Introduction to the theory of mixing for incompressible flows

TL;DR

This paper introduces the mathematical theory of mixing in incompressible flows, discussing both Lagrangian and Eulerian perspectives, and explores bounds on how quickly mixing occurs, with recent results on their sharpness and implications.

Contribution

It provides a comprehensive PDE-based introduction to mixing scales, establishes universal lower bounds, and discusses recent advances on their optimality and flow regularity.

Findings

Universal lower bounds on mixing scale evolution

Sharpness of these bounds demonstrated

Connections to flow regularity and geometry

Abstract

In these lecture notes, we provide an 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.