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

TL;DR

This paper introduces a rough path solution framework for the 3D Navier-Stokes equation with fractional Brownian motion-driven noise, linking it to multi-scale dynamics and demonstrating broader applicability to nonlinear SPDEs.

Contribution

It defines a rigorous rough path solution for Navier-Stokes with fractional transport noise and connects it to multi-scale system limits, expanding solution concepts for nonlinear SPDEs.

Findings

Solutions characterized as limits of slow/fast systems

Equivalence with incremental solution notions

Broader applicability to nonlinear SPDEs

Abstract

We define a bona fide rough path solution for the Navier-Stokes equation with an additional rough transport term, and show that the SPDE on the three-dimensional torus driven by a fractional Brownian motion on $H^\sigma$ has solutions characterised as the effective limits of a slow/fast system. We further show that this rough path solution is equivalent to the widely used incremental notion of solution (the unbounded rough driver formulation), demonstrating broader applicability to other nonlinear SPDEs.