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Finite time mixing and enhanced dissipation for 2D Navier--Stokes equations by Ornstein--Uhlenbeck flow

Chang Liu, Dejun Luo

TL;DR

The paper analyzes the vorticity formulation of the 2D Navier–Stokes equations on the torus perturbed by a divergence-free Ornstein–Uhlenbeck transport flow, treated in a pathwise sense. By choosing the OU flow so that the Itô correction reduces to a simple diffusion term, the authors compare the stochastic system to a deterministic equation with augmented viscosity $( u+ ext{diffusion})$ and establish two main results: a quantitative mixing bound in negative Sobolev norms and, for suitably large perturbation strength $\nu$ and carefully tuned parameters $(\alpha,\theta)$, an almost-sure exponential dissipation of energy. The analysis hinges on a random distribution $f$ capturing deviations from the deterministic dynamics, a Young integral framework, and a meticulous decomposition of nonlinear terms into manageable components, balanced against the covariance structure of the OU flow. The results provide a rigorous pathwise mechanism for mixing and enhanced dissipation under OU perturbations, with implications for eddy-viscosity-type effects in fluid models.

Abstract

We consider the vorticity form of 2D Navier--Stokes equations perturbed by an Ornstein--Uhlenbeck flow of transport type. Contrary to previous works where the random perturbation was interpreted as Stratonovich transport noise, here we understand the equation in a pathwise manner and show the properties of mixing and enhanced dissipation for suitable choice of the flow.

Finite time mixing and enhanced dissipation for 2D Navier--Stokes equations by Ornstein--Uhlenbeck flow

TL;DR

The paper analyzes the vorticity formulation of the 2D Navier–Stokes equations on the torus perturbed by a divergence-free Ornstein–Uhlenbeck transport flow, treated in a pathwise sense. By choosing the OU flow so that the Itô correction reduces to a simple diffusion term, the authors compare the stochastic system to a deterministic equation with augmented viscosity and establish two main results: a quantitative mixing bound in negative Sobolev norms and, for suitably large perturbation strength and carefully tuned parameters , an almost-sure exponential dissipation of energy. The analysis hinges on a random distribution capturing deviations from the deterministic dynamics, a Young integral framework, and a meticulous decomposition of nonlinear terms into manageable components, balanced against the covariance structure of the OU flow. The results provide a rigorous pathwise mechanism for mixing and enhanced dissipation under OU perturbations, with implications for eddy-viscosity-type effects in fluid models.

Abstract

We consider the vorticity form of 2D Navier--Stokes equations perturbed by an Ornstein--Uhlenbeck flow of transport type. Contrary to previous works where the random perturbation was interpreted as Stratonovich transport noise, here we understand the equation in a pathwise manner and show the properties of mixing and enhanced dissipation for suitable choice of the flow.
Paper Structure (14 sections, 21 theorems, 244 equations)

This paper contains 14 sections, 21 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Preparations
  3. Useful Estimates
  4. Proofs of main results
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['thm2']}
  7. Proof of Proposition \ref{['main proposition']}
  8. Decomposition
  9. The terms $I_{21}(h),I_{22}(h),I_{24}(h),I_{31}(h)$
  10. The term $I_{231}(h)$
  11. The term $I_{25}(h)$
  12. The term $I_1(h)+\delta (u_{h\delta} \cdot \nabla \xi_{h\delta})$
  13. The terms $I_a$ and $I_b$
  14. Proof of Proposition \ref{['main proposition']}

Key Result

Theorem 1.1

Let $\xi_0\in L^2(\mathbb{T}^2)$ and $\xi,\, \bar{\xi}$ be the unique solutions of SPDE and PDE respectively. Then for any $\gamma \in (0,\frac{1}{3})$, $\vartheta>0$ and $T \geq 1$, there exist $\zeta \in (0,1)$ and $\epsilon>0$ such that for $\alpha$ sufficiently large, it holds where $C_1>0$ is a constant depending on $\kappa, \nu, \zeta, \gamma, T$ and $C_2>0$ only depends on $\kappa,\nu, T$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Interpolation inequality
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Example 2.7
  • ...and 33 more