Global Smoothing for the Davey-Stewartson System on $\mathbb{R}^2$

TL;DR

This paper investigates the regularity and smoothing effects of solutions to the Davey-Stewartson system, demonstrating that the nonlinear component becomes smoother over time and analyzing the system's long-term behavior.

Contribution

It establishes new smoothing estimates for solutions and provides a novel proof of the existence and smoothness of global attractors in the energy space.

Findings

Nonlinear part of solutions is smoother than initial data for all times.

Sobolev norm of the nonlinear evolution grows at most polynomially.

Existence and smoothness of global attractors are confirmed in the energy space.

Abstract

In this paper we study the regularity properties of solutions to the Davey-Stewartson system. It is shown that for initial data in a Sobolev space, the nonlinear part of the solution flow resides in a smoother space than the initial data for all times. We also obtain that the Sobolev norm of the nonlinear part of the evolution grows at most polynomially. As an application of the smoothing estimate, we study the long term dynamics of the forced and weakly damped Davey-Stewartson system. In this regard we give a new proof for the existence and smoothness of the global attractors in the energy space.