To the theory of entropy sub-solutions of degenerate non-linear parabolic equations

TL;DR

This paper establishes the existence of extremal entropy sub- and super-solutions for degenerate nonlinear parabolic equations, proving uniqueness and a comparison principle under periodic conditions.

Contribution

It introduces new existence results for extremal entropy solutions and extends the comparison principle to cases with periodic initial data.

Findings

Existence of the largest entropy sub-solution and smallest entropy super-solution.

Uniqueness of entropy solutions with periodic initial data.

A generalized comparison principle for periodic initial functions.

Abstract

We prove existence of the largest entropy sub-solution and the smallest entropy super-solution to the Cauchy problem for a nonlinear degenerate parabolic equation with only continuous flux and diffusion functions. Applying this result, we establish the uniqueness of entropy solution with periodic initial data. The more general comparison principle is also proved in the case when at least one of the initial functions is periodic.