A refined global well-posedness result for Schrodinger equations with derivative

Abstract

In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in $H^s$, for $s>\frac{1}{2}$ for data small in $L^2$. To understand the strength of this result one should recall that for $s<\frac{1}{2}$ the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the ``I-method'' used by the same authors to obtain global well-posedness for $s>\frac{2}{3}$. The same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in $H^s$, for $s>\frac{1}{2}$.

21 pages, no figures, submitted, Siam J. Math