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

### Herbert Koch

Mathematisches Institut, Universität Bonn, Germany### Daniel Tataru

Department of Mathematics, University of California, Berkeley, United States

## 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.

## Cite this article

Herbert Koch, Daniel Tataru, Energy and local energy bounds for the 1-d cubic NLS equation in $H^{- \frac{1}{4}}$. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, pp. 955–988

DOI 10.1016/J.ANIHPC.2012.05.006