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Divergence-free drifts decrease concentration

Elias Hess-Childs, Renaud Raquépas, Keefer Rowan

TL;DR

This work establishes a universal comparison principle for concentration between advection-diffusion with bounded divergence-free drifts and the heat equation on $\mathbb{R}^d$: if $\mu \preceq \nu$ with $\nu$ symmetric decreasing, then $\mathcal{P}_t^u \mu \preceq e^{t\Delta}\nu$ for all $t\ge0$, for any bounded divergence-free drift $u$. Consequently, for symmetric decreasing initial data, the advection-diffusion evolution has at least as large variance and entropy and at most as large $L^p$ norms as the pure-diffusion evolution (with no prefactor), with precise consequences for the entire spectrum of concentration via the modulus of absolute continuity. The analysis hinges on symmetric decreasing rearrangements and a split-operator approach that alternates diffusion and advection, leveraging the fact that advection preserves concentration while diffusion is order-preserving in the rearrangement order. A torus $\mathbb{T}^d$ counterexample shows the inequality can fail on compact manifolds due to broken rotational symmetry, highlighting the domain’s geometry as a fundamental factor. Overall, the results provide a robust, quantitative framework for understanding how divergence-free drifts modulate concentration, variance, entropy, and $L^p$ norms in passive scalars relevant to turbulence and mixing.

Abstract

We show that bounded divergence-free vector fields $u : [0,\infty) \times \mathbb{R}^d \to\mathbb{R}^d$ decrease the ''concentration'', quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to the associated advection-diffusion equation when compared to solutions to the heat equation. In particular, for symmetric decreasing initial data, the solution to the advection-diffusion equation has (without a prefactor constant) larger variance, larger entropy, and smaller $L^p$ norms for all $p \in [1,\infty]$ than the solution to the heat equation. We also note that the same is not true on $\mathbb{T}^d$.

Divergence-free drifts decrease concentration

TL;DR

This work establishes a universal comparison principle for concentration between advection-diffusion with bounded divergence-free drifts and the heat equation on : if with symmetric decreasing, then for all , for any bounded divergence-free drift . Consequently, for symmetric decreasing initial data, the advection-diffusion evolution has at least as large variance and entropy and at most as large norms as the pure-diffusion evolution (with no prefactor), with precise consequences for the entire spectrum of concentration via the modulus of absolute continuity. The analysis hinges on symmetric decreasing rearrangements and a split-operator approach that alternates diffusion and advection, leveraging the fact that advection preserves concentration while diffusion is order-preserving in the rearrangement order. A torus counterexample shows the inequality can fail on compact manifolds due to broken rotational symmetry, highlighting the domain’s geometry as a fundamental factor. Overall, the results provide a robust, quantitative framework for understanding how divergence-free drifts modulate concentration, variance, entropy, and norms in passive scalars relevant to turbulence and mixing.

Abstract

We show that bounded divergence-free vector fields decrease the ''concentration'', quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to the associated advection-diffusion equation when compared to solutions to the heat equation. In particular, for symmetric decreasing initial data, the solution to the advection-diffusion equation has (without a prefactor constant) larger variance, larger entropy, and smaller norms for all than the solution to the heat equation. We also note that the same is not true on .

Paper Structure

This paper contains 14 sections, 27 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Discussion and background
  4. Motivation and interpretation of results
  5. Symmetric decreasing rearrangements
  6. Related literature on PDE and rearrangements
  7. Overview of the argument
  8. Proof of Theorem \ref{['thm:main-result']}
  9. Proof of the time-independent case by operator splitting
  10. Reduction to the case of L2 data and a smooth time-independent vector field
  11. Proofs of Propositions \ref{['prop:advection-maintains-concentration']} and \ref{['prop:heat-increases-diffuseness']}
  12. Proof of Corollary \ref{['cor:main-cor']}
  13. Proofs of Propositions \ref{['prop:convex-dom']} and \ref{['prop:test-with-increasing-rearrange']}
  14. Proof of Theorem \ref{['thm:torus']}

Key Result

Theorem 1.7

Let $\mu, \nu \in \mathcal{M}_+(\mathds{R}^d)$ and let $u : [0,\infty) \times \mathds{R}^d\to \mathds{R}^d$ be a bounded, measurable, divergence-free vector field. If $\nu$ is symmetric decreasing and $\mu \preceq \nu$, then for all $t\geq 0$.

Theorems & Definitions (63)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 53 more