October  2014, 34(10): 4259-4290. doi: 10.3934/dcds.2014.34.4259

The Penrose-Fife phase-field model with coupled dynamic boundary conditions

1. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex

2. 

Dipartimento di Matematica, Università di Milano, Via Saldini, 50, I-20133 Milano

3. 

Dipartimento di Matematica, Università di Pavia, Via Ferrata, 1, I-27100 Pavia

4. 

Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy

Received  January 2013 Revised  April 2013 Published  April 2014

In this paper we derive, starting from the basic principles of ther- modynamics, an extended version of the nonconserved Penrose-Fife phase tran- sition model, in which dynamic boundary conditions are considered in order to take into account interactions with walls. Moreover, we study the well- posedness and the asymptotic behavior of the initial-boundary value problem for the PDE system associated to the model, allowing the phase con guration of the material to be described by a singular function.
Citation: Alain Miranville, Elisabetta Rocca, Giulio Schimperna, Antonio Segatti. The Penrose-Fife phase-field model with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4259-4290. doi: 10.3934/dcds.2014.34.4259
References:
[1]

H. Attouch, Variational Convergence for Functions and Operators,, Applicable Mathematics Series, (1984). Google Scholar

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (1976). Google Scholar

[3]

M. Bonforte and J.L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations,, Adv. Math., 223 (2010), 529. doi: 10.1016/j.aim.2009.08.021. Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Éspaces de Hilbert,, North-Holland Math. Studies, (1973). Google Scholar

[5]

F. Brezzi and G. Gilardi, FEM Mathematics,, in Finite Element Handbook (Ed. H. Kardestuncer), (1987), 1. Google Scholar

[6]

G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar

[7]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions,, Nonlinear Anal., 72 (2010), 2375. doi: 10.1016/j.na.2009.11.002. Google Scholar

[8]

L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials,, J. Math. Anal. Appl., 343 (2008), 557. doi: 10.1016/j.jmaa.2008.01.077. Google Scholar

[9]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials,, Appl. Math., 54 (2009), 89. doi: 10.1007/s10492-009-0008-6. Google Scholar

[10]

P. Colli and Ph. Laurençot, Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws,, Phys. D, 111 (1998), 311. doi: 10.1016/S0167-2789(97)80018-8. Google Scholar

[11]

M. Conti, S. Gatti and A. Miranville, Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions,, Discrete Contin. Dyn. Syst. S, 5 (2012), 485. doi: 10.3934/dcdss.2012.5.485. Google Scholar

[12]

E. Feireisl and G. Schimperna, Large time behaviour of solutions to Penrose-Fife phase change models,, Math. Methods Appl. Sci., 28 (2005), 2117. doi: 10.1002/mma.659. Google Scholar

[13]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, Phys. Rev. Letters, 79 (1997), 893. doi: 10.1103/PhysRevLett.79.893. Google Scholar

[14]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, Europhys. Letters, 42 (1998), 49. Google Scholar

[15]

H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall,, J. Chem. Phys., 108 (1998), 3028. doi: 10.1063/1.475690. Google Scholar

[16]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 689. doi: 10.3934/cpaa.2009.8.689. Google Scholar

[17]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 881. doi: 10.3934/cpaa.2009.8.881. Google Scholar

[18]

G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, Chin. Ann. Math. Ser. B, 31 (2010), 679. doi: 10.1007/s11401-010-0602-7. Google Scholar

[19]

G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non-permeable walls,, Phys. D, 240 (2011), 754. doi: 10.1016/j.physd.2010.12.007. Google Scholar

[20]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials,, Discrete Contin. Dyn. Syst., 28 (2010), 67. doi: 10.3934/dcds.2010.28.67. Google Scholar

[21]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential,, Z. Anal. Anwend., 25 (2006), 51. doi: 10.4171/ZAA/1277. Google Scholar

[22]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988). Google Scholar

[23]

W. Horn, J. Sprekels and S. Zheng, Global existence of smooth solutions to the Penrose-Fife model for Ising ferromagnets,, Adv. Math. Sci. Appl., 6 (1996), 227. Google Scholar

[24]

A. Ito, N. Kenmochi and M. Kubo, Non-isothermal phase transition models with Neumann boundary conditions,, Nonlinear Anal., 53 (2003), 977. doi: 10.1016/S0362-546X(03)00032-4. Google Scholar

[25]

A. Ito and N. Kenmochi, Inertial set for a phase transition model of Penrose-Fife type,, Adv. Math. Sci. Appl., 10 (2000), 353. Google Scholar

[26]

A. Ito, N. Kenmochi and M. Niezgódka, Phase separation model of Penrose-Fife type with Signorini boundary condition,, Adv. Math. Sci. Appl., 17 (2007), 337. Google Scholar

[27]

R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dieterich, Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions,, Comput. Phys. Commun., 133 (2001), 139. doi: 10.1016/S0010-4655(00)00159-4. Google Scholar

[28]

Ph. Laurençot, Solutions to a Penrose-Fife model of phase-field type,, J. Math. Anal. Appl., 185 (1994), 262. doi: 10.1006/jmaa.1994.1247. Google Scholar

[29]

Ph. Laurençot, Weak solutions to a Penrose-Fife model for phase transitions,, Adv. Math. Sci. Appl., 5 (1995), 117. Google Scholar

[30]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, (French) Dunod, (1969). Google Scholar

[31]

A. Miranville, Some Mathematical Models in Phase Transition,, Lecture Notes, (2009). doi: 10.3934/dcdss.2014.7.271. Google Scholar

[32]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar

[33]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, Discrete Contin. Dyn. Syst., 28 (2010), 275. doi: 10.3934/dcds.2010.28.275. Google Scholar

[34]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions,, Electron. J. Diff. Equ., 2006 (). Google Scholar

[35]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Phys. D, 43 (1990), 44. doi: 10.1016/0167-2789(90)90015-H. Google Scholar

[36]

E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems,, Phys. D, 192 (2004), 279. doi: 10.1016/j.physd.2004.01.024. Google Scholar

[37]

E. Rocca and G. Schimperna, Universal attractor for a Penrose-Fife system with special heat flux law,, Mediterr. J. Math., 1 (2004), 109. doi: 10.1007/s00009-004-0007-5. Google Scholar

[38]

G. Savaré and A. Visintin, Variational convergence of nonlinear diffusion equations: Applications to concentrated capacity problems with change of phase,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8 (1997), 49. Google Scholar

[39]

G. Schimperna, Weak solution to a phase-field transmission problem in a concentrated capacity,, Math. Methods Appl. Sci., 22 (1999), 1235. doi: 10.1002/(SICI)1099-1476(19990925)22:14<1235::AID-MMA82>3.0.CO;2-W. Google Scholar

[40]

G. Schimperna, Global and exponential attractors for the Penrose-Fife system,, Math. Models Methods Appl. Sci., 19 (2009), 969. doi: 10.1142/S0218202509003681. Google Scholar

[41]

G. Schimperna, A. Segatti and S. Zelik, Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model,, J. Evol. Equ., 12 (2012), 863. doi: 10.1007/s00028-012-0159-x. Google Scholar

[42]

G. Schimperna, A. Segatti, and S. Zelik, On a singular heat equation with dynamic boundary conditions,, submitted, (2013). Google Scholar

[43]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[44]

J. Sprekels and S. Zheng, Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions,, J. Math. Anal. Appl., 176 (1993), 200. doi: 10.1006/jmaa.1993.1209. Google Scholar

[45]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations,, Oxford Lecture Series in Mathematics and its Applications, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar

show all references

References:
[1]

H. Attouch, Variational Convergence for Functions and Operators,, Applicable Mathematics Series, (1984). Google Scholar

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (1976). Google Scholar

[3]

M. Bonforte and J.L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations,, Adv. Math., 223 (2010), 529. doi: 10.1016/j.aim.2009.08.021. Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Éspaces de Hilbert,, North-Holland Math. Studies, (1973). Google Scholar

[5]

F. Brezzi and G. Gilardi, FEM Mathematics,, in Finite Element Handbook (Ed. H. Kardestuncer), (1987), 1. Google Scholar

[6]

G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar

[7]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions,, Nonlinear Anal., 72 (2010), 2375. doi: 10.1016/j.na.2009.11.002. Google Scholar

[8]

L. Cherfils, S. Gatti and A. Miranville, Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials,, J. Math. Anal. Appl., 343 (2008), 557. doi: 10.1016/j.jmaa.2008.01.077. Google Scholar

[9]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials,, Appl. Math., 54 (2009), 89. doi: 10.1007/s10492-009-0008-6. Google Scholar

[10]

P. Colli and Ph. Laurençot, Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws,, Phys. D, 111 (1998), 311. doi: 10.1016/S0167-2789(97)80018-8. Google Scholar

[11]

M. Conti, S. Gatti and A. Miranville, Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions,, Discrete Contin. Dyn. Syst. S, 5 (2012), 485. doi: 10.3934/dcdss.2012.5.485. Google Scholar

[12]

E. Feireisl and G. Schimperna, Large time behaviour of solutions to Penrose-Fife phase change models,, Math. Methods Appl. Sci., 28 (2005), 2117. doi: 10.1002/mma.659. Google Scholar

[13]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, Phys. Rev. Letters, 79 (1997), 893. doi: 10.1103/PhysRevLett.79.893. Google Scholar

[14]

H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, Europhys. Letters, 42 (1998), 49. Google Scholar

[15]

H. P. Fischer, J. Reinhard, W. Dieterich, J.-F. Gouyet, P. Maass, A. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall,, J. Chem. Phys., 108 (1998), 3028. doi: 10.1063/1.475690. Google Scholar

[16]

C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 689. doi: 10.3934/cpaa.2009.8.689. Google Scholar

[17]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 881. doi: 10.3934/cpaa.2009.8.881. Google Scholar

[18]

G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, Chin. Ann. Math. Ser. B, 31 (2010), 679. doi: 10.1007/s11401-010-0602-7. Google Scholar

[19]

G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non-permeable walls,, Phys. D, 240 (2011), 754. doi: 10.1016/j.physd.2010.12.007. Google Scholar

[20]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials,, Discrete Contin. Dyn. Syst., 28 (2010), 67. doi: 10.3934/dcds.2010.28.67. Google Scholar

[21]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential,, Z. Anal. Anwend., 25 (2006), 51. doi: 10.4171/ZAA/1277. Google Scholar

[22]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988). Google Scholar

[23]

W. Horn, J. Sprekels and S. Zheng, Global existence of smooth solutions to the Penrose-Fife model for Ising ferromagnets,, Adv. Math. Sci. Appl., 6 (1996), 227. Google Scholar

[24]

A. Ito, N. Kenmochi and M. Kubo, Non-isothermal phase transition models with Neumann boundary conditions,, Nonlinear Anal., 53 (2003), 977. doi: 10.1016/S0362-546X(03)00032-4. Google Scholar

[25]

A. Ito and N. Kenmochi, Inertial set for a phase transition model of Penrose-Fife type,, Adv. Math. Sci. Appl., 10 (2000), 353. Google Scholar

[26]

A. Ito, N. Kenmochi and M. Niezgódka, Phase separation model of Penrose-Fife type with Signorini boundary condition,, Adv. Math. Sci. Appl., 17 (2007), 337. Google Scholar

[27]

R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dieterich, Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions,, Comput. Phys. Commun., 133 (2001), 139. doi: 10.1016/S0010-4655(00)00159-4. Google Scholar

[28]

Ph. Laurençot, Solutions to a Penrose-Fife model of phase-field type,, J. Math. Anal. Appl., 185 (1994), 262. doi: 10.1006/jmaa.1994.1247. Google Scholar

[29]

Ph. Laurençot, Weak solutions to a Penrose-Fife model for phase transitions,, Adv. Math. Sci. Appl., 5 (1995), 117. Google Scholar

[30]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, (French) Dunod, (1969). Google Scholar

[31]

A. Miranville, Some Mathematical Models in Phase Transition,, Lecture Notes, (2009). doi: 10.3934/dcdss.2014.7.271. Google Scholar

[32]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar

[33]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, Discrete Contin. Dyn. Syst., 28 (2010), 275. doi: 10.3934/dcds.2010.28.275. Google Scholar

[34]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions,, Electron. J. Diff. Equ., 2006 (). Google Scholar

[35]

O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Phys. D, 43 (1990), 44. doi: 10.1016/0167-2789(90)90015-H. Google Scholar

[36]

E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems,, Phys. D, 192 (2004), 279. doi: 10.1016/j.physd.2004.01.024. Google Scholar

[37]

E. Rocca and G. Schimperna, Universal attractor for a Penrose-Fife system with special heat flux law,, Mediterr. J. Math., 1 (2004), 109. doi: 10.1007/s00009-004-0007-5. Google Scholar

[38]

G. Savaré and A. Visintin, Variational convergence of nonlinear diffusion equations: Applications to concentrated capacity problems with change of phase,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8 (1997), 49. Google Scholar

[39]

G. Schimperna, Weak solution to a phase-field transmission problem in a concentrated capacity,, Math. Methods Appl. Sci., 22 (1999), 1235. doi: 10.1002/(SICI)1099-1476(19990925)22:14<1235::AID-MMA82>3.0.CO;2-W. Google Scholar

[40]

G. Schimperna, Global and exponential attractors for the Penrose-Fife system,, Math. Models Methods Appl. Sci., 19 (2009), 969. doi: 10.1142/S0218202509003681. Google Scholar

[41]

G. Schimperna, A. Segatti and S. Zelik, Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model,, J. Evol. Equ., 12 (2012), 863. doi: 10.1007/s00028-012-0159-x. Google Scholar

[42]

G. Schimperna, A. Segatti, and S. Zelik, On a singular heat equation with dynamic boundary conditions,, submitted, (2013). Google Scholar

[43]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[44]

J. Sprekels and S. Zheng, Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions,, J. Math. Anal. Appl., 176 (1993), 200. doi: 10.1006/jmaa.1993.1209. Google Scholar

[45]

J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations,, Oxford Lecture Series in Mathematics and its Applications, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar

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