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Introduction to the theory of mixing for incompressible flows

Gianluca Crippa

TL;DR

This work develops a PDE-based framework for understanding mixing of a passive scalar by incompressible flows on the torus, introducing geometric and functional mixing scales to quantify homogenization. It derives universal lower bounds on mixing scales under natural energy constraints, using both Eulerian energy estimates and Lagrangian flow methods, and extends to Sobolev velocity fields via the regular Lagrangian flow theory. The text provides sharpness results through constructions like Bressan’s mixing scheme and explores long-time behavior, regularity, and potential limitations of the theory in non-Lipschitz settings, including connections to the DiPerna–Lions–Ambrosio framework. It also outlines a harmonic-analysis toolkit and a logarithmic flow-regularity functional that enable quantitative estimates and the demonstration of exponential decay rates under broad hypotheses, while highlighting open problems such as the BV case and universal mixer constructions. Overall, the notes establish a rigorous, quantitative PDE perspective on mixing that links geometric and spectral measures, flow regularity, and dynamical systems insights to characterize mixing rates and their optimality.

Abstract

In these lecture notes, we provide a pedagogical introduction to the theory of mixing for incompressible flows from a PDE perspective. We discuss both the Lagrangian (ODE) and Eulerian (PDE, continuity equation) viewpoints, and introduce suitable notions of mixing scales that quantify the degree to which a scalar field transported by a velocity field becomes mixed. We then address the problem of establishing universal lower bounds on the time evolution of the mixing scale. This is first done in the smooth setting, using energy estimates and flow-based arguments, and later in the Sobolev setting, relying on quantitative estimates for regular Lagrangian flows. Finally, we present recent results concerning the sharpness of these lower bounds, their implications for the geometry and regularity of regular Lagrangian flows, and connections with more recent developments in the literature.

Introduction to the theory of mixing for incompressible flows

TL;DR

This work develops a PDE-based framework for understanding mixing of a passive scalar by incompressible flows on the torus, introducing geometric and functional mixing scales to quantify homogenization. It derives universal lower bounds on mixing scales under natural energy constraints, using both Eulerian energy estimates and Lagrangian flow methods, and extends to Sobolev velocity fields via the regular Lagrangian flow theory. The text provides sharpness results through constructions like Bressan’s mixing scheme and explores long-time behavior, regularity, and potential limitations of the theory in non-Lipschitz settings, including connections to the DiPerna–Lions–Ambrosio framework. It also outlines a harmonic-analysis toolkit and a logarithmic flow-regularity functional that enable quantitative estimates and the demonstration of exponential decay rates under broad hypotheses, while highlighting open problems such as the BV case and universal mixer constructions. Overall, the notes establish a rigorous, quantitative PDE perspective on mixing that links geometric and spectral measures, flow regularity, and dynamical systems insights to characterize mixing rates and their optimality.

Abstract

In these lecture notes, we provide a pedagogical introduction to the theory of mixing for incompressible flows from a PDE perspective. We discuss both the Lagrangian (ODE) and Eulerian (PDE, continuity equation) viewpoints, and introduce suitable notions of mixing scales that quantify the degree to which a scalar field transported by a velocity field becomes mixed. We then address the problem of establishing universal lower bounds on the time evolution of the mixing scale. This is first done in the smooth setting, using energy estimates and flow-based arguments, and later in the Sobolev setting, relying on quantitative estimates for regular Lagrangian flows. Finally, we present recent results concerning the sharpness of these lower bounds, their implications for the geometry and regularity of regular Lagrangian flows, and connections with more recent developments in the literature.

Paper Structure

This paper contains 20 sections, 92 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The geometric mixing scale and the functional mixing scale
  3. The geometric mixing scale
  4. The functional mixing scale
  5. A brief overview of fractional Sobolev spaces
  6. Energy estimates for the functional mixing scale
  7. Velocity fields with uniformly-in-time Lipschitz regularity
  8. Velocity fields with uniformly-in-time bounded kinetic energy
  9. Velocity fields with uniformly-in-time bounded enstrophy
  10. Estimates using the flow for uniformly-in-time Lipschitz velocity fields
  11. Lower bound for the functional mixing scale
  12. Lower bound for the geometric mixing scale
  13. Bressan's mixing scheme and a glimpse of DiPerna-Lions-Ambrosio theory
  14. An integral functional measuring the logarithmic regularity of the flow
  15. Some tools from harmonic analysis
  16. ...and 5 more sections

Figures (6)

  • Figure 1: Numerical simulation of a mixing evolution (taken from MR2801050).
  • Figure 2: An initial configuration, and a more mixed final configuration.
  • Figure 3: Finding the optimal resolution at which the two phases are balanced.
  • Figure 4: Bressan's mixing scheme
  • Figure 5: Intermediate stages and shear flows in Bressan's mixing scheme.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Remark 1.4
  • Remark 1.5
  • Remark 4.4
  • Remark 4.5
  • Remark 7.4
  • Remark 7.5
  • Remark 8.3
  • Remark 8.5
  • Remark 8.6
  • Remark 8.7