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

2022/02/10 by Spitz, Martin
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.05224

Abstract

In this article we study the defocusing energy-critical nonlinear wave equation on ℝ⁴ with scaling supercritical data. We prove almost sure scattering for randomized initial data in H^s(ℝ⁴) × H^s-1(ℝ⁴) with (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 L¹_t L^∞_x.

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