Global well-posedness and scattering  for the focusing, cubic Schrödinger  equation in dimension $d = 4$

Dodson, Benjamin

doi:10.24033/asens.2385

Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension

d = 4

[Existence globale et diffusion des ondes pour l'équation de Schrödinger cubique focalisante en dimension 4]

Dodson, Benjamin

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 139-180.

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Résumé
Abstract

Nous prouvons l'existence globale et la diffusion des ondes pour l'équation de Schrödinger cubique focalisante en dimension quatre. Des travaux antérieurs ont montré de tels résultats en dimension supérieure ou égale à cinq. Nous utilisons ici la méthode de Duhamel double et les estimations de Strichartz en temps long.

In this paper we prove global well-posedness and scattering for the focusing, cubic Schrödinger equation in four dimensions below the ground state. Previous work proved this in five dimensions and higher. To prove this, we combine the double Duhamel method with the long time Strichartz estimates.

DOI : 10.24033/asens.2385

Classification : 35Q55.
Keywords: Scattering, Schrödinger equation, focusing.
Mot clés : Diffusion des ondes, équation de Schrödinger, focalisation.

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     author = {Dodson, Benjamin},
     title = {Global well-posedness and scattering  for the focusing, cubic {Schr\"odinger}  equation in dimension $d = 4$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {139--180},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {1},
     year = {2019},
     doi = {10.24033/asens.2385},
     mrnumber = {3940908},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2385/}
}

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Dodson, Benjamin. Global well-posedness and scattering  for the focusing, cubic Schrödinger  equation in dimension $d = 4$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 1, pp. 139-180. doi : 10.24033/asens.2385. http://www.numdam.org/articles/10.24033/asens.2385/

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