We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier--Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most -1/2-\kappa for any \kappa>0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions.Our result applies to any divergence free initial condition in L^{2}\cup B^{-1+\kappa}_{\infty,\infty}, \kappa>0, and implies also non-uniqueness in law. Publication: Archive for Rational Mechanics and Analysis volume 247, Article number: 46 (2023)
Author: Martina Hofmanová, Fakulttfür Mathematik, Universitt Bielefeld; Rongchan Zhu, Department of Mathematics, Beijing Institute of Technology; Xiangchan Zhu, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (Email: zhuxiangchan@126.com)
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