American Institute of Mathematical Sciences

Advanced
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 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.

show all references

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.

[1]

Sami Injrou, Morgan Pierre. Stable discretizations of the Cahn-Hilliard-Gurtin equations. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1065-1080. doi: 10.3934/dcds.2008.22.1065
[2]

Alain Miranville, Giulio Schimperna. On a doubly nonlinear Cahn-Hilliard-Gurtin system. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 675-697. doi: 10.3934/dcdsb.2010.14.675
[3]

Mathias Wilke. $L_p$-theory for a Cahn-Hilliard-Gurtin system. Evolution Equations and Control Theory, 2012, 1 (2) : 393-429. doi: 10.3934/eect.2012.1.393
[4]

Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419
[5]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113
[6]

Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581
[7]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[8]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[9]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[10]

Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855
[11]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6387-6403. doi: 10.3934/dcdsb.2021024
[12]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093
[13]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations and Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
[14]

Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069
[15]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[16]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[17]

Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625
[18]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2135-2172. doi: 10.3934/dcdss.2022001
[19]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[20]

Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (114)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy