Sharp global well-posedness for 1D NLS with derivatives

We show that the 1d derivative nonlinear Schrödinger equation (\ref{equ}) is globally well-posed in $H^s(\mathbb{R})$ for $s\geq 1/2$. 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.

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