# American Institute of Mathematical Sciences

April  2013, 6(2): 387-400. doi: 10.3934/dcdss.2013.6.387

## A Cahn-Hilliard-Gurtin model with dynamic boundary conditions

 1 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States 2 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex, France

Received  September 2011 Revised  October 2011 Published  November 2012

Our aim in this paper is to define proper dynamic boundary conditions for a generalization of the Cahn-Hilliard system proposed by M. Gurtin. Such boundary conditions take into account the interactions with the walls in confined systems. We then study the existence and uniqueness of weak solutions.
Citation: Gisèle Ruiz Goldstein, Alain Miranville. A Cahn-Hilliard-Gurtin model with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 387-400. doi: 10.3934/dcdss.2013.6.387
##### References:
 [1] A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, Nonlinear Anal., 47 (2001), 3455-3466. [2] J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. [3] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [4] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. [5] R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. [6] C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in "Mathematical Models for Phase Change Problems" (ed. J. F. Rodrigues), International Series of Numerical Mathematics, Birkhäuser, Basel, 88 (1989). [7] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. [8] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. doi: 10.2986/tren.001-0349. [9] 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), 3028-3037. [10] 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), 881-912. [11] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712. [12] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. [13] G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non-permeable walls, Physica D, 240 (2011), 754-766. [14] 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), 139-157. [15] A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptotic Anal., 22 (2000), 235-259. [16] A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Physica D, 158 (2001), 233-257. [17] A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., 2003 (2003), 165-185. [18] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. [19] A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl, 8 (1998), 965-985. [20] A. Novick-Cohen, The Cahn-Hilliard equation, in "Handbook of Differential Equations, Evolutionary Partial Differential Equations" (eds. C. M. Dafermos and M. Pokorny), 4, Elsevier, Amsterdam, (2008), 201-228. [21] J. Prüss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl. (4), 185 (2006), 627-648. [22] R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Diff. Eqns., 8 (2003), 83-110. [23] H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Diff. Eqns., 204 (2004), 511-531.

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##### References:
 [1] A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, Nonlinear Anal., 47 (2001), 3455-3466. [2] J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. [3] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [4] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. [5] R. Chill, E. Fašangová and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. [6] C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in "Mathematical Models for Phase Change Problems" (ed. J. F. Rodrigues), International Series of Numerical Mathematics, Birkhäuser, Basel, 88 (1989). [7] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. [8] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. doi: 10.2986/tren.001-0349. [9] 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), 3028-3037. [10] 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), 881-912. [11] G. Gilardi, A. Miranville and G. Schimperna, Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Ann. Math., Ser. B, 31 (2010), 679-712. [12] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. [13] G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non-permeable walls, Physica D, 240 (2011), 754-766. [14] 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), 139-157. [15] A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptotic Anal., 22 (2000), 235-259. [16] A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Physica D, 158 (2001), 233-257. [17] A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., 2003 (2003), 165-185. [18] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. [19] A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl, 8 (1998), 965-985. [20] A. Novick-Cohen, The Cahn-Hilliard equation, in "Handbook of Differential Equations, Evolutionary Partial Differential Equations" (eds. C. M. Dafermos and M. Pokorny), 4, Elsevier, Amsterdam, (2008), 201-228. [21] J. Prüss, R. Racke and S. Zheng, Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Ann. Mat. Pura Appl. (4), 185 (2006), 627-648. [22] R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Diff. Eqns., 8 (2003), 83-110. [23] H. Wu and S. Zheng, Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions, J. Diff. Eqns., 204 (2004), 511-531.
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