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

Abstract

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 $u-z\in C([0,\infty );L^p)\cap L_{\mathrm{loc}}^2([0,\infty );H^{\zeta })\cap L_{\mathrm{loc}}^1([0,\infty );W^{\frac{1}{3},1})$ 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 $u-z\in C([0,\infty );L^2)\cap L_{\mathrm{loc}}^2([0,\infty );H^{\zeta })$ for some $\zeta \in (0,1)$ is unique.

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