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

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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.

GLOBAL EXISTENCE AND NON-UNIQUENESS FOR 3D NAVIER–STOKES EQUATIONS WITH SPACE-TIME WHITE NOISE arXiv:2112.14093v1 [math.AP] 28 Dec 2021 MARTINA HOFMANOVÁ, RONGCHAN ZHU, AND XIANGCHAN ZHU Abstract. 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…

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

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