Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Harris theorem for enhanced dissipation, and an example of Pierrehumbert

William Cooperman, Gautam Iyer, Seungjae Son

TL;DR

This work links enhanced dissipation for the advection–diffusion equation to mixing properties of a random, divergence-free flow by employing Harris-type ergodicity for the associated two-point chain. The authors develop a κ-independent Lyapunov function and a κ-independent small set to obtain V-geometric ergodicity of the two-point chain, which then yields almost-sure exponential mixing of the stochastic flows and sharp short-time dissipation bounds on $\mathbb{T}^d$ with high probability. Their framework produces κ-dependent bounds on the dissipation and a κ-independent long-time convergence rate, and they explicitly verify these results for Pierrehumbert’s randomly shifted alternating shear flows, even constructing an explicit Lyapunov function in that case. The results provide a robust probabilistic mechanism for enhanced dissipation, with precise mixing-time estimates and broad applicability to pulsed and randomly forced advection–diffusion systems. Overall, the paper advances the understanding of how random dynamical system structure controls rapid convergence to equilibrium in dissipative PDEs, and concretely demonstrates this for a classical Floquet-type flow.

Abstract

In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium -- a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If $κ$ denotes the strength of the diffusion, then we show that with probability at least $1 - o(κ^N)$ enhanced dissipation occurs on time scales of order $|\ln κ|$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94.

A Harris theorem for enhanced dissipation, and an example of Pierrehumbert

TL;DR

This work links enhanced dissipation for the advection–diffusion equation to mixing properties of a random, divergence-free flow by employing Harris-type ergodicity for the associated two-point chain. The authors develop a κ-independent Lyapunov function and a κ-independent small set to obtain V-geometric ergodicity of the two-point chain, which then yields almost-sure exponential mixing of the stochastic flows and sharp short-time dissipation bounds on with high probability. Their framework produces κ-dependent bounds on the dissipation and a κ-independent long-time convergence rate, and they explicitly verify these results for Pierrehumbert’s randomly shifted alternating shear flows, even constructing an explicit Lyapunov function in that case. The results provide a robust probabilistic mechanism for enhanced dissipation, with precise mixing-time estimates and broad applicability to pulsed and randomly forced advection–diffusion systems. Overall, the paper advances the understanding of how random dynamical system structure controls rapid convergence to equilibrium in dissipative PDEs, and concretely demonstrates this for a classical Floquet-type flow.

Abstract

In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium -- a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If denotes the strength of the diffusion, then we show that with probability at least enhanced dissipation occurs on time scales of order , a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94.
Paper Structure (15 sections, 15 theorems, 186 equations)

This paper contains 15 sections, 15 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Motivation and Literature Review
  4. Notation and Preliminaries.
  5. Notation and setup.
  6. Two point and projective chains.
  7. Harris Conditions.
  8. Checkable Conditions that Guarantee the Harris Conditions.
  9. Proof of Theorem \ref{['t:EDIntro']}
  10. The existence of a k-independent Lyapunov function.
  11. A stable small set.
  12. Verification of the Harris Assumptions (Lemma \ref{['l:harrisassumptions']}).
  13. V-geometric ergodicity (Lemma \ref{['l: uge']}).
  14. Exponential Mixing of the Stochastic Flows (Lemma \ref{['l:X-exp-mix']}).
  15. An explicit Lyapunov function for Pierrehumbert flows.

Key Result

Theorem 1.1

Suppose the flow of $u$ generates a random dynamical system that satisfies the Harris conditions stated in Assumptions a:fts--a:flow, below. For any $\alpha > 0$, $q < \infty$ there exists $\gamma > 0$ and a $\kappa$-dependent random variable $D_\kappa$ such that for every initial data $\rho_0 \in L Moreover, there exist a $\kappa$-independent, deterministic, constant $\bar{D}_q$ such that

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6: Pulsed diffusions
  • Corollary 1.7
  • Remark 1.8
  • Proposition 2.9
  • proof : Proof of Proposition \ref{['p:checkable-conditions']}
  • ...and 24 more