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 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 . 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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Jean Bourgain, Aynur Bulut, Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case. J. Eur. Math. Soc. 16 (2014), no. 6, pp. 1289–1325DOI 10.4171/JEMS/461