Sharp non-uniqueness of solutions to stochastic Navier-Stokes equations

In this paper we establish a sharp non-uniqueness result for stochastic $d$-dimensional ($d\geq2$) incompressible Navier-Stokes equations. First, for every divergence free initial condition in $L^2$ we show existence of infinite many global in time probabilistically strong and analytically weak solutions in the class $L^\alpha\big(\Omega,L^p_tL^\infty\big)$ for any $1\leq p<2,\alpha\geq1$. Second, we prove the above result is sharp in the sense that pathwise uniqueness holds in the class of $L^p_tL^q$ for some $p\in[2,\infty],q\in(2,\infty]$ such that $\frac2{p}+\frac{d}{q}\leq1$, which is a stochastic version of Ladyzhenskaya-Prodi-Serrin criteria. Moreover, for stochastic $d$-dimensional incompressible Euler equation, existence of infinitely many global in time probabilistically strong and analytically weak solutions is obtained. Compared to the stopping time argument used in \cite{HZZ19, HZZ21a}, we developed a new stochastic version of the convex integration. More precisely, we introduce expectation during convex integration scheme and construct directly solutions on the whole time interval $[0,\infty)$.