Selected topics in transport phenomena: deterministic and probabilistic aspects

Critical non Sobolev regularity for continuity equations with rough force fields

Pierre-Emmanuel Jabin

University of Maryland

We prove quantitative regularity estimates for the solutions to non-linear continuity equations and their discretized numerical approximations when advected by a rough force field. This allow us to recover the known regularity for linear transport equations but also to obtain the convergence of a wide range of numerical schemes. We will also discuss recent examples testing the optimality of those regularity estimates.