Rough analysis of two scale systems

TL;DR

This paper studies a coupled slow-fast Navier--Stokes system with stochastic perturbations, proving convergence of the slow component to a limit driven by rough path theory, revealing new insights into stochastic fluid dynamics.

Contribution

It provides a new, more general proof of convergence for a stochastic Navier--Stokes system using rough path theory, identifying the limiting rough path as geometric.

Findings

Convergence of the slow component to a rough path driven Navier--Stokes system.

Identification of the limiting rough path as a geometric rough path.

An alternative proof to previous results using rough path techniques.

Abstract

We address a slow-fast system of coupled three dimensional Navier--Stokes equations where the fast component is perturbed by an additive Brownian noise. By means of the rough path theory, we establish the convergence in law of the slow component towards a Navier--Stokes system with an It{\^o}--Stokes drift and a rough path driven transport noise. This gives an alternative, more general and direct proof to \cite{DP22}. Notably, the limiting rough path is identified as a geometric rough path, which does not necessarily coincide with the Stratonovich lift of the Brownian motion.