Global well-posedness and long-time behavior of the fractional NLS

Mouhamadou Sy, Xueying Yu

Published 2020-11-27Version 1

We establish probabilistic global well-posedness results for the cubic Schr\"odinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities on both radial (in dimension $\geq 2$) and periodic settings (in all dimensions). For the high regularities, an {\it Inviscid - Infinite dimensional (IID) limit} is used while we use the Skorokhod representation for low regularity global well-posedness. The IID limit is presented in details as an independent method. Also, our discussion mainly focuses on equations with energy supercritical nonlinearities.