We derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitly calculable constants and establish the convergence to thermodynamical equilibrium, first in entropy and later in L1 norm using Cziszár–Kullback–Pinsker type inequalities.
Bibliographical noteFunding Information:
The work of the first and third authors was supported by KAUST baseline funds and grant 1000000193. The work of the second author was supported by the Austrian Science Fund via the Hertha-Firnberg project T-764, and the previous funding by the Austrian Academy of Sciences ÖAW via the New Frontiers project NST-000. The work of the fourth author was partially supported by Einstein-Stiftung Berlin through the Matheon-Project OT1.
© 2018 Society for Industrial and Applied Mathematics.
- Gradient flows
- Maximum entropy principle
- Onsager system
- Thermodynamical reaction-diffusion systems
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics