An Interesting Class of Partial Differential Equations

TL;DR

This paper identifies three structural properties common to many PDEs in physics, linking entropy, thermodynamics, and relaxation processes, and proposes an approximation method validated across diverse systems.

Contribution

It introduces a new observation connecting PDE properties to thermodynamic principles and develops an approximation method for relaxation problems.

Findings

Properties imply entropy dissipation conditions

Approximation method effectively solves relaxation problems

Validated across eight different physical systems

Abstract

This paper presents an observation that under reasonable conditions, many partial differential equations from mathematical physics possess three structural properties. One of them can be understand as a variant of the celebrated Onsager reciprocal relation in Modern Thermodynamics. It displays a direct relation of irreversible processes to the entropy change. We show that the properties imply various entropy dissipation conditions for hyperbolic relaxation problems. As an application of the observation, we propose an approximation method to solve relaxation problems. Moreover, the observation is interpreted physically and verified with eight (sets of) systems from different fields.