American Institute of Mathematical Sciences

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.
 [1] Roman Shvydkoy. Lectures on the Onsager conjecture. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 473-496. doi: 10.3934/dcdss.2010.3.473 [2] Wenji Chen, Jianfeng Zhou. Global existence of weak solutions to inhomogeneous Doi-Onsager equations. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021257 [3] Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940 [4] Yanheng Ding, Fukun Zhao. On a diffusion system with bounded potential. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1073-1086. doi: 10.3934/dcds.2009.23.1073 [5] Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206 [6] Myoungjean Bae, Hyangdong Park. Three-dimensional supersonic flows of Euler-Poisson system for potential flow. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2421-2440. doi: 10.3934/cpaa.2021079 [7] Manuel Falconi, E. A. Lacomba, C. Vidal. The flow of classical mechanical cubic potential systems. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 827-842. doi: 10.3934/dcds.2004.11.827 [8] Jin-Zan Liu, Xin-Wei Liu. A dual Bregman proximal gradient method for relatively-strongly convex optimization. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021028 [9] Gui-Qiang Chen, Bo Su. A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1587-1606. doi: 10.3934/dcds.2003.9.1587 [10] Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573 [11] Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377 [12] Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389 [13] Volker W. Elling. Shock polars for non-polytropic compressible potential flow. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1581-1594. doi: 10.3934/cpaa.2022032 [14] Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial and Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167 [15] Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems and Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031 [16] Bin Li, Hai Huyen Dam, Antonio Cantoni. A low-complexity zero-forcing Beamformer design for multiuser MIMO systems via a dual gradient method. Numerical Algebra, Control and Optimization, 2016, 6 (3) : 297-304. doi: 10.3934/naco.2016012 [17] Xiayang Zhang, Yuqian Kong, Shanshan Liu, Yuan Shen. A relaxed parameter condition for the primal-dual hybrid gradient method for saddle-point problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022008 [18] Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747 [19] Lu Chen, Zhao Liu, Guozhen Lu. Qualitative properties of solutions to an integral system associated with the Bessel potential. Communications on Pure and Applied Analysis, 2016, 15 (3) : 893-906. doi: 10.3934/cpaa.2016.15.893 [20] Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195

2020 Impact Factor: 2.425