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

Bringmann, Bjoern

doi:10.2140/apde.2020.13.1011

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

Bjoern Bringmann

Vol. 13 (2020), No. 4, 1011–1050

DOI: 10.2140/apde.2020.13.1011

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.

Keywords

nonlinear wave equation, probabilistic well-posedness, scattering, spherical symmetry

Mathematical Subject Classification 2010

Primary: 35L05, 35L15, 35L71

Milestones

Received: 24 April 2018

Accepted: 18 April 2019

Published: 13 June 2020

Authors

Bjoern Bringmann
Department of Mathematics University of California Los Angeles, CA United States

Vol. 13, No. 4, 2020