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Mixing Estimates for Passive Scalar Transport by $BV$ Vector Fields

Lucas Huysmans, Ayman Rimah Said

TL;DR

The paper establishes the first explicit quantitative mixing bounds for passive scalar transport driven by divergence-free vector fields in the BV class. By quantifying Ambrosio's local regularisation with anisotropic Alberti mollifiers and employing a five- and seven-term error decomposition, the authors derive explicit, BV-specific harmonic estimates and leverage a pigeonhole principle to reduce to a finite parameter cover. A tetration-type bound emerges, reflecting the local nature of the regularisation and the difficulty of high-frequency control for BV fields. The results bridge a gap between BV regularity and mixing rates, providing both a forward-macing estimate and a backward-macing corollary that recover Ambrosio's seminal well-posedness in a quantitative framework.

Abstract

We prove a quantitative mixing estimate for the Cauchy problem for transport along divergence-free vector fields with bounded variation. By developing a framework that quantifies Ambrosio's regularisation scheme, we derive the first explicit bounds on the mixing rate for general $BV$ vector fields. Our analysis reveals that tetration (repeated exponentiation) emerges in the mixing rate from the local nature of Ambrosio's regularisation.

Mixing Estimates for Passive Scalar Transport by $BV$ Vector Fields

TL;DR

The paper establishes the first explicit quantitative mixing bounds for passive scalar transport driven by divergence-free vector fields in the BV class. By quantifying Ambrosio's local regularisation with anisotropic Alberti mollifiers and employing a five- and seven-term error decomposition, the authors derive explicit, BV-specific harmonic estimates and leverage a pigeonhole principle to reduce to a finite parameter cover. A tetration-type bound emerges, reflecting the local nature of the regularisation and the difficulty of high-frequency control for BV fields. The results bridge a gap between BV regularity and mixing rates, providing both a forward-macing estimate and a backward-macing corollary that recover Ambrosio's seminal well-posedness in a quantitative framework.

Abstract

We prove a quantitative mixing estimate for the Cauchy problem for transport along divergence-free vector fields with bounded variation. By developing a framework that quantifies Ambrosio's regularisation scheme, we derive the first explicit bounds on the mixing rate for general vector fields. Our analysis reveals that tetration (repeated exponentiation) emerges in the mixing rate from the local nature of Ambrosio's regularisation.

Paper Structure

This paper contains 13 sections, 7 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline of the proof
  4. Quantifying the Ambrosio regularisation
  5. Harmonic estimates in $BV$
  6. The pigeonhole principle
  7. Main results
  8. Proof of Proposition \ref{['prop:fiveterms']}
  9. Proof of Proposition \ref{['prop:seventerms']}
  10. Proof of Theorem \ref{['thm:quantitativeBV']} and Corollary \ref{['thm:mixingBV']}
  11. Appendix
  12. Proof of Proposition \ref{['prop:nonquantitativeBV']}
  13. Proof of Corollary \ref{['cor: proof of Ambrosio']}

Key Result

Theorem A

Let $\kappa>0$. For constants $A,B>0$ depending only on the dimension $d\ge2$, if where $\exp^n(x)$ refers to repeated exponentiation $\underbrace{\exp(\exp(\dots(\exp(x))))}_{n}$ (tetration), then

Theorems & Definitions (13)

  • Theorem A
  • Corollary B: Mixing bound
  • Proposition B
  • Proposition B
  • Corollary B
  • Proposition B
  • Theorem \ref{thm:quantitativeBV}
  • Remark \ref{thm:quantitativeBV}
  • proof
  • proof
  • ...and 3 more