Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on ${\mathbb{R}}^d$, $d=4$ and $5$

Abstract

We consider the energy-critical defocusing nonlinear wave equation (NLW) on ${\mathbb{R}}^d$, $d=4$ and $5$. We prove almost sure global existence and uniqueness for NLW with rough random initial data in $H^s({\mathbb{R}}^d)\times H^{s-1}({\mathbb{R}}^d)$, with $0<s\le 1$ if $d=4$, and $0\le s\le 1$ if $d=5$. The randomization we consider is naturally associated with the Wiener decomposition and with modulation spaces. The proof is based on a probabilistic perturbation theory. Under some additional assumptions, for $d=4$, we also prove the probabilistic continuous dependence of the flow with respect to the initial data (in the sense proposed by Burq and Tzvetkov).