Degenerate parabolic equations -- compactness and regularity of solutions

TL;DR

This paper introduces a novel method for establishing regularity and compactness of entropy solutions to nonlinear degenerate parabolic equations, addressing both homogeneous and heterogeneous cases using kinetic formulations and advanced measure techniques.

Contribution

The paper presents a new approach that reduces degenerate parabolic PDEs to kinetic formulations, enabling regularity and existence results under non-degeneracy conditions.

Findings

Established regularity of entropy solutions for homogeneous equations

Proved existence of weak solutions for heterogeneous equations

Developed a kinetic reformulation method for degenerate parabolic PDEs

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.