Global Existence and Non-Uniqueness for 3D Navier–Stokes Equations with Space-Time White Noise (Xiangchan Zhu)

　　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

　　Fakult？t für Mathematik, Universit？t Bielefeld

　　Rongchan Zhu

　　Department of Mathematics, Beijing Institute of Technology

　　XiangchanZhu

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences

　　Email: zhuxiangchan@126.com