Almost sure scattering for the radial energy critical nonlinear wave equation in three dimensions

TL;DR

This paper proves that, with high probability, solutions to the radial energy critical nonlinear wave equation in three dimensions scatter, using probabilistic methods and refined estimates to handle initial data below the energy space.

Contribution

It introduces a radial randomization technique and probabilistic Strichartz estimates to establish almost sure scattering for initial data below the energy space.

Findings

Almost sure scattering for radial initial data below energy space.

Development of a radial randomization based on annular Fourier multipliers.

Use of a triple bootstrap argument to control nonlinear energy growth.

Abstract

We study the Cauchy problem for the radial energy critical nonlinear wave equation in three dimensions. Our main result proves almost sure scattering for radial initial data below the energy space. In order to preserve the spherical symmetry of the initial data, we construct a radial randomization that is based on annular Fourier multipliers. We then use a refined radial Strichartz estimate to prove probabilistic Strichartz estimates for the random linear evolution. The main new ingredient in the analysis of the nonlinear evolution is an interaction flux estimate between the linear and nonlinear components of the solution. We then control the energy of the nonlinear component by a triple bootstrap argument involving the energy, the Morawetz term, and the interaction flux estimate.