On the almost sure scattering for the energy-critical cubic wave equation with supercritical data

Abstract

In this article we study the defocusing energy-critical nonlinear wave equation on R 4 with scaling supercritical data. We prove almost sure scattering for randomized initial data in H s (R 4 ) × H s−1 (R 4 ) with 5 6 < s < 1. The proof relies on new probabilistic estimates for the linear flow of the wave equation with randomized data, where the randomization is based on a unitscale decomposition in frequency space, a decomposition in the angular variable, and a unit-scale decomposition of physical space. In particular, we show that the solution to the linear wave equation with randomized data almost surely belongs to L 1 t L ∞ x .