Global well-posedness for Schr\"odinger equation with derivative in $H^{{1/2}}(\R)$

TL;DR

This paper proves global well-posedness for the cubic nonlinear Schrödinger equation with derivative in the critical Sobolev space H^{1/2}( ), extending previous local and global results to the critical case using advanced analytical techniques.

Contribution

It establishes the global well-posedness of the Schrödinger equation with derivative in H^{1/2}( ), employing a novel combination of the third generation I-method and resonant decomposition.

Findings

Global well-posedness in H^{1/2}( ) established

Advanced analytical techniques successfully control resonant interactions

Extends previous results from s > 1/2 to the critical case s=1/2

Abstract

In this paper, we consider the Cauchy problem of the cubic nonlinear Schr\"{o}dinger equation with derivative in $H^s(\R)$. This equation was known to be the local well-posedness for $s\geq \frac12$ (Takaoka,1999), ill-posedness for $s<\frac12$ (Biagioni and Linares, 2001, etc.) and global well-posedness for $s>\frac12$ (I-team, 2002). In this paper, we show that it is global well-posedness in $H^{1/2(\R)$. The main approach is the third generation I-method combined with some additional resonant decomposition technique. The resonant decomposition is applied to control the singularity coming from the resonant interaction.