Structure theorems for 2D linear and nonlinear Schrödinger equations

Hajer Bahouri

doi:10.1016/j.crma.2015.01.009

Partial differential equations

Structure theorems for 2D linear and nonlinear Schrödinger equations
[Théorèmes de structure pour les équations de Schrödinger linéaire et non linéaire en dimension deux d'espace]

Présenté par : Jean-Michel Bony

Hajer Bahouri ¹

¹ Université Paris-Est Créteil, UMR 8050, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France

Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 235-240.

Résumé
Abstract

On étudie le comportement des solutions de l'équation de Schrödinger non linéaire à croissance exponentielle, où la norme d'Orlicz joue un rôle crucial. L'approche qu'on a adoptée dans ce travail consiste à comparer des suites de solutions des équations de Schrödinger linéaires et non linéaires issues de la même suite de données de Cauchy, moyennant un terme de reste petit à la fois en normes de Strichartz et d'Orlicz. Cette analyse, qui est basée sur les décompositions en profils, met en lumière le rôle distingué de la composante 1-oscillante de la suite des données initiales. Ce phénomène est complètement différent de ceux obtenus dans le cadre des équations semi-linéaires dispersives critiques, comme dans [2,13], où toutes les composantes oscillantes créent le même effet non linéaire, à un changement d'échelle près.

We investigate the behavior of solutions to 2D nonlinear Schrödinger equations with exponential growth, where the Orlicz norm plays a crucial role. The approach we adopted in this paper consists in comparing the evolution of oscillations and concentration effects displayed by sequences of solutions to linear and nonlinear Schrödinger equations associated with the same sequence of Cauchy data, up to small remainder terms both in Strichartz and Orlicz norms. The analysis we conducted in this work emphasizes that the nonlinear effect highlighted in this framework only arises from the 1-oscillating component of the sequence of the Cauchy data. This phenomenon is strikingly different from those obtained for critical semilinear dispersive equations, such as for instance in [2,13], where all the oscillating components induce the same nonlinear effect, up to a change of scale.

Reçu le : 2014-12-02
Accepté le : 2015-01-21
Publié le : 2015-01-28

DOI : 10.1016/j.crma.2015.01.009

Affiliations des auteurs :

Hajer Bahouri ¹

¹ Université Paris-Est Créteil, UMR 8050, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France

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     author = {Hajer Bahouri},
     title = {Structure theorems for {2D} linear and nonlinear {Schr\"odinger} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {235--240},
     publisher = {Elsevier},
     volume = {353},
     number = {3},
     year = {2015},
     doi = {10.1016/j.crma.2015.01.009},
     language = {en},
}

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Hajer Bahouri. Structure theorems for 2D linear and nonlinear Schrödinger equations. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 235-240. doi : 10.1016/j.crma.2015.01.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.009/

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