Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case
Jean Bourgain
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, United StatesAynur Bulut
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, United States
![Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-aihpc-volume-31-issue-6.png&w=3840&q=90)
Abstract
Our first purpose is to extend the results from [14] on the radial defocusing NLS on the disc in 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 -norm restriction. A phase transition is established. For sufficiently small -norm, the Gibbs measure is absolutely continuous with respect to the free measure, and moreover we have a well-defined dynamics. For sufficiently large -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.
Cite this article
Jean Bourgain, Aynur Bulut, Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 6, pp. 1267–1288
DOI 10.1016/J.ANIHPC.2013.09.002