# 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 (2014)

- Volume: 016, Issue: 6, page 1289-1325
- ISSN: 1435-9855

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topBourgain, 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 -

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