Global existence and well-posedness of the 2D viscous shallow water system in Besov spaces. (August 2015) - Liu, Yanan; Yin, Zhaoyang
Abstract: 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 B p, r s ( R 2 ), 1 ≤ p ≤ ∞, s > m a x 1, 2 p , 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 B p, r s ( R 2 ), 1 ≤ p ≤ 2, s > 2 p, 1 ≤ r < ∞ . Our obtained results improve considerably the recent results in Wang and Xu (2005), and Liu and Yin (2014).