Two Dimensional NLS Equation With Random Radial Data

In this paper we study radial solutions of certain two-dimensional nonlinear Schr"odinger equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schr "odinger 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.