Abstract
The nonlinear Schrödinger equation (NLS), as a typical dispersive PDE, is sensitive to the underlying geometry. While the study of NLS on R^d is quite complete, our understanding of its dynamics on compact manifolds remains limited. In this talk, I address the Cauchy problem for the nonlinear Schrödinger equation on the two-dimensional sphere. I will first review the semi-linear well-posedness theory developed by Burq, Gérard, and Tzvetkov. To go beyond the semi-linear regime, I then consider the probabilistic Cauchy theory, in which the initial data are distributed according to Gaussian measures with positive regularity in Sobolev spaces. In particular, I will explain how to resolve the Cauchy problem using a refined resolution ansatz (in the spirit of Deng, Nahmod, and Yue). This part is based on the collaboration with Nicolas Burq, Nicolas Camps, and Nikolay Tzvetkov.