Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions

Alexander Mielke

doi:10.3934/dcdss.2013.6.479

Article Contents

2013, Volume 6, Issue 2: 479-499. Doi: 10.3934/dcdss.2013.6.479

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Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions

Alexander Mielke^1,

1.
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

Received: October 31, 2011

Revised: December 31, 2011

Published: March 2013

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Abstract

We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,Φ,K)$ defining the evolution $\dot U=-K(U)D Φ(U)$. Here $Φ$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G =K ^{-1}$ exists, the triple $(X,Φ,G)$ defines a gradient system.
Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi)=1/2\langle \Xi, K(U)\Xi\rangle$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.

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Mathematics Subject Classification: Primary: 35K57, 80A17, 82B35; Secondary: 35Q72, 74F25, 82B35.

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