Almost sure global well posedness for the radial nonlinear Schr\"odinger equation on the unit ball II: the 3D case

TL;DR

This paper proves the almost sure global well-posedness of the three-dimensional nonlinear Schrödinger equation on the unit ball for supercritical initial data using probabilistic methods and Gibbs measure invariance.

Contribution

It extends probabilistic well-posedness results to 3D NLS with supercritical data, a first in this setting, employing novel probabilistic bounds and trilinear estimates.

Findings

First probabilistic global well-posedness for 3D NLS with supercritical data

Uses Gibbs measure invariance to establish almost sure convergence

Develops probabilistic a priori bounds and trilinear estimates

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 Schr\"odinger (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.