A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting

Abstract

Abstract. In this note, we prove almost sure global well-posedness of the energy-critical defocusing nonlinear wave equation on T d , d = 3, 4, and 5, with random initial data below the energy space.

Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity

“…While the proof of Lemma 1.9 is a straightforward application of the Wiener chaos estimate (Lemma 2.5), we point out that the set of probability one on which the conclusion of Lemma 1.9 holds depends on the choice of deterministic initial data (X 0 , X 1 ) ∈ H s 1 (T 3 ). This is analogous to the situation for the recent study of nonlinear dispersive PDEs with randomized initial data [14,15,60,7,80,73,74,8], where a set of probability one for local-in-time or global-in-time well-posedness depends on the choice of deterministic initial data (to which a randomization is applied). See [15] for a further discussion.…”

Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three

“…For higher dimensional case d ≥ 4, the global infinite energy solution to the cubic wave equation was constructed by Burq-Thomann-Tzvetkov [5], where the conditionally continuous dependence on the initial data is left unknown. But Oh-Pocovnicu succeeded in proving this uniqueness result in [16].…”

Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions

“…In the context of NLW, see the work [9,10] by Burq and the third author for almost sure local well-posedness. There are also globalization arguments in this probabilistic setting; see [10,32,23,24]. See also a general review [3] on the subject.…”