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

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.

arXiv comment49 pages, 5 figures

References in corpus (8)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Random data Cauchy theory for supercritical wave equations II : A global existence result
• Ill-posedness for nonlinear Schrodinger and wave equations
• Generalized and weighted Strichartz estimates
• Almost sure local well-posedness for the supercritical quintic NLS
• On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
• On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on ${\mathbb{R}}^3$
• Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^3$

Cited by in corpus (2)

• Probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation
• Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases