Global dynamics for the two-dimensional stochastic nonlinear wave equations
Abstract
We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the -method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
Cite
@article{arxiv.2005.10570,
title = {Global dynamics for the two-dimensional stochastic nonlinear wave equations},
author = {Massimiliano Gubinelli and Herbert Koch and Tadahiro Oh and Leonardo Tolomeo},
journal= {arXiv preprint arXiv:2005.10570},
year = {2021}
}
Comments
33 pages. To appear in Internat. Math. Res. Not. Minor typos corrected