An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation
Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 793-841.

We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures under the dynamics of the dispersionless model.

Nous étudions le transport de mesures gaussiennes par le flot de l’équation de Schrödinger non linéaire d’ordre 4. La nouveauté principale est une estimation d’énergie améliorée faisant appel à un nombre infini de transformations de forme normale sur la fonctionnelle d’énergie. De plus, nous démontrons que la dispersion est essentielle dans cette problématique en prouvant qu’en son absence le même résultat de quasi-invariance ne peut être vrai.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.83
Classification: 35Q55
Keywords: Fourth order nonlinear Schrödinger equation, biharmonic nonlinear Schrödinger equation, quasi-invariant measure, normal form method
Mot clés : Équation de Schrödinger non linéaire, mesures quasi-invariantes, méthode de forme normale

Tadahiro Oh 1; Philippe Sosoe 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, Cornell University 584 Malott Hall, Ithaca, New York 14853, USA
3 Université de Cergy-Pontoise 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {An optimal regularity result on the~quasi-invariant {Gaussian} measures for the~cubic fourth order nonlinear {Schr\"odinger~equation}},
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Tadahiro Oh; Philippe Sosoe; Nikolay Tzvetkov. An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation. Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 793-841. doi : 10.5802/jep.83. https://jep.centre-mersenne.org/articles/10.5802/jep.83/

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