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Averaging Dynamics and Wong-Zakai approximations for a Fast-Slow Navier-Stokes System Driven by fractional Brownian Motion

Eliseo Luongo, Francesco Triggiano

TL;DR

This work analyzes a slow-fast Navier–Stokes system driven by fractional Brownian motion with $H>\tfrac{1}{3}$, using rough path methods to control the fast-scale dynamics and pass to a limiting equation for the slow scale as $\varepsilon\to0$. Depending on $H$, the limiting dynamics exhibit distinct features: for $H>\tfrac{1}{2}$ the slow equation converges to a transport-noise NS equation with no Itô–Stokes drift, while for $H<\tfrac{1}{2}$ the limit is a deterministic NS equation augmented by an Itô–Stokes drift term; at the critical case $H=\tfrac{1}{2}$ both effects coexist. The analysis relies on a precise decomposition of the fast variable, unbounded rough-driver techniques, and a detailed study of long-time behavior of quadratic functionals of fractional stochastic convolutions, including an infinite-dimensional Itô–Stokes drift. The results provide a rigorous bridge between Wong–Zakai-type limits and fractional multiscale phenomena in fluid dynamics, with implications for stochastic model reduction and understanding transport-type noise in turbulent flows.

Abstract

We study a slow-fast system of coupled two- and three-dimensional Navier-Stokes equations in which the fast component is perturbed by an additive fractional Brownian noise with Hurst parameter $H>\frac{1}{3}$. The system is analyzed using rough path theory, and the limiting behaviour strongly depends on the value of $H$. We prove convergence in law of the slow component to a Navier-Stokes system with an additional Itô-Stokes drift when $H<\frac{1}{2}$. In contrast, for $H\in (\frac{1}{2},1)$, the limit equation features only a transport noise driven by a rough path.

Averaging Dynamics and Wong-Zakai approximations for a Fast-Slow Navier-Stokes System Driven by fractional Brownian Motion

TL;DR

This work analyzes a slow-fast Navier–Stokes system driven by fractional Brownian motion with , using rough path methods to control the fast-scale dynamics and pass to a limiting equation for the slow scale as . Depending on , the limiting dynamics exhibit distinct features: for the slow equation converges to a transport-noise NS equation with no Itô–Stokes drift, while for the limit is a deterministic NS equation augmented by an Itô–Stokes drift term; at the critical case both effects coexist. The analysis relies on a precise decomposition of the fast variable, unbounded rough-driver techniques, and a detailed study of long-time behavior of quadratic functionals of fractional stochastic convolutions, including an infinite-dimensional Itô–Stokes drift. The results provide a rigorous bridge between Wong–Zakai-type limits and fractional multiscale phenomena in fluid dynamics, with implications for stochastic model reduction and understanding transport-type noise in turbulent flows.

Abstract

We study a slow-fast system of coupled two- and three-dimensional Navier-Stokes equations in which the fast component is perturbed by an additive fractional Brownian noise with Hurst parameter . The system is analyzed using rough path theory, and the limiting behaviour strongly depends on the value of . We prove convergence in law of the slow component to a Navier-Stokes system with an additional Itô-Stokes drift when . In contrast, for , the limit equation features only a transport noise driven by a rough path.
Paper Structure (18 sections, 11 theorems, 233 equations)

This paper contains 18 sections, 11 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Methodologies and Main Results
  3. Preliminaries
  4. Notation
  5. Small Scales Assumptions
  6. Rough path formulation
  7. Long-Time Behavior of Quadratic Functionals of Fractional Stochastic Convolutions
  8. One-dimensional case
  9. Infinite-dimensional case
  10. Unbounded rough drivers convergence
  11. A Priori Estimates
  12. Space Compactness
  13. Time Compactness
  14. Compactness and passage to the limit
  15. Multiscale Model
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{H}\in (\frac{1}{3},1)$, and let the initial data $u_0^{\epsilon},v_0^\epsilon$ be divergence-free, of zero mean, and such that Assume that the operators $Q$ and $C$ satisfy the conditions stated in sec_hp_noise. Then there exist probabilistically weak rough path solutions $(u^{\epsilon},(-C)^{-1}dW^{\mathcal{H},\epsilon})$ to system_prelimit that converge in law to a probabilistical

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 19 more