Vol. 13, No. 4, 2020

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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
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