Mixing and Enhanced Dissipation in Measure Preserving Dynamical Systems

Abstract

The movement of particles and energy in a fluid is governed by the advection-diffusion equation. Given an underlying velocity field, a common question in fluid mechanics is to understand the motion described by the advection-diffusion equation. An interesting notion in fluid systems is the concept of mixing, the irreversible thermodynamic process seen by the mixing paint, mixing water of different temperatures, or the behavior of smoke in a smoke-filled rooms. In order to mathematically quantify mixing, we can view fluid systems as a measure preserving dynamical system. This paper will introduce the notion of measure preserving dynamical systems, quantify mixing and enhanced dissipation, and the study long term behavior of solutions to the advection diffusion equation. In particular, we provide an explicit instance of a smooth velocity field that exhibits enhanced dissipation at a rate of $\nu^{\frac13}$