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 \leq 1 if d=4 , and 0\leq s\leq 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).