Sharp global well-posedness for 1D NLS with derivatives

Abstract

We show that the 1d derivative nonlinear Schrödinger equation (\ref{equ}) is globally well-posed in $H^s(\mathbb{R})$ for $s\ge 1/2$. We use the linear-nonlinear decomposition method to take advantage of the local smoothing effect of the nonlinearity, which enables us to establish a refined version of the almost conservation law. Note that $H^{1/2}$ is the endpoint that we have uniform continuous for the solution map and hence our result is sharp.

The same result has been obtained by other authors using third generation modified energy

References in corpus (1)

• Global well-posedness for the defocusing, cubic, nonlinear Schrodinger equation when n = 3 via a linear-nonlinear decomposition