Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index

We show non-uniqueness of local strong solutions to stochastic fractional Navier-Stokes equations with linear multiplicative noise and some certain deterministic force. Such non-uniqueness holds true even if we perturb such deterministic force in appropriate sense.This is closely related to a critical condition on force under which Leray-Hopf solution to the stochastic equations is locally unique. Meanwhile, by a new idea, we show that for some stochastic force the system admits two different global Leray-Hopf solutions smooth on any compact subset of $(0,\infty) \times \mathbb{R}^d$.