Invariant measures for the Nonlinear Schrodinger equation on the disc

2006/03/04 by Nikolay Tzvetkov, Tzvetkov, Nikolay · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #advanced mathematical theories #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.math/0603112

Abstract

We study Gibbs measures invariant under the flow of the NLS on the unit disc of \R². For that purpose, we construct the dynamics on a phase space of limited Sobolev regularity and a wighted Wiener measure invariant by the NLS flow. The density of the measure is integrable with respect to the Wiener measure for sub cubic nonlinear interactions. The existence of the dynamics is obtained in Bourgain spaces of low regularity. The key ingredient are bilinear Strichartz estimates for the free evolution. The bilinear effect in our analysis results from simple properties of the Bessel functions and estimates on series of Bessel functions.

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• A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation
• Invariant Gibbs dynamics for the nonlinear Schrödinger equations on the disc

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