$L^1$-stability of periodic stationary solutions of scalar convection-diffusion equations

TL;DR

This paper investigates the $L^1$-stability of periodic stationary solutions in scalar convection-diffusion equations, establishing dispersion in $L^2$ for all dimensions and $L^1$-stability in one dimension through sign change estimates.

Contribution

It provides new $L^2$ dispersion results for all dimensions and a novel $L^1$-stability proof in one dimension using sign change analysis.

Findings

Dispersion in $L^2$ for all space dimensions.

$L^1$-stability established in one dimension.

Sign change estimates are key to stability analysis.

Abstract

The aim of this paper is to study the $L^1$-stability of periodic stationary solutions of scalar convection-diffusion equations. We obtain dispersion in $L^2$ for all space dimensions using Kru\v{z}kov type entropy. And when the space dimension is one, we estimate the number of sign changes of a solution to obtain $L^1$-stability.