Probabilistic well-posedeness for the nonlinear Schr\"odinger equation on the $2d$ sphere I: positive regularities

TL;DR

This paper proves that the nonlinear Schrödinger equation on the 2D sphere is well-posed with high probability for initial data with positive regularity, surpassing deterministic limitations.

Contribution

It introduces a probabilistic approach to establish well-posedness for initial data in $H^{s}$ with $s>0$, beyond previous deterministic results.

Findings

Probabilistic well-posedness established for $H^{s}$ with $s>0$

Flow map cannot be extended uniformly continuously in this regime

Results apply to Gaussian-distributed initial data on the sphere

Abstract

We establish the probabilistic well-posedness of the nonlinear Schr\"odinger equation on the $2d$ sphere $\mathbb{S}^{2}$. The initial data are distributed according to Gaussian measures with typical regularity $H^{s}(\mathbb{S}^{2})$, for $s>0$. This level of regularity goes significantly beyond existing deterministic results, in a regime where the flow map cannot be extended uniformly continuously.