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

Abstract

In this paper, we consider the Cauchy problem of the cubic nonlinear Schrödinger equation with derivative in $H^s(\mathbb{R})$. This equation was known to be the local well-posedness for $s\ge \frac{1}{2}$ (Takaoka,1999), ill-posedness for $s<\frac{1}{2}$ (Biagioni and Linares, 2001, etc.) and global well-posedness for $s>\frac{1}{2}$ (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.

31pages; In this version, we change some expressions in English

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• The low regularity global solutions for the critical generalized KdV equation
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• Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in $H^{\frac{1}{2}}$