  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.
