Separation of time-scales in drift-diffusion equations on $\mathbb{R}^2$

TL;DR

This paper investigates the separation of time-scales and filamentation in a linear drift-diffusion equation on 2, revealing a faster mixing time-scale due to flow-diffusion interaction, using hypocoercivity methods.

Contribution

It identifies a specific fast time-scale for mixing in radial flows and adapts hypocoercivity techniques to establish semigroup estimates in weighted spaces.

Findings

Mixing occurs at a faster time-scale than diffusion due to flow interaction.

The fast time-scale depends only on the flow's behavior at the origin.

The method employs an adapted hypocoercivity scheme for analysis.

Abstract

We deal with the problem of separation of time-scales and filamentation in a linear drift-diffusion problem posed on the whole space $\mathbb{R}^2$. The passive scalar considered is stirred by an incompressible flow with radial symmetry. We identify a time-scale, much faster than the diffusive one, at which mixing happens along the streamlines, as a result of the interaction between transport and diffusion. This effect is also known as enhanced dissipation. For power-law circular flows, this time-scale only depends on the behavior of the flow at the origin. The proofs are based on an adaptation of a hypocoercivity scheme and yield a linear semigroup estimate in a suitable weighted $L^2$-based space.