Global Analysis for Rough Solutions to the Davey-Stewartson System

Abstract

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is in $H^s$ with s > 2/5, then there exists a global solution in time, and the $H^s$ norm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-time $L_{t,x}^4$ estimate for nonlinear equation with the relatively general defocusing power nonlinearity.