Velocity averaging for diffusive transport equations with discontinuous flux

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