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

Our first purpose is to extend the results from [14] on the radial defocusing NLS on the disc in 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. (C) 2013 Elsevier Masson SAS. All rights reserved.