Energy and local energy bounds for the 1-d cubic NLS equation in ${H}^{-\frac{1}{4}}$
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 955-988.

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.

@article{AIHPC_2012__29_6_955_0,
author = {Koch, Herbert and Tataru, Daniel},
title = {Energy and local energy bounds for the 1-d cubic NLS equation in ${H}^{-\frac{1}{4}}$},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {955--988},
publisher = {Elsevier},
volume = {29},
number = {6},
year = {2012},
doi = {10.1016/j.anihpc.2012.05.006},
zbl = {1280.35137},
language = {en},
url = {http://archive.numdam.org/item/AIHPC_2012__29_6_955_0/}
}
Koch, Herbert; Tataru, Daniel. Energy and local energy bounds for the 1-d cubic NLS equation in ${H}^{-\frac{1}{4}}$. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 955-988. doi : 10.1016/j.anihpc.2012.05.006. http://archive.numdam.org/item/AIHPC_2012__29_6_955_0/

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