Velocity averaging for diffusive transport equations with discontinuous flux

TL;DR

This paper proves velocity averaging results for diffusive transport equations with discontinuous flux, introducing new micro-local defect functionals, and establishes existence of weak solutions and strong traces under non-degeneracy conditions.

Contribution

It introduces a novel variant of micro-local defect functionals capable of recognizing equation type changes, enabling new existence and regularity results.

Findings

Proved velocity averaging under non-degeneracy conditions.

Established existence of weak solutions for nonlinear degenerate parabolic equations.

Demonstrated existence of strong traces at t=0 for quasi-solutions.

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.