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Quantitative Estimates in Passive Scalar Transport: A Unified Approach in $W^{1,p}$ via Christ-Journé Commutator Estimates

Lucas Huysmans, Ayman Rimah Said

TL;DR

This work develops a unified harmonic-analytic framework based on Christ-Journé commutator estimates to study passive scalar transport by divergence-free fields in $W^{1,p}$, with $p>1$. It derives a quantitative Cauchy-problem stability bound via frequency cutoffs, yielding a dimensionless mixing rate that depends on the initial scalar/velocity ratio and the accumulated gradient norm. The authors further establish propagation of a full logarithm of the derivative for the passive scalar and connect their harmonic estimates to decay properties of the DiPerna–Lions commutator, giving integral decay bounds and a Besov-regularity perspective. These results unify and extend previous quantitative analyses of mixing, vanishing-diffusion limits, and regularity, and lay groundwork for BV extensions. Overall, the paper provides a robust, scale-aware framework for stability, mixing, and diffusion limits in passive-scalar transport under rough velocity fields.

Abstract

This paper establishes new applications of the Christ-Journé singular integral estimate to the transport equation for divergence-free vector fields in the Sobolev class $W^{1,p}$ with $p>1$. Our main result is a stability estimate for the Cauchy problem, which quantifies the continuous dependence of solutions on initial data in the weak topology by giving bounds on the transfer of mass from high frequencies to low frequencies. Among other applications to stability and vanishing diffusion, this implies a mixing bound valid for all initial data in the DiPerna-Lions well-posedness class, with the mixing rate expressed in terms of a dimensionless ratio of the passive scalar and vector field. We also demonstrate that the passive scalar propagates a full "logarithm of a derivative" and finally discuss connections between the commutator estimates developed in this paper and uniform decay rates for the standard DiPerna-Lions commutator.

Quantitative Estimates in Passive Scalar Transport: A Unified Approach in $W^{1,p}$ via Christ-Journé Commutator Estimates

TL;DR

This work develops a unified harmonic-analytic framework based on Christ-Journé commutator estimates to study passive scalar transport by divergence-free fields in , with . It derives a quantitative Cauchy-problem stability bound via frequency cutoffs, yielding a dimensionless mixing rate that depends on the initial scalar/velocity ratio and the accumulated gradient norm. The authors further establish propagation of a full logarithm of the derivative for the passive scalar and connect their harmonic estimates to decay properties of the DiPerna–Lions commutator, giving integral decay bounds and a Besov-regularity perspective. These results unify and extend previous quantitative analyses of mixing, vanishing-diffusion limits, and regularity, and lay groundwork for BV extensions. Overall, the paper provides a robust, scale-aware framework for stability, mixing, and diffusion limits in passive-scalar transport under rough velocity fields.

Abstract

This paper establishes new applications of the Christ-Journé singular integral estimate to the transport equation for divergence-free vector fields in the Sobolev class with . Our main result is a stability estimate for the Cauchy problem, which quantifies the continuous dependence of solutions on initial data in the weak topology by giving bounds on the transfer of mass from high frequencies to low frequencies. Among other applications to stability and vanishing diffusion, this implies a mixing bound valid for all initial data in the DiPerna-Lions well-posedness class, with the mixing rate expressed in terms of a dimensionless ratio of the passive scalar and vector field. We also demonstrate that the passive scalar propagates a full "logarithm of a derivative" and finally discuss connections between the commutator estimates developed in this paper and uniform decay rates for the standard DiPerna-Lions commutator.
Paper Structure (9 sections, 17 theorems, 169 equations)

This paper contains 9 sections, 17 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Well-posedness and stability of the Cauchy problem
  3. Mixing
  4. Regularity of the passive scalar
  5. Connection with the standard commutator
  6. Outline of paper
  7. Quantitative well-posedness estimates
  8. Mixing, regularity, and other applications
  9. Estimates on the DiPerna-Lions commutator

Key Result

Lemma \ref{thm:calderon}

For the commutator we have the bound where $\frac{1}{r} = \frac{1}{p}+\frac{1}{q}$ with $1\le r<\infty$, and where $\left\|\cdot\right\|_{CZ}$ refers to the Calderon-Zygmund norm, i.e. the minimum constant such that

Theorems & Definitions (34)

  • Lemma \ref{thm:calderon}
  • Theorem \ref{thm:quantitative2}
  • Theorem \ref{thm:mixing}
  • Lemma \ref{thm:mixing}
  • proof
  • Definition \ref{thm:mixing}: Mollifier
  • Definition \ref{thm:mixing}: Frequency cutoff
  • Theorem \ref{thm:mixing}
  • proof
  • Definition \ref{thm:mixing}: $W_x^{-1,r}$
  • ...and 24 more