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, p. 1267-1288
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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.

@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/

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