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Exponential mixing for Hamiltonian shear flow

Weili Zhang

TL;DR

This work analyzes passive scalar advection by a time-periodic, analytic velocity field that randomly switches between two Hamiltonian shear flows on the torus. By formulating the dynamics as random Markov chains on the configuration space and its projective and two-point extensions, the authors establish $V$-uniform geometric ergodicity of the one-point chain, show that the top Lyapunov exponent is positive, and prove exponential mixing in Bressan's sense for the advection equation. The framework is then applied to the Pierrehumbert model with randomized time and to a Chirikov-type map, demonstrating that these systems exhibit robust chaotic mixing under random switching. The results rely on controllability via Lie brackets, strong Feller smoothing, and Lyapunov–Foster drift conditions, yielding quantitative mixing rates and validating exponential mixing for a broad class of random Hamiltonian shear flows.

Abstract

We consider the advection equation on $\mathbb{T}^2$ with a real analytic and time-periodic velocity field that alternates between two Hamiltonian shears. Randomness is injected by alternating the vector field randomly in time between just two distinct shears. We prove that, under general conditions, these models have a positive top Lyapunov exponent and exhibit exponential mixing. This framework is then applied to the Pierrehumbert model with randomized time and to a model analogous to the Chirikov standard map.

Exponential mixing for Hamiltonian shear flow

TL;DR

This work analyzes passive scalar advection by a time-periodic, analytic velocity field that randomly switches between two Hamiltonian shear flows on the torus. By formulating the dynamics as random Markov chains on the configuration space and its projective and two-point extensions, the authors establish -uniform geometric ergodicity of the one-point chain, show that the top Lyapunov exponent is positive, and prove exponential mixing in Bressan's sense for the advection equation. The framework is then applied to the Pierrehumbert model with randomized time and to a Chirikov-type map, demonstrating that these systems exhibit robust chaotic mixing under random switching. The results rely on controllability via Lie brackets, strong Feller smoothing, and Lyapunov–Foster drift conditions, yielding quantitative mixing rates and validating exponential mixing for a broad class of random Hamiltonian shear flows.

Abstract

We consider the advection equation on with a real analytic and time-periodic velocity field that alternates between two Hamiltonian shears. Randomness is injected by alternating the vector field randomly in time between just two distinct shears. We prove that, under general conditions, these models have a positive top Lyapunov exponent and exhibit exponential mixing. This framework is then applied to the Pierrehumbert model with randomized time and to a model analogous to the Chirikov standard map.

Paper Structure

This paper contains 13 sections, 25 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Uniformly geometrically ergodic']}
  3. Markov chain preliminaries
  4. Uniform geometric ergodicity of the one-point Markov chain
  5. Proof of Theorem \ref{['Positivity of the top Lyapunov exponent']}
  6. Consequences of $2\lambda_1=\lambda_{\Sigma}$
  7. Strong Feller
  8. Ruling out the degeneracy
  9. Proof of Theorem \ref{['Exponential mixing']}
  10. Uniform geometric ergodicity for projective chain
  11. Uniform geometric ergodicity for two-point chain
  12. Exponential mixing
  13. Proofs of Corollary \ref{['The Pierrehumbert model with randomized time']} and Corollary \ref{['Analog model to the Chirikov standard map']}

Key Result

Theorem 1.1

Let $\{\Phi_{\underline{\tau}}^m\}$ be as above. Assume $(H1)$ holds. Then there exists a continuous function $V:\mathcal{O}\to [1,\infty)$ such that the transition kernel $P$ of $\{\Phi_{\underline{\tau}}^m\}$ is $V$-uniformly geometrically ergodic.

Theorems & Definitions (49)

  • Theorem 1.1: Uniformly geometrically ergodic
  • Theorem 1.2: Positivity of the top Lyapunov exponent
  • Theorem 1.3: Exponential mixing
  • Corollary 1.4: The Pierrehumbert model with randomized time
  • Corollary 1.5: Analog model to the Chirikov standard map
  • Remark 1.6
  • Theorem 2.1: BCZG, Theorem 2.3
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 39 more