Nicolas Camps, Université de Nantes
This talk is devoted to a statistical approach to nonlinear waves on surfaces. Specifically, we consider the cubic Schrödinger equation on the 2 dimensional sphere and we study the collective behavior of solutions on the support of the Gibbs measure. We first evidence strong instabilities due to concentrated spherical harmonics around a great circle, which make the Cauchy problem very challenging. Then, we present a probabilistic quasi-linear resolution scheme to prove long time existence of solutions on the support of the Gibbs measure.
This is ongoing joint work with Nicolas Burq, Chenmin Sun and Nikolay Tzvetkov.