Global existence and well-posedness of 2D viscous shallow water system in Sobolev spaces with low regularity

TL;DR

This paper establishes local and global well-posedness of the 2D viscous shallow water equations in Sobolev spaces with low regularity, using harmonic analysis techniques, and improves previous results.

Contribution

It proves the global existence of solutions for small initial data in Sobolev spaces with low regularity, advancing the understanding of the system's well-posedness.

Findings

Local well-posedness in $H^s$ for $s>1$

Global existence for small initial data

Improves previous regularity requirements

Abstract

In this paper we consider the Cauchy problem for 2D viscous shallow water system in $H^s(\mathbb{R}^2)$, $s>1$. 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 $H^s(\mathbb{R}^2)$, $s>1$. Our obtained result improves the recent result in \cite{W}