Velocity averaging for diffusive transport equations with discontinuous flux

Abstract

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which are able to ``recognise'' changes of the type of the equation. As a corollary, we show the existence of a weak solution for the Cauchy problem for nonlinear degenerate parabolic equation with discontinuous flux. We also show existence of strong traces at $t=0$ for so-called quasi-solutions to degenerate parabolic equations under non-degeneracy conditions on the diffusion term.