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, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics, engineering and numerical analysis.

A discontinuous Galerkin finite element method for an optimalcontrol problem related to semilinear parabolic PDE's is examined.The schemes under consideration are discontinuous in time butconforming in space. Convergence of discrete schemes of arbitraryorder is proven. In addition, the convergence of discontinuousGalerkin approximations of the associated optimality system to thesolutions of the continuous optimality system is shown. The proofis based on stability estimates at arbitrary time points underminimal regularity assumptions, and a discrete compactnessargument for discontinuous Galerkin schemes (see Walkington[SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).