Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Abstract

We study the cubic defocusing nonlinear Schrödinger equation on R4 with supercritical initial data. For randomized initial data in Hs(R4), we prove almost sure local wellposedness for 17<s<1 and almost sure scattering for 57<s<1. The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and – for the almost sure scattering result – an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, where we effectively combine the advantages of the different decompositions.