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 $u-z\in C([0,\infty);L^p)\cap L^2_{\rm{loc}}([0,\infty);H^\zeta)\cap L^1_{\rm{loc}}([0,\infty);W^{\frac13,1})$ 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 $u-z\in C([0,\infty);L^2)\cap L^2_{\rm{loc}}([0,\infty);H^\zeta)$ for some $\zeta\in(0,1)$ is unique.