A Boundary Value Problem for a Class of Anisotropic Degenerate Parabolic-Hyperbolic Equations

TL;DR

This paper studies a boundary value problem for a class of degenerate parabolic-hyperbolic equations, establishing existence, uniqueness, and boundary trace properties of entropy solutions in a mixed boundary setting.

Contribution

It introduces a new strong trace property for solutions using a boundary trace formula for divergence-measure fields, and proves well-posedness for the mixed boundary problem.

Findings

Established a strong boundary trace property for solutions.

Proved existence and uniqueness of entropy solutions.

Applied boundary trace formula to mixed boundary conditions.

Abstract

We consider a mixed type boundary value problem for a class of degenerate parabolic-hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for $L^2$-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove existence and uniqueness of the entropy solution. This is a revised corrected version of the paper published in Arch.\ Ration.\ Mech.\ Anal.\ {\bf 226} (2017), no. 3, 975--1008.