Polynomial mixing under a certain stationary Euler flow

TL;DR

This paper investigates how a scalar quantity mixed over a 2D unit ball under a specific stationary Euler flow decays over time, showing polynomial decay for the geometric mixing scale using a physical space approach.

Contribution

It introduces a physical space method to analyze the decay of the geometric mixing scale, extending previous Fourier-based results for the functional mixing scale.

Findings

Polynomial decay of geometric mixing scale for large initial data

Extension of previous Fourier-based results to physical space analysis

Quantitative decay estimates for scalar mixing under stationary Euler flows

Abstract

We study the mixing properties of a scalar $\rho$ advected by a certain incompressible velocity field $u$ on the two dimensional unit ball, which is a stationary radial solution of the Euler equation. The scalar $\rho$ solves the continuity equation with velocity field $u$ and we can measure the degree of mixedness of~$\rho$ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted for the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a large class of initial data. This extends previous results for the functional mixing scale, based on the explicit expression for the solution in Fourier variable, results that are also partially recovered by our approach.