An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation

Oh, Tadahiro; Sosoe, Philippe; Tzvetkov, Nikolay

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 793-841.

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.

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.

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

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