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 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.
"Two-dimensional nonlinear Schrödinger equation with random radial data." Anal. PDE 5 (5) 913 - 960, 2012. https://doi.org/10.2140/apde.2012.5.913