This talk is devoted to a probabilistic approach to the cubic nonlinear Schrödinger equation (NLS) on the two-dimensional sphere, where we study the collective behavior of random initial data supported below the deterministic threshold for the Cauchy theory.
We begin with a brief overview of the Cauchy problem for NLS on compact surfaces, and with an introduction to the Gibbs measure problem and Bourgain’s resolution scheme in the case of the flat torus. We show that on the sphere, strong instabilities arise due to the concentration of spherical harmonics around great circles, preventing a direct extension of Bourgain’s method.
We present a probabilistic quasi-linear approach, inspired by recent works from Deng, Nahmod, and Yue, designed to overcome these instabilities.
This talk is based on joint work with Nicolas Burq, Chenmin Sun, and Nikolay Tzvetkov.
Jérémy Heleine, David Lafontaine
