Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs

Regularity estimates for transport equations through "dual norms"

Pierre-Emmanuel Jabin

University of Maryland

We present a new theory for well posedness of transport equations with rough velocity fields. The theory is based on critical regularity estimates on the solution; those estimates are obtained for norms that are defined through a solution to the dual equation.

This new theory is applied to compressible fluid mechanics and other related models (inspired by biology) and allows to relax some strong assumptions on the pressure term, in particular monotonicity.