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 bounds in terms of the size of the initial data for . 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://www.numdam.org/articles/10.1016/j.anihpc.2012.05.006/} }
TY - JOUR AU - Koch, Herbert AU - Tataru, Daniel TI - Energy and local energy bounds for the 1-d cubic NLS equation in $ {H}^{-\frac{1}{4}}$ JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 955 EP - 988 VL - 29 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2012.05.006/ DO - 10.1016/j.anihpc.2012.05.006 LA - en ID - AIHPC_2012__29_6_955_0 ER -
%0 Journal Article %A Koch, Herbert %A Tataru, Daniel %T Energy and local energy bounds for the 1-d cubic NLS equation in $ {H}^{-\frac{1}{4}}$ %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 955-988 %V 29 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2012.05.006/ %R 10.1016/j.anihpc.2012.05.006 %G en %F 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, Volume 29 (2012) no. 6, pp. 955-988. doi : 10.1016/j.anihpc.2012.05.006. http://www.numdam.org/articles/10.1016/j.anihpc.2012.05.006/
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