Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion

Tadahiro Oh; Yoshio Tsutsumi; Nikolay Tzvetkov

doi:10.1016/j.crma.2019.04.001

Partial differential equations/Probability theory

Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion
[Mesures gaussiennes quasi invariantes pour l'équation de Schrödinger non linéaire cubique avec dispersion d'ordre trois]

Présenté par : Haïm Brézis

Tadahiro Oh ¹ ; Yoshio Tsutsumi ² ; Nikolay Tzvetkov ³

¹ School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
² Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
³ Université de Cergy-Pontoise, 2, av. Adolphe-Chauvin, 95302 Cergy-Pontoise cedex, France

Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 366-381.

Résumé
Abstract

Dans cet article, nous considérons l'équation de Schrödinger non linéaire cubique avec dispersion d'ordre trois sur le cercle. Dans le cas non résonant, nous prouvons que les mesures gaussiennes de moyenne nulle sur les espaces de Sobolev $H^{s} (T)$ , $s > \frac{3}{4}$ , sont quasi invariantes par le flot. En établissant le résultat, nous appliquons des transformations de gauge pour éliminer la partie résonante de la dynamique, et nous utilisons l'invariance des mesures gaussiennes par rapport à ces transformations de gauge.

In this paper, we consider the cubic nonlinear Schrödinger equation with third-order dispersion on the circle. In the non-resonant case, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^{s} (T)$ , $s > \frac{3}{4}$ , are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.

Reçu le : 2017-11-08
Accepté le : 2019-04-05
Publié le : 2019-04-01

DOI : 10.1016/j.crma.2019.04.001

Affiliations des auteurs :

Tadahiro Oh ¹ ; Yoshio Tsutsumi ² ; Nikolay Tzvetkov ³

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     author = {Tadahiro Oh and Yoshio Tsutsumi and Nikolay Tzvetkov},
     title = {Quasi-invariant {Gaussian} measures for the cubic nonlinear {Schr\"odinger} equation with third-order dispersion},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {366--381},
     publisher = {Elsevier},
     volume = {357},
     number = {4},
     year = {2019},
     doi = {10.1016/j.crma.2019.04.001},
     language = {en},
}

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Tadahiro Oh; Yoshio Tsutsumi; Nikolay Tzvetkov. Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 366-381. doi : 10.1016/j.crma.2019.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.04.001/

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