#### 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 ${L}^{p}$ 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