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}, publisher = {Elsevier}, volume = {31}, number = {6}, year = {2014}, pages = {1267-1288}, doi = {10.1016/j.anihpc.2013.09.002}, zbl = {1307.35272}, language = {en}, url = {http://www.numdam.org/item/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://www.numdam.org/item/AIHPC_2014__31_6_1267_0/

[1] Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 no. 2 (1993), 107 -156 | MR 1209299 | Zbl 0787.35097

,[2] Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 no. 3 (1993), 209 -262 | MR 1215780 | Zbl 0787.35098

,[3] Periodic nonlinear Schrödinger equation in invariant measures, Commun. Math. Phys. 166 (1994), 1 -24 | MR 1309539 | Zbl 0822.35126

,[4] Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys. 176 (1996), 421 -445 | MR 1374420 | Zbl 0852.35131

,[5] Invariant measures for the Gross–Piatevskii equation, J. Math. Pures Appl. 76 (1997), 649 -702 | MR 1470880 | Zbl 0906.35095

,[6] Nonlinear Schrödinger equations, Hyperbolic Equations and Frequency Interactions, IAS/Park City Math. Ser. vol. 5 , Amer. Math. Soc., Providence, RI (1999), 3 -157 | MR 1662829 | Zbl 0952.35127

,[7] Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball, C. R. Math. 350 no. 11–12 (2012), 571 -575 | MR 2956145 | Zbl 1251.35141

, ,[8] J. Bourgain, A. Bulut, Invariant Gibbs measure evolution for the 3D radial NLW on the ball, preprint, 2012. | MR 3150162

[9] Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3D case, arXiv:1302.5409 (2013) | MR 3226743 | Zbl 1301.35145

, ,[10] Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation, Commun. Math. Phys. 182 (1996), 485 -504 | MR 1447302 | Zbl 0867.35090

, ,[11] Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 no. 3 (2008), 449 -475 , , Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math. 173 no. 3 (2008), 477 -496 | MR 2425133 | Zbl 1156.35062

, ,[12] A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Am. Math. Soc. 299 no. 1 (1987), 193 -203 | MR 869407 | Zbl 0638.35043

,[13] Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys. 50 (1988), 657 -687 | MR 939505 | Zbl 1084.82506

, , ,[14] Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier 58 no. 7 (2008), 2543 -2604 | Numdam | MR 2498359 | Zbl 1171.35116

,[15] Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ. 3 no. 2 (2006), 111 -160 | MR 2227040 | Zbl 1142.35090

,