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

Abstract

We establish probabilistic global well-posedness results for the cubic Schrödinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities on both radial (in dimension ≥ 2) and periodic settings (in all dimensions). For the high regularities, an 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.