Vol. 5, No. 5, 2012

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Two-dimensional nonlinear Schrödinger equation with random radial data

Yu Deng

Vol. 5 (2012), No. 5, 913–960
Abstract

We study radial solutions of a certain two-dimensional nonlinear Schrödinger (NLS) equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schrödinger equation with Lp estimates of Laguerre functions, we are able to prove an almost-sure global well-posedness result and the invariance of the Gibbs measure. We also discuss an application to the NLS equation without harmonic potential.

Keywords
nonlinear Schrödinger equation, supercritical NLS, random data, Gibbs measure, global well-posedness
Mathematical Subject Classification 2010
Primary: 35Q55, 37L40, 37L50
Secondary: 37K05
Milestones
Received: 16 November 2010
Revised: 14 February 2011
Accepted: 3 June 2011
Published: 29 December 2012
Authors
Yu Deng
Department of Mathematics
Princeton University
Princeton, NJ 08544 United States