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Exponential Lower Bounds for the Advection-Diffusion Equation with Shear Flows

Yupei Huang, Xiaoqian Xu

TL;DR

This work analyzes the 2D advection–diffusion equation on the torus with shear flows and proves an exponential lower bound on the $L^2$ norm of mean-zero solutions, showing diffusion can suppress filamentation rather than always enhancing mixing. It combines Fourier analysis, resolvent estimates, and constructive shear-flow designs to demonstrate both the inevitability of exponential decay bounds under certain conditions and the possibility of diffusion-induced suppression of mixing even for flows that are strong mixers in $H^{-1}$. The authors also contrast simultaneous advection–diffusion with pulsed-diffusion scenarios and present a 1D toy model indicating that, in some simplified settings, faster-than-exponential decay is not attainable. Collectively, the results refine our understanding of the balance between filamentation and homogenization and provide rigorous support for diffusion’s stabilizing role in passive-scalar mixing.

Abstract

In this paper, we prove that the $L^2$ norm of spatial mean-free solutions to the advection--diffusion equation on $\mathbb{T}^2$ with shear drifts satisfies an \emph{exponential lower bound} in time. This lower bound shows that diffusion can fundamentally suppress passive-scalar mixing.

Exponential Lower Bounds for the Advection-Diffusion Equation with Shear Flows

TL;DR

This work analyzes the 2D advection–diffusion equation on the torus with shear flows and proves an exponential lower bound on the norm of mean-zero solutions, showing diffusion can suppress filamentation rather than always enhancing mixing. It combines Fourier analysis, resolvent estimates, and constructive shear-flow designs to demonstrate both the inevitability of exponential decay bounds under certain conditions and the possibility of diffusion-induced suppression of mixing even for flows that are strong mixers in . The authors also contrast simultaneous advection–diffusion with pulsed-diffusion scenarios and present a 1D toy model indicating that, in some simplified settings, faster-than-exponential decay is not attainable. Collectively, the results refine our understanding of the balance between filamentation and homogenization and provide rigorous support for diffusion’s stabilizing role in passive-scalar mixing.

Abstract

In this paper, we prove that the norm of spatial mean-free solutions to the advection--diffusion equation on with shear drifts satisfies an \emph{exponential lower bound} in time. This lower bound shows that diffusion can fundamentally suppress passive-scalar mixing.

Paper Structure

This paper contains 6 sections, 15 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Exponential Lower bound of the advection diffusion equation
  3. Exponential mixing by shear flow
  4. One counterexample for pulsed-diffusion method
  5. One 1D model equation
  6. Appendix

Key Result

Theorem 2.1

Consider the initial value problem on the torus $\mathbb{T}^2=[0,2\pi]^2$: Suppose that $\mathbf{u}$ is divergence-free, $\mathbf{u}\in L^{\infty}([0,T];L^2(\mathbb{T}^2))$, and that $\rho(0,\cdot)\in L^2(\mathbb{T}^2)$. Then there exists a unique distributional solution $\rho$ to eqn:ad such that Moreover, for all $T\in (0,\infty)$, we have

Theorems & Definitions (31)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • proof : Proof of Theorem \ref{['main']}
  • Lemma 2.1
  • proof : Proof of the Lemma \ref{['lemma1']}
  • Corollary 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.1
  • ...and 21 more