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 \cite{T} 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 \cite{BB-1} 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 \cite{T2}) where the Gibbs measure is subject to an $L^2$-norm restriction. A phase transition is established, of the same nature as studied in the work of Lebowitz-Rose-Speer \cite{LRS} on the torus. 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.