Energy and local energy bounds for the 1-d cubic NLS equation in $H^{-\frac{1}{4}}$

Abstract

We consider the cubic nonlinear Schrödinger equation (NLS) in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s⩾-\frac{1}{4}$. This improves earlier results of Christ, Colliander and Tao [3] and of the authors (Koch and Tataru, 2007 [13]). The new ingredients are a localization in space and local energy decay, which we hope to be of independent interest.