Selected topics in transport phenomena: deterministic and probabilistic aspects

Optimal stability estimates for continuity equations

Christian Seis

Universit├Ąt Bonn


In their ground breaking '89 paper, R. J. DiPerna and P.-L. Lions study uniqueness and stability of solutions to linear continuity equations with Sobolev coefficients. What is not contained in their "theory of renormalized solutions" are quantitative stability estimates which allow to control the distance of two solutions if the data are varied. Such estimates, however, are indispensable in the analysis of a number of applications in fluid dynamics. In this talk, I will present a new quantitative approach to continuity equations. This approach is based on stability estimates that are formulated in terms of Kantorovich--Rubinstein distances with logarithmic costs. I will show how the new estimates can be applied to obtain optimal bounds on the order of convergence of the numerical upwind scheme or on the rate of mixing by stirring of fluids. This is partially joint work with A. Schlichting.