Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform-in-diffusivity mixing by shear flows: stochastic and dynamical perspectives

Kyle L. Liss, Kunhui Luan

Abstract

We study passive scalar mixing by parallel shear flows in the presence of weak molecular diffusion. We recover the sharp uniform-in-diffusivity mixing rate for shear flows with finitely many critical points, recently proven in [1]. Our approach is based on the stochastic representation formula of the associated advection-diffusion equation and yields two short proofs. The first uses a stochastic integration-by-parts argument and gives optimal mixing under the weakest regularity assumption required in the zero-diffusion case, answering Question II in [1, Section 4]. The second adopts a dynamical systems perspective and provides a proof of shear-induced mixing that, to our knowledge, is new even in the zero-diffusivity setting.

Uniform-in-diffusivity mixing by shear flows: stochastic and dynamical perspectives

Abstract

We study passive scalar mixing by parallel shear flows in the presence of weak molecular diffusion. We recover the sharp uniform-in-diffusivity mixing rate for shear flows with finitely many critical points, recently proven in [1]. Our approach is based on the stochastic representation formula of the associated advection-diffusion equation and yields two short proofs. The first uses a stochastic integration-by-parts argument and gives optimal mixing under the weakest regularity assumption required in the zero-diffusion case, answering Question II in [1, Section 4]. The second adopts a dynamical systems perspective and provides a proof of shear-induced mixing that, to our knowledge, is new even in the zero-diffusivity setting.
Paper Structure (6 sections, 8 theorems, 112 equations)

This paper contains 6 sections, 8 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Discussion of the proof
  4. Key stochastic lemmas
  5. Stochastic Integration by parts: Proof of Theorem \ref{['thm:main']}
  6. Dynamical argument: Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1.1

Let $b: \mathbb{T} \to \mathbb{R}$ satisfy Assumption assumption. There exists $C > 0$, depending only on $b$, such that for all $\nu > 0$ and $f_0 \in \ell^2_k(W_y^{1,1})$ satisfying eq:meanzero, the associated smooth solution of eq:ADE satisfies

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 2.3
  • ...and 9 more