Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case

Abstract

Our first purpose is to extend the results from [14] on the radial defocusing NLS on the disc in ${\mathbb{R}}^2$ to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs measure (which is the natural extension of the classical flow for smooth data). We follow a similar approach as in [8] exploiting certain additional a priori space–time bounds that are provided by the invariance of the Gibbs measure.

Next, we consider the radial focusing equation with cubic nonlinearity (the mass-subcritical case was studied in [15]) where the Gibbs measure is subject to an $L^2$-norm restriction. A phase transition is established. For sufficiently small $L^2$-norm, the Gibbs measure is absolutely continuous with respect to the free measure, and moreover we have a well-defined dynamics. For sufficiently large $L^2$-norm cutoff, the Gibbs measure concentrates on delta functions centered at 0. This phenomenon is similar to the one observed in the work of Lebowitz, Rose, and Speer [13] on the torus.