Lower bounds on mixing norms for the advection diffusion equation in $\mathbb{R}^d$

TL;DR

This paper establishes algebraic lower bounds on energy decay and mixing norms for solutions to the advection-diffusion equation in two and three dimensions, advancing understanding of scalar mixing in fluids.

Contribution

It introduces new algebraic lower bounds on energy decay and mixing norms for the advection-diffusion equation using Fourier splitting and interpolation methods.

Findings

Derived algebraic lower bounds on energy decay in 2D and 3D.

Established lower bounds on the $L^2$ norm of the inverse gradient of solutions.

Provided insights into passive scalar mixing in fluid flows.

Abstract

An algebraic lower bound on the energy decay for solutions of the advection-diffusion equation in $\mathbb{R}^d$ with $d=2,3$ is derived using the Fourier splitting method. Motivated by a conjecture on mixing of passive scalars in fluids, a lower bound on the $L^2-$ norm of the inverse gradient of the solution is obtained via gradient estimates and interpolation.