On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on $\mathbb{R}^3$

Abstract

We consider energy sub-critical defocusing nonlinear wave equations on $\mathbb{R}^3$ and establish the existence of unique global solutions almost surely with respect to a unit-scale randomization of the initial data on Euclidean space. In particular, we provide examples of initial data at super-critical regularities which lead to unique global solutions. The proof is based on probabilistic growth estimates for a new modified energy functional. This work improves upon the authors' previous results in [25] by significantly lowering the regularity threshold and strengthening the notion of uniqueness.