Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case

Bourgain, Jean, and Bulut, Aynur. "Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case." Journal of the European Mathematical Society 016.6 (2014): 1289-1325. <http://eudml.org/doc/277714>.

@article{Bourgain2014,
abstract = {We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in $\mathbb \{R\}^d$ to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in $\mathbb \{R\}^3$. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation, and results are obtained almost surely with respect to this probability measure. The key tools used include a class of probabilistic a priori bounds for finite-dimensional projections of the equation and a delicate trilinear estimate on the nonlinearity, which – when combined with the invariance of the Gibbs measure – enables the a priori bounds to be enhanced to obtain convergence of the sequence of approximate solutions.},
author = {Bourgain, Jean, Bulut, Aynur},
journal = {Journal of the European Mathematical Society},
keywords = {Gibbs measure; global well-posedness; defocusing nonlinear Schrödinger equation; Gibbs measure; global well-posedness; defocusing nonlinear Schrödinger equation},
language = {eng},
number = {6},
pages = {1289-1325},
publisher = {European Mathematical Society Publishing House},
title = {Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case},
url = {http://eudml.org/doc/277714},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Bourgain, Jean
AU - Bulut, Aynur
TI - Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 6
SP - 1289
EP - 1325
AB - We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in $\mathbb {R}^d$ to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in $\mathbb {R}^3$. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation, and results are obtained almost surely with respect to this probability measure. The key tools used include a class of probabilistic a priori bounds for finite-dimensional projections of the equation and a delicate trilinear estimate on the nonlinearity, which – when combined with the invariance of the Gibbs measure – enables the a priori bounds to be enhanced to obtain convergence of the sequence of approximate solutions.
LA - eng
KW - Gibbs measure; global well-posedness; defocusing nonlinear Schrödinger equation; Gibbs measure; global well-posedness; defocusing nonlinear Schrödinger equation
UR - http://eudml.org/doc/277714
ER -