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 ${\mathbb{R}}^4$ with scaling supercritical data. We prove almost sure scattering for randomized initial data in $H^s({\mathbb{R}}^4)\times H^{s-1}({\mathbb{R}}^4)$ with $\frac{5}{6}<s<1$. 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 unit-scale 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 $L_t^1{L}_x^{\infty }$.

30 pages