An Interesting Class of Partial Differential Equations

2007/07/25 by Wen‐An Yong, Yong, Wen-an · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.0707.3708

openalex publication_date 2007/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

Cited by

• Validity of relaxation models arising from numerical schemes for hyperbolic-parabolic systems
• Numerical boundary control of multi-dimensional discrete-velocity kinetic models

Related

• Minimum Rate of Dissipation Principle and Linear Corrections to the Onsager Matrix
• Hyperbolic systems of conservation laws in one space dimension
• Onsager relations and Eulerian hydrodynamics for systems with several conservation laws
• The Dynamics of Thermodynamics
• Entropy-Time Geodesics as a Universal Framework for Transport and Transition Phenomena