Mielke, Alexander Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. (English) Zbl 1262.35127 Discrete Contin. Dyn. Syst., Ser. S 6, No. 2, 479-499 (2013). Summary: 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 \((X, \Phi, K)\) defining the evolution \(\dot{U} = -K(U)D\Phi(U)\). Here \(\Phi\) is the driving functional, while the Onsager operator \(K(U)\) is symmetric and positive semidefinite. If the inverse \(G = K^{-1}\) exists, the triple \((X, \Phi, G)\) defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential \(\Psi^\ast(U, \Xi) = 1/2 \langle \Xi, K(U) \Xi \rangle\). Then, the two functionals \(\Phi\) and \(\Psi^\ast\) 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. Cited in 33 Documents MSC: 35K57 Reaction-diffusion equations 80A17 Thermodynamics of continua 82B35 Irreversible thermodynamics, including Onsager-Machlup theory 74F25 Chemical and reactive effects in solid mechanics 35K51 Initial-boundary value problems for second-order parabolic systems Keywords:gradient flow; Onsager system; Onsager operator; dual dissipation potential; dual entropy-production potential; thermionic emission; reversible reactions Software:GENERIC PDFBibTeX XMLCite \textit{A. Mielke}, Discrete Contin. Dyn. Syst., Ser. S 6, No. 2, 479--499 (2013; Zbl 1262.35127) Full Text: DOI