Journal
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
Volume 16, Issue 6, Pages 1289-1325Publisher
EUROPEAN MATHEMATICAL SOC
DOI: 10.4171/JEMS/461
Keywords
Gibbs measure; global well-posedness; defocusing nonlinear Schrodinger equation
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Funding
- NSF [DMS-0808042, DMS-0835373]
- Fernholz Foundation
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We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrodinger (NLS) and nonlinear wave (NLW) equations on the unit ball in 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 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.
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