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

We establish that 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 of at most - 1 / 2 - κ for any κ> 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up until 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 L2∪B∞,∞-1+κ, κ> 0 , and also implies non-uniqueness in law.