Global existence and well-posedness of the 2D viscous shallow water system in Besov spaces. (August 2015)

• Record Type: Journal Article
• Authors: Liu, Yanan; Yin, Zhaoyang
• Is Part Of: Nonlinear analysis. Volume 24(2015)
• Page Start: 1
• Page End: 17
• Publication Date: 2015-08
• Subjects: The 2D viscous shallow water system -- Littlewood–Paley theory -- Besov spaces -- Local well-posedness -- Global existence; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2014.12.005
• Languages: English
• ISSNs: 1468-1218

Abstract

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).