Sharp Non-uniqueness of Solutions to 2D Navier-Stokes Equations with Space-Time White Noise

In this paper we are concerned with the 2D incompressible Navier-Stokes equations driven by space-time white noise. We establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions $u$ for every divergence free initial condition $u_0\in L^p\cup C^{-1+\delta },p\in (1,2),\delta >0$. More precisely, there exist infinitely many solutions such that $u-z\in C([0,\mathrm{\infty });L^p)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2([0,\mathrm{\infty });H^{\zeta })\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1([0,\mathrm{\infty });W^{\frac{1}{3},1})$ for some $\zeta \in (0,1)$, where $z$ is the solution to the linear equation. This result in particular implies non-uniqueness in law. Our result is sharp in the sense that the solution satisfying $u-z\in C([0,\mathrm{\infty });L^2)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2([0,\mathrm{\infty });H^{\zeta })$ for some $\zeta \in (0,1)$ is unique.