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 & 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.   Google Scholar

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, (1993), 9.   Google Scholar

[3]

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

[4]

Dick Bedeaux, Nonequilibrium thermodynamics and statistical physics of surfaces,, in, (1986), 47.   Google Scholar

[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.   Google Scholar

[6]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics,", Dover Publ., (1984).  doi: 10.1063/1.34645.  Google Scholar

[7]

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

[8]

Laurent Desvillettes and Klemens Fellner, Entropy methods for reaction-diffusion systems,, in, (2007), 304.   Google Scholar

[9]

Michael Ederer, "Thermokinetic Modeling and Model Reduction of Reaction Networks,", PhD thesis, (2009).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

Michel Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002).   Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[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).   Google Scholar

[17]

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

[18]

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

[19]

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

[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., (2008).   Google Scholar

[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., (1968).   Google Scholar

[22]

A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case,, in, (2006), 351.   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

Lars Onsager, Reciprocal relations in irreversible processes, I+{II},, Physical Review, 37 (1931), 405.   Google Scholar

[27]

Hans Christian Öttinger, "Beyond Equilibrium Thermodynamics,", John Wiley, (2005).   Google Scholar

[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.   Google Scholar

[29]

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

[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.   Google Scholar

[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.   Google Scholar

[32]

Wen-An Yong, An interesting class of partial differential equations,, J. Math. Phys., 49 (2008).   Google Scholar

show all references

References:
[1]

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

[2]

Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, (1993), 9.   Google Scholar

[3]

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

[4]

Dick Bedeaux, Nonequilibrium thermodynamics and statistical physics of surfaces,, in, (1986), 47.   Google Scholar

[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.   Google Scholar

[6]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics,", Dover Publ., (1984).  doi: 10.1063/1.34645.  Google Scholar

[7]

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

[8]

Laurent Desvillettes and Klemens Fellner, Entropy methods for reaction-diffusion systems,, in, (2007), 304.   Google Scholar

[9]

Michael Ederer, "Thermokinetic Modeling and Model Reduction of Reaction Networks,", PhD thesis, (2009).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

Michel Frémond, "Non-Smooth Thermomechanics,", Springer-Verlag, (2002).   Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[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).   Google Scholar

[17]

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

[18]

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

[19]

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

[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., (2008).   Google Scholar

[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., (1968).   Google Scholar

[22]

A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case,, in, (2006), 351.   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

Lars Onsager, Reciprocal relations in irreversible processes, I+{II},, Physical Review, 37 (1931), 405.   Google Scholar

[27]

Hans Christian Öttinger, "Beyond Equilibrium Thermodynamics,", John Wiley, (2005).   Google Scholar

[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.   Google Scholar

[29]

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

[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.   Google Scholar

[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.   Google Scholar

[32]

Wen-An Yong, An interesting class of partial differential equations,, J. Math. Phys., 49 (2008).   Google Scholar

[1]

Roman Shvydkoy. Lectures on the Onsager conjecture. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 473-496. doi: 10.3934/dcdss.2010.3.473

[2]

Yanheng Ding, Fukun Zhao. On a diffusion system with bounded potential. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1073-1086. doi: 10.3934/dcds.2009.23.1073

[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]

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 & Continuous Dynamical Systems - A, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206

[5]

Manuel Falconi, E. A. Lacomba, C. Vidal. The flow of classical mechanical cubic potential systems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 827-842. doi: 10.3934/dcds.2004.11.827

[6]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

[7]

Gui-Qiang Chen, Bo Su. A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1587-1606. doi: 10.3934/dcds.2003.9.1587

[8]

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 & Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389

[9]

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

[10]

Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial & Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167

[11]

Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems & Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031

[12]

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 & Optimization, 2016, 6 (3) : 297-304. doi: 10.3934/naco.2016012

[13]

Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297

[14]

Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014

[15]

Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006

[16]

Lu Chen, Zhao Liu, Guozhen Lu. Qualitative properties of solutions to an integral system associated with the Bessel potential. Communications on Pure & Applied Analysis, 2016, 15 (3) : 893-906. doi: 10.3934/cpaa.2016.15.893

[17]

Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195

[18]

Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385

[19]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control & Related Fields, 2016, 6 (1) : 95-112. doi: 10.3934/mcrf.2016.6.95

[20]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a conserved phase field system with a possibly singular potential. Evolution Equations & Control Theory, 2018, 7 (1) : 95-116. doi: 10.3934/eect.2018006

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]