Invariant measures for the nonlinear Schrödinger equation on the disc

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.

Keywords

nonlinear Schrödinger, eigenfunctions, dispersive equations, invariant measures

2010 Mathematics Subject Classification

35Bxx, 35Q55, 37K05, 37L50, 81Q20

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Published 1 January 2006