Open mathematical aspects of continuum thermodynamics: hyperbolicity, boundaries and nonlinearities

TL;DR

This paper reviews the mathematical properties of continuum thermodynamics models, focusing on hyperbolicity, boundary conditions, and nonlinearities, comparing classical and extended approaches to identify open questions.

Contribution

It systematically analyzes mathematical challenges in various continuum thermodynamics frameworks, highlighting differences and open problems in their formulation and properties.

Findings

Comparison of CIT, NET-IV, and RET approaches

Identification of specific mathematical issues in each framework

Discussion of how boundary conditions and nonlinearities affect models

Abstract

Thermodynamics is continuously spreading in the engineering practice, which is especially true for non-equilibrium models in continuum problems. Although there are concepts and approaches beyond the classical knowledge, which are known for decades, their mathematical properties and consequences of the generalizations are less-known and are still of high interest in current researches. Therefore, we found it essential to collect the most important and still open mathematical questions related to different continuum thermodynamic approaches. First, we start with the example of Classical Irreversible Thermodynamics (CIT) in order to provide the basis for the more general and complex frameworks, such as the Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and Rational Extended Thermodynamics (RET). Here, we aim to present that each approach has its specific problems, such as how the initial and boundary conditions can be formulated, how the coefficients in the partial differential equations are connected to each other, and how it affects the appearance of nonlinearities. We present these properties and comparing the approach of NET-IV and RET to each other from these points of view.