Quantitative dependence of the Pierrehumbert flow's mixing rate on the amplitude

TL;DR

The paper addresses how the exponential mixing rate of the Pierrehumbert flow on the torus depends on the flow amplitude $A$. It constructs explicit Lyapunov functions and coupling trajectories for the associated two-point Markov chain and applies a quantitative Harris theorem to obtain geometric ergodicity with amplitude-dependent constants, yielding the bound $\,\|\phi_n\|_{H^{-1}} \le D_A \, e^{-e^{-A^{96}} n} \|\phi_0\|_{H^1}$ (a.s. for large $A$) and moment bounds on $D_A$. This provides the first explicit quantitative upper bound on the Pierrehumbert flow’s mixing rate and clarifies how amplitude controls mixing through uniformly controlled small sets and a Lyapunov drift. The results open avenues toward sharper scalings (potentially $\,\gamma(A) \sim A$) and showcase a rigorous route for extracting explicit amplitude effects in chaotic advection models, with potential implications for geophysical and engineering mixing processes.

Abstract

We quantitatively study the mixing rate of randomly shifted alternating shears on the torus. This flow was introduced by Pierrehumbert '94, and was recently shown to be exponentially mixing. In this work, we quantify the dependence of the exponential mixing rate on the flow amplitude. Our approach is based on constructing an explicit Lyapunov function and a coupling trajectory for the associated two-point Markov chain, together with an application of the quantitative Harris theorem.

Theorems & Definitions (26)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['l:R1-uniform-small']}
• Lemma 3.3: Quantitative Inverse Function Theorem Christ85
• Lemma 3.4: Quantitative Implicit Function Theorem