Explosion pour l'équation de Schrödinger au régime du « log log »  [ Blow-up for the non linear Schrödinger equation in the “log log” regime ]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 953, p. 33-54

In this talk we present some recent results by Merle and Raphael on analysis of blow-up solution for the ${L}^{2}$ critical non linear Schrödinger equation. In particular, we focus on their proof of the fact that initial data with negative energy (which had been known to blow up by the viriel identity) and with ${L}^{2}$ norm close to the ground states’s ${L}^{2}$ norm, do blow up in the “log log” regime and that this behaviour is stable.

On présente dans cet exposé des résultats récents de Merle et Raphael sur l’analyse des solutions explosives de l’équation de Schrödinger ${L}^{2}$ critique. On s’intéresse en particulier à leur preuve du fait que les solutions d’énergie négative (dont on savait qu’elles explosaient par l’argument du viriel) et dont la norme ${L}^{2}$ est proche de celle de l’état fondamental, explosent au régime du “log log”et que ce comportement est stable.

Classification:  35B30,  35B35,  35B65
Keywords: non linear Schrödinger equations, blow-up
@incollection{SB_2005-2006__48__33_0,
author = {Burq, Nicolas},
title = {Explosion pour l'\'equation de Schr\"odinger au r\'egime du \guillemotleft{} log log \guillemotright{}},
booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
author = {Collectif},
series = {Ast\'erisque},
publisher = {Soci\'et\'e math\'ematique de France},
number = {311},
year = {2007},
note = {talk:953},
pages = {33-54},
zbl = {1194.35400},
language = {fr},
url = {http://www.numdam.org/item/SB_2005-2006__48__33_0}
}

Burq, Nicolas. Explosion pour l'équation de Schrödinger au régime du « log log », in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 953, pp. 33-54. http://www.numdam.org/item/SB_2005-2006__48__33_0/

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