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

TL;DR

This paper proves that for the defocusing energy-critical cubic wave equation in four dimensions, solutions with supercritical initial data almost surely scatter, using novel probabilistic estimates and randomization techniques.

Contribution

It introduces new probabilistic estimates for the linear wave flow with randomized data in supercritical regimes, establishing almost sure scattering.

Findings

Almost sure scattering for supercritical data in 4D wave equation

New probabilistic estimates for linear wave flow with randomized data

Solutions belong to $L^1_t L^_x$ almost surely

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 $L^1_t L^\infty_x$.