Sharp global well-posedness for 1D NLS with derivatives

Details Sharp global well-posedness for 1D NLS with derivatives Sharp global well-posedness for 1D NLS with derivatives

2012-01-03

arxiv.org/abs/1201.0727v2

Mathematics

Analysis of PDEs

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

Scientific paper

We show that the 1d derivative nonlinear Schr\"{o}dinger 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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