Optimal stability estimates and a new uniqueness result for advection-diffusion equations

TL;DR

This paper establishes optimal stability estimates for advection-diffusion equations with Sobolev regular velocity fields and extends these results to cases with singular integral gradients, leading to new well-posedness insights.

Contribution

It introduces the first optimal stability estimates using Kantorovich-Rubinstein distances for Sobolev regular velocity fields and extends these to more singular cases with new well-posedness results.

Findings

Optimal stability estimates in Sobolev settings.

Extension to velocity fields with singular integral gradients.

New well-posedness results for complex advection-diffusion equations.

Abstract

This paper contains two main contributions. First, it provides optimal stability estimates for advection-diffusion equations in a setting in which the velocity field is Sobolev regular in the spatial variable. This estimate is formulated with the help of Kantorovich--Rubinstein distances with logarithmic cost functions. Second, the stability estimates are extended to the advection-diffusion equations with velocity fields whose gradients are singular integrals of $L^1$ functions entailing a new well-posedness result.