedness of the 2D viscous shallow water system in Besov spaces

In this paper we consider the Cauchy problem for the 2D viscous shallow water system in Besov spaces. We first establish the local well-posedness of this problem in Bp,rs(R2), 1≤p≤∞1≤p≤∞, s>max{1,2p}, 1≤r<∞1≤r<∞ by using the Littlewood–Paley theory, the Bony decomposition and the theories of transport equations and transport–diffusion equations. Then by the obtained local well-posedness result, we can prove the global existence of the system with small enough initial data in Bp,rs(R2), 1≤p≤21≤p≤2, s>2p, 1≤r<∞1≤r<∞. Our obtained results improve considerably the recent results in Wang and Xu (2005), and Liu and Yin (2014).