April  2013, 6(2): 479-499. doi: 10.3934/dcdss.2013.6.479

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

1. 

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

Received  November 2011 Revised  January 2012 Published  November 2012

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.
Citation: Alexander Mielke. Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 479-499. doi: 10.3934/dcdss.2013.6.479
References:
[1]

Günter Albinus, Herbert Gajewski and Rolf Hünlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002), 367-383.

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (editors,,H.-J. Schmeisser and et al.), Teubner-Texte Math. 133, 9-126. Teubner, Stuttgart, (1993).

[3]

Denis Anders, Kerstin Weinberg and Roland Reichardt, Isogeometric analysis of thermal diffusion in binary blends, Computational Materials Science, 52 (2012), 182-188.

[4]

Dick Bedeaux, Nonequilibrium thermodynamics and statistical physics of surfaces, in "Advance in Chemical Physics" (editors, I. Prigogine and S. A. Rice), \textbf LXIV, 47-109. John Wiley & Sons, Inc., 1986.

[5]

Dieter Bothe and Michel Pierre, The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction, Discr. Cont. Dynam. Systems Ser. S, 8 (2011), 49-59.

[6]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics," Dover Publ., New York, 1984. doi: 10.1063/1.34645.

[7]

Laurent Desvillettes and Klemens Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.

[8]

Laurent Desvillettes and Klemens Fellner, Entropy methods for reaction-diffusion systems, in "Discrete Contin. Dyn. Syst. (suppl). Dynamical Systems and Differential Equations," Proceedings of the 6th AIMS International Conference, (2007), 304-312.

[9]

Michael Ederer, "Thermokinetic Modeling and Model Reduction of Reaction Networks," PhD thesis, Universität Stuttgart, Nov. 2009.

[10]

Brian J. Edwards, An analysis of single and double generator thermodynamics formalisms for the macroscopic description of complex fluids, J. Non-Equilib. Thermodyn., 23 (1998), 301-333.

[11]

Eduard Feireisl and Giulio Schimperna, Large time behaviour of solutions to Penrose-Fife phase change models, Math. Methods Appl. Sci. (MMAS), 28 (2005), 2117-2132.

[12]

Michel Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.

[13]

Vincent Giovangigli and Marc Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Methods Appl. Sci. (MMAS), 27 (2004), 739-768.

[14]

A. Glitzky and R. Hünlich, Global existence result for pair diffusion models, SIAM J. Math. Analysis, 36 (2005), 1200-1225. (electronic).

[15]

Annegret Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, Discr. Cont. Dynam. Systems Ser. A, 25 (2009), 159-174.

[16]

Annegret Glitzky and Alexander Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. angew. Math. Phys. (ZAMP), 2011. Submitted. WIAS preprint 1603.

[17]

Klaus Hackl, Generalized standard media and variational principles in classical and finite strain elastoplasticity, J. Mech. Phys. Solids, 45 (1997), 667-688.

[18]

Bernard Halphen and Quoc Son Nguyen, Sur les matériaux standards généralisés, J. Mécanique}, 14 (1975), 39-63.

[19]

Richard Jordan, David Kinderlehrer, and Felix Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17.

[20]

Signe Kjelstrup and Dick Bedeaux, "Non-equilibrium Thermodynamics of Heterogeneous Systems," volume 16 of Series on Advances in Statistical Mechanics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs. Amer. Math. Soc., Providence, R. I., 1968.

[22]

A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case, in "Multifield Problems in Solid and Fluid Mechanics" (editors, R. Helmig, A. Mielke and B. I. Wohlmuth), 351-379. Springer-Verlag, Berlin, 2006.

[23]

Alexander Mielke, Formulation of thermoelastic dissipative material behavior using GENERIC, Contin. Mech. Thermodyn., 23 (2011), 233-256. doi: 10.1007/s00161-010-0179-0.

[24]

Alexander Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.

[25]

Alexander Mielke, On thermodynamically consistent models and gradient structures for thermoplasticity, GAMM Mitt., 34 (2011), 51-58.

[26]

Lars Onsager, Reciprocal relations in irreversible processes, I+{II}, Physical Review, 37 (1931), 405-426. (part II, 38, 2265-227).

[27]

Hans Christian Öttinger, "Beyond Equilibrium Thermodynamics," John Wiley, New Jersey, 2005.

[28]

Hans Christian Öttinger and Miroslav Grmela, Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E (3), 56 (1997), 6633-6655.

[29]

Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.

[30]

Oliver Penrose and Paul C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62.

[31]

Oliver Penrose and Paul C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model, Physica D, 69 (1993), 107-113.

[32]

Wen-An Yong, An interesting class of partial differential equations, J. Math. Phys., 49 (2008) pp. 21. 033503.

show all references

References:
[1]

Günter Albinus, Herbert Gajewski and Rolf Hünlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002), 367-383.

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (editors,,H.-J. Schmeisser and et al.), Teubner-Texte Math. 133, 9-126. Teubner, Stuttgart, (1993).

[3]

Denis Anders, Kerstin Weinberg and Roland Reichardt, Isogeometric analysis of thermal diffusion in binary blends, Computational Materials Science, 52 (2012), 182-188.

[4]

Dick Bedeaux, Nonequilibrium thermodynamics and statistical physics of surfaces, in "Advance in Chemical Physics" (editors, I. Prigogine and S. A. Rice), \textbf LXIV, 47-109. John Wiley & Sons, Inc., 1986.

[5]

Dieter Bothe and Michel Pierre, The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction, Discr. Cont. Dynam. Systems Ser. S, 8 (2011), 49-59.

[6]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics," Dover Publ., New York, 1984. doi: 10.1063/1.34645.

[7]

Laurent Desvillettes and Klemens Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176.

[8]

Laurent Desvillettes and Klemens Fellner, Entropy methods for reaction-diffusion systems, in "Discrete Contin. Dyn. Syst. (suppl). Dynamical Systems and Differential Equations," Proceedings of the 6th AIMS International Conference, (2007), 304-312.

[9]

Michael Ederer, "Thermokinetic Modeling and Model Reduction of Reaction Networks," PhD thesis, Universität Stuttgart, Nov. 2009.

[10]

Brian J. Edwards, An analysis of single and double generator thermodynamics formalisms for the macroscopic description of complex fluids, J. Non-Equilib. Thermodyn., 23 (1998), 301-333.

[11]

Eduard Feireisl and Giulio Schimperna, Large time behaviour of solutions to Penrose-Fife phase change models, Math. Methods Appl. Sci. (MMAS), 28 (2005), 2117-2132.

[12]

Michel Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002.

[13]

Vincent Giovangigli and Marc Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Methods Appl. Sci. (MMAS), 27 (2004), 739-768.

[14]

A. Glitzky and R. Hünlich, Global existence result for pair diffusion models, SIAM J. Math. Analysis, 36 (2005), 1200-1225. (electronic).

[15]

Annegret Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, Discr. Cont. Dynam. Systems Ser. A, 25 (2009), 159-174.

[16]

Annegret Glitzky and Alexander Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. angew. Math. Phys. (ZAMP), 2011. Submitted. WIAS preprint 1603.

[17]

Klaus Hackl, Generalized standard media and variational principles in classical and finite strain elastoplasticity, J. Mech. Phys. Solids, 45 (1997), 667-688.

[18]

Bernard Halphen and Quoc Son Nguyen, Sur les matériaux standards généralisés, J. Mécanique}, 14 (1975), 39-63.

[19]

Richard Jordan, David Kinderlehrer, and Felix Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17.

[20]

Signe Kjelstrup and Dick Bedeaux, "Non-equilibrium Thermodynamics of Heterogeneous Systems," volume 16 of Series on Advances in Statistical Mechanics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Transl. Math. Monographs. Amer. Math. Soc., Providence, R. I., 1968.

[22]

A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case, in "Multifield Problems in Solid and Fluid Mechanics" (editors, R. Helmig, A. Mielke and B. I. Wohlmuth), 351-379. Springer-Verlag, Berlin, 2006.

[23]

Alexander Mielke, Formulation of thermoelastic dissipative material behavior using GENERIC, Contin. Mech. Thermodyn., 23 (2011), 233-256. doi: 10.1007/s00161-010-0179-0.

[24]

Alexander Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.

[25]

Alexander Mielke, On thermodynamically consistent models and gradient structures for thermoplasticity, GAMM Mitt., 34 (2011), 51-58.

[26]

Lars Onsager, Reciprocal relations in irreversible processes, I+{II}, Physical Review, 37 (1931), 405-426. (part II, 38, 2265-227).

[27]

Hans Christian Öttinger, "Beyond Equilibrium Thermodynamics," John Wiley, New Jersey, 2005.

[28]

Hans Christian Öttinger and Miroslav Grmela, Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E (3), 56 (1997), 6633-6655.

[29]

Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.

[30]

Oliver Penrose and Paul C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62.

[31]

Oliver Penrose and Paul C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model, Physica D, 69 (1993), 107-113.

[32]

Wen-An Yong, An interesting class of partial differential equations, J. Math. Phys., 49 (2008) pp. 21. 033503.

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