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On the cubic NLS on 3D compact domains

Published online by Cambridge University Press:  04 February 2013

Fabrice Planchon*
Affiliation:
Laboratoire J. A. Dieudonné, UMR 7351, Université de Nice Sophia-Antipolis, Parc Valrose, F-06108 Nice Cedex 02, et Institut universitaire de France (fabrice.planchon@unice.fr)

Abstract

We prove bilinear estimates for the Schrödinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the ${ \mathbb{R} }^{3} $ case, while on bounded domains they match the generic boundaryless manifold case. We obtain, as an application, global well-posedness for the defocusing cubic NLS for data in ${ H}_{0}^{s} (\Omega )$, $1\lt s\leq 3$, with $\Omega $ any bounded domain with smooth boundary.

Résumé

On démontre des estimations bilinéaires pour l’équation de Schrödinger sur des domaines tridimensionnels, avec condition de Dirichlet au bord. Dans le cas non-captant, on retrouve les estimations connues dans ${ \mathbb{R} }^{3} $, et sur un domaine borné on obtient des estimations similaires à celles du cas d’une variété compacte générique sans bord. Une application est donnée à l’existence de solutions globales dans ${ H}_{0}^{s} (\Omega )$, $1\lt s\leq 3$, avec $\Omega $ un domaine borné régulier.

Type
Research Article
Copyright
©Cambridge University Press 2013 

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On the cubic NLS on 3D compact domains
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