Recent Advances on Boundary Conditions for Equations in Nonequilibrium Thermodynamics

Wen-An Yong,Yizhou Zhou

doi:10.3390/sym13091710

Abstract

This paper is concerned with modeling nonequilibrium phenomena in spatial domains with boundaries. The resultant models consist of hyperbolic systems of first-order partial differential equations with boundary conditions (BCs). Taking a linearized moment closure system as an example, we show that the structural stability condition and the uniform Kreiss condition do not automatically guarantee the compatibility of the models with the corresponding classical models. This motivated the generalized Kreiss condition (GKC)—a strengthened version of the uniform Kreiss condition. Under the GKC and the structural stability condition, we show how to derive the reduced BCs for the equilibrium systems as the classical models. For linearized problems, the validity of the reduced BCs can be rigorously verified. Furthermore, we use a simple example to show how thus far developed theory can be used to construct proper BCs for equations modeling nonequilibrium phenomena in spatial domains with boundaries.

Highlights

Irreversible thermodynamics is a theory in physics for the mathematical modeling of nonequilibrium processes
Under the generalized Kreiss condition (GKC) and the structural stability condition, we show how to derive the reduced boundary conditions (BCs) for the equilibrium systems as the classical models
We presented a systematical review on boundary conditions (BCs) for partial differential equations (PDEs) from nonequilibrium thermodynamics

Summary

Introduction

Irreversible thermodynamics is a theory in physics for the mathematical modeling of nonequilibrium processes. (1) can be written as (2) with A j (U ) = ∂Uj (U ) For such first-order PDEs, the first fundamental requirement corresponds to the hyperbolicity of Equation (2) [11,12]. Kreiss condition (UKC) [18] to ensure the well-posedness of the complete model (PDEs together with BCs)—the first fundamental requirement. The rest requirements apply to the complete model This suggests to study the zero relaxation limit of initial-boundary-value problems (IBVPs) for hyperbolic systems (1) or (2).

Preliminaries

Generalized Kreiss Condition

Reduced Boundary Conditions

Construction of BCs

Concluding Remarks