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.

@article{AIHPC_2014__31_6_1267_0, author = {Bourgain, Jean and Bulut, Aynur}, title = {Almost sure global well posedness for the radial nonlinear {Schr\"odinger} equation on the unit ball {I:} {The} {2D} case}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1267--1288}, publisher = {Elsevier}, volume = {31}, number = {6}, year = {2014}, doi = {10.1016/j.anihpc.2013.09.002}, zbl = {1307.35272}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.002/} }

TY - JOUR AU - Bourgain, Jean AU - Bulut, Aynur TI - Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 1267 EP - 1288 VL - 31 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.002/ DO - 10.1016/j.anihpc.2013.09.002 LA - en ID - AIHPC_2014__31_6_1267_0 ER -

%0 Journal Article %A Bourgain, Jean %A Bulut, Aynur %T Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 1267-1288 %V 31 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.002/ %R 10.1016/j.anihpc.2013.09.002 %G en %F AIHPC_2014__31_6_1267_0

Bourgain, Jean; Bulut, Aynur. Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, pp. 1267-1288. doi : 10.1016/j.anihpc.2013.09.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.002/

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