Degenerate parabolic equations -- compactness and regularity of solutions

Abstract

We introduce a new method which resolves the problem of regularity and compactness of entropy solutions for nonlinear degenerate parabolic equations under non-degeneracy conditions on the sphere. In particular, we address a problem of regularity of entropy solutions to homogeneous (autonomous) degenerate parabolic PDEs and existence of weak solutions to heterogeneous degenerate parabolic PDEs (non-autonomous PDEs -- with flux and diffusion explicitly depending on the space and time variables). The method of proof is reduction of the equation to a specific kinetic formulation involving two transport equations, one of the second and one of the first order. In the heterogeneous situation, this enables us to use (variants of) the H-measures to get velocity averaging lemmas and then, consequently, existence of a weak solution. In the homogeneous case, the kinetic reformulation makes it possible to get necessary estimates of the solution on the Littlewood-Paley dyadic blocks of the dual space.