Optimal stability estimates for continuity equations

TL;DR

This paper reviews stability estimates for the continuity equation with Sobolev regular velocity fields, comparing recent optimal bounds with previous results and demonstrating their applications in physics, engineering, and numerical analysis.

Contribution

It provides a comprehensive comparison of recent optimal stability estimates with earlier bounds and illustrates their practical applications.

Findings

Recent estimates are shown to be optimal.

Comparison highlights improvements over previous bounds.

Applications demonstrate relevance in various scientific fields.

Abstract

This review paper is concerned with the stability analysis of the continuity equation in the DiPerna--Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently (Seis 2016), but optimality was not discussed. In this paper, we revisit the results from (Seis 2016), compare the new estimates with previously known estimates for Lagrangian flows, e.g.\ (Crippa & De Lellis 2008), and finally demonstrate how those can be applied to produce optimal bounds in applications from physics, engineering or numerics.