The Schrödinger equation on a compact manifold : Strichartz estimates and applications
Journées équations aux dérivées partielles, (2001), article no. 5, 18 p.

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

Nous établissons des estimations de Strichartz avec perte de dérivée fractionnaire pour l'équation de Schrödinger sur toute variété riemannienne compacte. Nous en déduisons des théorèmes d'existence globale pour le problème de Cauchy d'équations de Schrödinger non-linéaires sur les surfaces dans le cas de non-linéarités polynomiales défocalisantes, et sur les variétés de dimension trois dans le cas de non-linéarités quadratiques. Nous discutons également l'optimalité de ces estimées de Strichartz sur les sphères.

@article{JEDP_2001____A5_0,
     author = {Burq, Nicolas and G\'erard, Patrick and Tzvetkov, Nikolay},
     title = {The Schr\"odinger equation on a compact manifold : Strichartz estimates and applications},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.589},
     zbl = {01808681},
     mrnumber = {1843406},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2001____A5_0}
}
Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay. The Schrödinger equation on a compact manifold : Strichartz estimates and applications. Journées équations aux dérivées partielles,  (2001), article  no. 5, 18 p. doi : 10.5802/jedp.589. http://www.numdam.org/item/JEDP_2001____A5_0/

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