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

Xue-Mei Li, Szymon Sobczak

TL;DR

This work develops a robust rough-path framework for the Navier–Stokes equation with a rough transport noise and proves that a slow/fast multi-scale model on the 3D torus converges, in the rough-path sense, to a limit SPDE driven by transport-type fractional Brownian noise. The authors establish an equivalence between rough-path solutions and the URD (unbounded rough driver) formulation, constructing the rough integral in a weak formulation and proving a rough functional limit theorem for the fast fractional OU component. They obtain uniform-in-$\varepsilon$ bounds, compactness, and convergence to a limit equation, and discuss extensions to non-Gaussian inputs via Hermite processes. The results provide a rigorous pathwise description of effective dynamics in turbulent-like systems with long-range dependence, enabling broader applicability to nonlinear SPDEs with rough transport terms.

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 $H^σ$ 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.

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

TL;DR

This work develops a robust rough-path framework for the Navier–Stokes equation with a rough transport noise and proves that a slow/fast multi-scale model on the 3D torus converges, in the rough-path sense, to a limit SPDE driven by transport-type fractional Brownian noise. The authors establish an equivalence between rough-path solutions and the URD (unbounded rough driver) formulation, constructing the rough integral in a weak formulation and proving a rough functional limit theorem for the fast fractional OU component. They obtain uniform-in- bounds, compactness, and convergence to a limit equation, and discuss extensions to non-Gaussian inputs via Hermite processes. The results provide a rigorous pathwise description of effective dynamics in turbulent-like systems with long-range dependence, enabling broader applicability to nonlinear SPDEs with rough transport terms.

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 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.
Paper Structure (15 sections, 26 theorems, 199 equations)

This paper contains 15 sections, 26 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Controls and rough path spaces
  4. Gaussian rough path lifts in Hilbert spaces
  5. Rough path solutions and their equivalence with URD solutions
  6. Construction of the rough integral in the weak formulation
  7. Unbounded rough drivers associated with fractional transport noise
  8. A-priori estimates for URD solutions
  9. Equivalence of solutions
  10. Rough functional limit theorem for the fast process
  11. Spectral decomposition and the stationary fOU
  12. First-level convergence
  13. Proof of the rough functional limit theorem
  14. Uniform bounds and proof of the main theorem
  15. Discussion on non-Gaussian inputs

Key Result

Theorem A

Assume that $M,Q$ satisfy Assumption assumption. Let $H\in \left(\frac{1}{3}, \frac{1}{2}\right)$, $q\in (1, \infty)$, $\sigma>2+ \frac{3}{2}$, and $\frac{1}{3}<\alpha<H$. Then $(X^{\epsilon}_{\cdot}, \mathbb{X}^{\epsilon}_{ \cdot\cdot})\to (M^{-1} Q^{\frac{1}{2}} B^H,\eta)$ in $L^{q}\bigl(\Omega;\

Theorems & Definitions (52)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem C
  • Proposition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 42 more