[Croissance de norme avec perte infinie de régularité pour les équations de Schrödinger périodiques en régularité négative]
In this paper we consider Schrödinger equations with nonlinearities of odd order on . We prove that for , they are strongly illposed in the Sobolev space for any , exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schrödinger equation and its renormalized version we prove such a result for with
Nous considérons des équations de Schrödinger avec des non-linéarités d’ordre impair sur le tore . Nous montrons que pour , ces équations sont fortement mal posées dans l’espace de Sobolev pour tout , avec en outre un phénomène de perte infinie de régularité. Dans le cas cubique mono-dimensionnel et sa version renormalisée, nous montrons le même résultat dans , sous l’hypothèse .
Accepté le :
Publié le :
DOI : 10.24033/bsmf.2749
Keywords: Nonlinear Schrödinger equation, periodic case, well-posedness, loss of regularity, geometric optics, semi-classical analysis
Mots-clés : Équation de Schrödinger non linéaire, cas périodique, caractère bien posé, perte de régularité, optique géométrique, analyse semi-classique
Carles, Rémi 1 ; Kappeler, Thomas 2
@article{BSMF_2017__145_4_623_0,
author = {Carles, R\'emi and Kappeler, Thomas},
title = {Norm-inflation with {Infinite} {Loss} of {Regularity} for {Periodic} {NLS} {Equations} in {Negative} {Sobolev} {Spaces}},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {623--642},
year = {2017},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {145},
number = {4},
doi = {10.24033/bsmf.2749},
mrnumber = {3770969},
zbl = {1428.35488},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2749/}
}
TY - JOUR AU - Carles, Rémi AU - Kappeler, Thomas TI - Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces JO - Bulletin de la Société Mathématique de France PY - 2017 SP - 623 EP - 642 VL - 145 IS - 4 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2749/ DO - 10.24033/bsmf.2749 LA - en ID - BSMF_2017__145_4_623_0 ER -
%0 Journal Article %A Carles, Rémi %A Kappeler, Thomas %T Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces %J Bulletin de la Société Mathématique de France %D 2017 %P 623-642 %V 145 %N 4 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2749/ %R 10.24033/bsmf.2749 %G en %F BSMF_2017__145_4_623_0
Carles, Rémi; Kappeler, Thomas. Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces. Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 4, pp. 623-642. doi: 10.24033/bsmf.2749
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