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

A Cahn-Hilliard-Gurtin model with dynamic boundary conditions

Gisèle Ruiz Goldstein 1, and  Alain Miranville 2,

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.
Keywords: Cahn-Hilliard-Gurtin equations, dynamic boundary conditions, well-posedness..
Mathematics Subject Classification: 35K5.
Citation: Gisèle Ruiz Goldstein, Alain Miranville. A Cahn-Hilliard-Gurtin model with dynamic boundary conditions. Discrete & 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.  Google Scholar

[2]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. Google Scholar

[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. Google Scholar

[4]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[7]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptotic Anal., 22 (2000), 235-259.  Google Scholar

[16]

A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Physica D, 158 (2001), 233-257.  Google Scholar

[17]

A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., 2003 (2003), 165-185.  Google Scholar

[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.  Google Scholar

[19]

A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl, 8 (1998), 965-985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Diff. Eqns., 8 (2003), 83-110.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, Nonlinear Anal., 47 (2001), 3455-3466.  Google Scholar

[2]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. Google Scholar

[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. Google Scholar

[4]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[7]

H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptotic Anal., 22 (2000), 235-259.  Google Scholar

[16]

A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Physica D, 158 (2001), 233-257.  Google Scholar

[17]

A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., 2003 (2003), 165-185.  Google Scholar

[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.  Google Scholar

[19]

A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl, 8 (1998), 965-985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Diff. Eqns., 8 (2003), 83-110.  Google Scholar

[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.  Google Scholar

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