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 ${\mathbb{R}}^4$ with supercritical initial data. For randomized initial data in $H^s({\mathbb{R}}^4)$, we prove almost sure local wellposedness for $\frac{1}{7}<s<1$ and almost sure scattering for $\frac{5}{7}<s<1$. 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.

33 pages. v2: Regularity threshold in Proposition 3.7 lowered, leading to an improvement of Theorem 1.2. Recent references added