Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples

Oana Ivanovici; Gilles Lebeau

doi:10.1016/j.crma.2017.05.011

Ordinary differential equations

Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples
[Estimations de dispersion pour l'équation des ondes et de Schrödinger à l'extérieur des obstacles strictement convexes et contre-exemples]

Présenté par : Jean-Michel Coron

Oana Ivanovici ¹ ; Gilles Lebeau ²

¹ CNRS et Université Côte d'Azur, Laboratoire J.-A.-Dieudonné, UMR CNRS 7351, parc Valrose, 06108 Nice cedex 02, France
² Université Côte d'Azur, Laboratoire J.-A.-Dieudonné, UMR CNRS 7351, parc Valrose, 06108 Nice cedex 02, France

Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 774-779.

Résumé
Abstract

L'objet de cette note est de démontrer des estimations de dispersion pour l'équation des ondes et de Schrödinger à l'extérieur d'un obstacle strictement convexe de $R^{d}$ . Si $d = 3$ , on démontre que, pour chacune des deux équations, le flot linéaire vérifie les estimations de dispersion comme dans $R^{3}$ . En dimension plus grande $d \geq 4$ , on démontre que des pertes dans la dispersion apparaissent à l'extérieur d'une boule de $R^{d}$ et que cela arrive au point de Poisson.

The purpose of this note is to prove dispersive estimates for the wave and the Schrödinger equations outside strictly convex obstacles in $R^{d}$ . If $d = 3$ , we show that, for both equations, the linear flow satisfies the (corresponding) dispersive estimates as in $R^{3}$ . In higher dimensions $d \geq 4$ and if the domain is the exterior of a ball in $R^{d}$ , we show that losses in dispersion do appear and this happens at the Poisson spot.

Reçu le : 2017-03-03
Accepté le : 2017-05-30
Publié le : 2017-06-13

DOI : 10.1016/j.crma.2017.05.011

Affiliations des auteurs :

Oana Ivanovici ¹ ; Gilles Lebeau ²

¹ CNRS et Université Côte d'Azur, Laboratoire J.-A.-Dieudonné, UMR CNRS 7351, parc Valrose, 06108 Nice cedex 02, France
² Université Côte d'Azur, Laboratoire J.-A.-Dieudonné, UMR CNRS 7351, parc Valrose, 06108 Nice cedex 02, France

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     author = {Oana Ivanovici and Gilles Lebeau},
     title = {Dispersion for the wave and the {Schr\"odinger} equations outside strictly convex obstacles and counterexamples},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {774--779},
     publisher = {Elsevier},
     volume = {355},
     number = {7},
     year = {2017},
     doi = {10.1016/j.crma.2017.05.011},
     language = {en},
}

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Oana Ivanovici; Gilles Lebeau. Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 774-779. doi : 10.1016/j.crma.2017.05.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.011/

Bibliographie
Cité par

[1] O. Ivanovici, G. Lebeau, Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples, preprint, 2017.

[2] D. Li; H. Smith; X. Zhang Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data, Math. Res. Lett., Volume 19 (2012) no. 1, pp. 213-232

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Oana Ivanovici; Gilles Lebeau Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples, Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, p. 774 | DOI:10.1016/j.crma.2017.05.011

Cité par 8 documents. Sources : Crossref

^☆ The authors were partially supported by ERC project SCAPDE (grant 320845). The authors would like to thank Centro di Giorgi, Pisa for the warm welcome during the summer 2015 when this article has started.

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