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

Abstract

In this paper, we consider the Cauchy problem for the 2D viscous shallow water system in , . 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 , . Our obtained result generalizes and improves considerably some recent results.