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Decay to equilibrium for energy-reaction-diffusion systems

Please always quote using this URN:urn:nbn:de:0296-matheon-13731
  • 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, at first in entropy and further in $L^1$ using Cziszar-Kullback-Pinsker type inequalities.

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Metadaten
Author:Jan Haskovec, Sabine Hittmeir, Peter Markowich, Alexander Mielke
URN:urn:nbn:de:0296-matheon-13731
Referee:Dietmar Hömberg
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2016/02/27
Release Date:2016/02/27
Tag:Energy-reaction-diffusion systems; dissipation functional; entropy entropy-production balance; entropy functional; log-Sobolev inequality
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
Project:D Optics and Electronics (Electronic and photonic devices) / D-OT1 Mathematical modeling, analysis, and optimization of strained Germanium-microbridges
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B40 Asymptotic behavior of solutions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K57 Reaction-diffusion equations
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Qxx Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] / 35Q80 PDEs in connection with classical thermodynamics and heat transfer
Preprint Number:1104
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