Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbb{R}^3$

TL;DR

This paper proves almost sure global existence for super-critical initial data in a nonlinear wave equation on , using randomization and advanced harmonic analysis techniques.

Contribution

It introduces a novel probabilistic approach to establish global solutions for super-critical data in nonlinear wave equations.

Findings

Almost sure global existence for super-critical initial data.

Effective use of Bourgain's high-low frequency decomposition.

Improved averaging effects for randomized initial data.

Abstract

We consider the defocusing nonlinear wave equation of power-type on $\mathbb{R}^3$. We establish an almost sure global existence result with respect to a suitable randomization of the initial data. In particular, this provides examples of initial data of super-critical regularity which lead to global solutions. The proof is based upon Bourgain's high-low frequency decomposition and improved averaging effects for the free evolution of the randomized initial data.