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

Tadahiro Oh; Philippe Sosoe; Nikolay Tzvetkov

doi:10.5802/jep.83

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

Tadahiro Oh, Philippe Sosoe, Nikolay Tzvetkov

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	793-841
Number of pages	49
Journal	Journal de l’École polytechnique — Mathématiques (JEP)
Volume	5
DOIs	https://doi.org/10.5802/jep.83
Publication status	Published - 30 Oct 2018

Keywords

fourth order nonlinear Schrödinger equation
biharmonic nonlinear Schrödinger equation
quasi-invariant measure
normal form method

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10.5802/jep.83Licence: Creative Commons: Attribution No Derivatives (CC-BY-ND)

OSTz_JEP_finalAccepted author manuscript, 586 KBLicence: Creative Commons: Attribution No Derivatives (CC-BY-ND)

https://arxiv.org/abs/1707.01666

ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
Oh, T.
EU government bodies
1/03/15 → 29/02/20
Project: Research

Tadahiro Oh
- School of Mathematics - Personal Chair of Dispersive Equations
Person: Academic: Research Active

Cite this

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title = "An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr{\"o}dinger equation",

abstract = "We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schr{\"o}dinger 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.",

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author = "Tadahiro Oh and Philippe Sosoe and Nikolay Tzvetkov",

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T1 - An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation

AU - Oh, Tadahiro

AU - Sosoe, Philippe

AU - Tzvetkov, Nikolay

PY - 2018/10/30

Y1 - 2018/10/30

N2 - 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.

AB - 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.

KW - fourth order nonlinear Schrödinger equation

KW - biharmonic nonlinear Schrödinger equation

KW - quasi-invariant measure

KW - normal form method

U2 - 10.5802/jep.83

DO - 10.5802/jep.83

M3 - Article

VL - 5

SP - 793

EP - 841

JO - Journal de l’École polytechnique — Mathématiques (JEP)

JF - Journal de l’École polytechnique — Mathématiques (JEP)

SN - 2429-7100

ER -

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

Abstract

Keywords

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ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations

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Tadahiro Oh

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