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.

Keywords:

• the 2D viscous shallow water system
• the Littlewood–Paley theory
• local well-posedness
• Sobolev spaces
• global existence

AMS Subject Classifications:

• 49K40
• 42B25
• 35A01
• 35B44
• 30H25

Acknowledgement

The authors thank the referee for valuable comments and suggestions.

Additional information

Funding

This work was partially supported by NNSFC (No.11271382), RFDP (No. 20120171110014), the Macao Science and Technology Development Fund (No. 098/2013/A3), and the key project of Sun Yat-sen University.