Koch, Herbert; Tataru, Daniel

Energy and local energy bounds for the 1-D cubic NLS equation in H^{-1/4}

2010

Abstract

We consider the cubic Nonlinear Schroedinger Equation (NLS) in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time Hs bounds in terms of the Hs size of the initial data for s >=-1/4 . This improves earlier results of Christ-Colliander-Tao [2] and of the authors [12]. The new ingredients are a localization in space and local energy decay, which we hope to be of independent interest.

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UC Berkeley

Energy and local energy bounds for the 1-D cubic NLS equation in H^{-1/4}