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

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.