Averaging dynamics for a two scales Navier-Stokes system driven by fractional Brownian Motion

The growing interest in stochastic fluid dynamics stems from the necessity to account for the unresolved scales of turbulent flows. As originally advocated in the seminal work by Brzeźniak et al. (1991), stochastic partial differential equations provide a rigorous framework to model the unpredictable evolution of small-scale features which, while not exactly trackable, significantly influence the statistical properties of the fluid. To rigorously investigate the interplay between large and small scales, we investigate a two-scale system derived from the Navier-Stokes equations, where the small scales are driven by a fractional additive noise. Our main objective is to characterize the asymptotic impact of this rough perturbation on the large-scale dynamics. Specifically, we analyze the role of the noise intensity scaling, identifying two distinct regimes that yield, respectively, a Wong-Zakai correction and an Averaging principle.

This talk is based on a joint ongoing work with Dr. Eliseo Luongo.