Non-unique ergodicity for stochastic 3D Navier--Stokes and Euler equations

We establish existence of infinitely many stationary solutions as well as ergodic stationary solutions to the three dimensional Navier--Stokes and Euler equations in the deterministic as well as stochastic setting, driven by an additive noise. The solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta })\cap C^{\vartheta }(\mathbb{R};L^2)$ for some $\vartheta >0$ and satisfy the equations in an analytically weak sense. The result is based on a stochastic variant of the convex integration method which provides uniform moment bounds locally in the aforementioned function spaces.