JournalsaihpcVol. 29, No. 6pp. 955–988

Energy and local energy bounds for the 1-d cubic NLS equation in H14 H^{- \frac{1}{4}}

  • Herbert Koch

    Mathematisches Institut, Universität Bonn, Germany
  • Daniel Tataru

    Department of Mathematics, University of California, Berkeley, United States
Energy and local energy bounds for the 1-d cubic NLS equation in \( H^{- \frac{1}{4}} \) cover
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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 HsH^{s} bounds in terms of the HsH^{s} size of the initial data for s14s⩾−\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 H14 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