Navier-Stokes with a fractional transport noise as a limit of multi-scale dynamics

We analyze a slow/fast system describing a velocity field of a fluid evolving on a three-dimensional torus. The fast equation is a fractional Ornstein-Uhlenbeck process, driven by a fractional Brownian motion with $H<\frac{1}{2}$. We show that the limit points of the slow process are solutions to a Navier-Stokes equation driven by a transport fractional noise. The main challenges arise from the need to use rough path techniques in PDE problems to establish a suitable notion of solution, as well as showing suitable convergence of an additive functional of the fast process in the absence of the usual ergodic properties.