Two-dimensional nonlinear Schrödinger equation with random radial data

Deng, Yu

doi:10.2140/apde.2012.5.913

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Two-dimensional nonlinear Schrödinger equation with random radial data

Yu Deng

Vol. 5 (2012), No. 5, 913–960

DOI: 10.2140/apde.2012.5.913

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 $L^{p}$ 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

Vol. 5, No. 5, 2012