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

Koch, Herbert; Tataru, Daniel

doi:10.1016/j.anihpc.2012.05.006

Energy and local energy bounds for the 1-d cubic NLS equation in

H^{- \frac{1}{4}}

Koch, Herbert ; Tataru, Daniel

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 6, pp. 955-988

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Résumé

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.

Zbl

DOI : 10.1016/j.anihpc.2012.05.006

@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://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.05.006/}
}

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%D 2012
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%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, Tome 29 (2012) no. 6, pp. 955-988. doi: 10.1016/j.anihpc.2012.05.006

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