Maximum principles for the relativistic heat equation

TL;DR

This paper investigates a relativistic heat equation that respects the finite propagation speed of signals, establishing maximum principles and comparison results that are absent in classical models, thus aligning heat conduction with relativity.

Contribution

It proves strong and weak maximum principles for a relativistic heat equation, and introduces a transformation linking it to the mean curvature operator for enhanced comparison results.

Findings

Maximum principles hold for stationary and time-varying solutions.

Transformation to mean curvature operator yields stronger comparison principles.

Relativistic heat equation aligns heat propagation with relativistic constraints.

Abstract

The classical heat equation is incompatible with relativity, since the strong maximum principle allows for disturbances to propagate instantaneously. Some authors have proposed limiting the propagation speed by adding a linear hyperbolic correction term, but then even a weak maximum principle fails to hold. We study a more recently introduced relativistic heat equation, which replaces the Laplace operator by a quasilinear elliptic operator, and show that strong and weak maximum principles hold for stationary and time-varying solutions, respectively, as well as for sub- and supersolutions. Moreover, by transforming the equation into an equivalent form, related to the mean curvature operator, we prove even stronger tangency and comparison principles.