Comparison and maximum principles for a class of flux-limited diffusions with external force fields

TL;DR

This paper studies a class of flux-limited diffusion equations compatible with relativity, establishing comparison and maximum principles for both stationary and evolutionary cases, using a transformation aligned with the gradient flow structure.

Contribution

It extends flux-limited diffusion models to include external forces and proves comparison and maximum principles for these equations.

Findings

Stationary solutions satisfy a comparison principle.

Evolutionary solutions satisfy a weaker maximum principle.

A transformation technique effectively links the equation to its gradient flow structure.

Abstract

In this paper, we are interested in a general equation that has finite speed of propagation compatible with Einstein's theory of special relativity. This equation without external force fields has been derived recently by means of optimal transportation theory. We first provide an argument to incorporate the external force fields. Then we are concerned with comparison and maximum principles for this equation. We consider both stationary and evolutionary problems. We show that the former satisfies a comparison principle and a strong maximum principle. While the latter fulfils weaker ones. The key technique is a transformation that matches well with the gradient flow structure of the equation.