Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-stokes equations: beyond the Lions exponents

We consider the 3D stochastic Navier-Stokes equations (NSE) on torus where the viscosity exponent can be larger than the Lions exponent 5/4. For arbitrarily prescribed divergence-free initial data in $L^{2}_x$, we construct infinitely many probabilistically strong and analytically weak solutions in the class $L^{r}_{\Omega}L_{t}^{\gamma}W_{x}^{s,p}$, where $r\geq1$ and $(s, \gamma, p)$ lie in two supercritical regimes with respect to the Lady\v{z}henskaya-Prodi-Serrin (LPS) criteria.It shows that even in the high viscosity regime beyond the Lions exponent, though solutions are unique in the Leray-Hopf class, the uniqueness fails in the mixed Lebesgue spaces and, actually, there exist infinitely manly non-Leray-Hopf solutions which can be very close to the Leray-Hopf solutions. Furthermore, we prove the vanishing noise limit result, which relates together the stochastic solutions and the deterministic solutions constructed by Buckmaster-Vicol [4] and the recent work [23].