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Annales scientifiques de l'ENS - Titles - série 4, 50 (2017)

Titles < série 4, 50

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 50, fascicule 3 (2017)

Thomas Boulenger, Enno Lenzmann
Blowup for biharmonic NLS
Annales scientifiques de l'ENS 50, fascicule 3 (2017), 503-544

Télécharger cet article : Fichier PDF

Résumé :
Phénomènes d'explosion pour NLS Biharmonique
On considère le problème de Cauchy pour NLS biharmonique (i.e., d'ordre quatre) focalisante définie par i _t u = ^2 u - u -|u|^2  u pour (t,x) [0,T) R ^d, avec 0 < < pour d 4 et 0 < 4/(d-4) pour d 5; et est un paramètre destiné à éventuellement inclure un terme dispersif d'ordre inférieur. Dans le cas sur-critique > 4/d, on prouve un résultat général d'explosion en temps fini pour des données radiales dans H^2(R ^d) en toute dimension d 2. On déduit par ailleurs une borne supérieure universelle pour la vitesse d'explosion moyennée en temps pour certains indices 4/d < < 4/(d-4). Dans le cas critique =4/d, on prouve ensuite un résultat général d'explosion en temps fini ou infini, toujours pour des solutions à données radiales H^2(R ^d). On utilise là de façon cruciale l'évolution temporelle d'une quantité positive, que nous baptisons la bivariance (locale) de Riesz pour NLS biharmonique. Cette quantité nous sert de substitut avantageux à la variance classiquement utilisée pour l'étude des problèmes NLS. On prouve enfin l'existence d'un ground state radial pour NLS biharmonique, qui pourra s'avérer utile pour l'étude du problème elliptique associé.

Mots-clefs : NLS biharmonique, NLS d'ordre quatre, phénomènes d'explosion, ground states.

Abstract:
We consider the Cauchy problem for the biharmonic (i.e., fourth-order) NLS with focusing nonlinearity given by i _t u = ^2 u - u -|u|^2  u for (t,x) [0,T) R ^d, where 0 < < for d 4 and 0 < 4/(d-4) for d 5; and R is some parameter to include a possible lower-order dispersion. In the mass-supercritical case > 4/d, we prove a general result on finite-time blowup for radial data in H^2(R ^d) in any dimension d 2. Moreover, we derive a universal upper bound for the blowup rate for suitable 4/d < < 4/(d-4). In the mass-critical case =4/d, we prove a general blowup result in finite or infinite time for radial data in H^2(R ^d). As a key ingredient, we utilize the time evolution of a nonnegative quantity, which we call the (localized) Riesz bivariance for biharmonic NLS. This construction provides us with a suitable substitute for the variance used for classical NLS problems. In addition, we prove a radial symmetry result for ground states for the biharmonic NLS, which may be of some value for the related elliptic problem.

Keywords: Biharmonic NLS, fourth-order NLS, blowup, ground states.

Class. math. : 35Q55, 35J48, 35A01, 31B30.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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