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June  2012, 5(3): 485-505. doi: 10.3934/dcdss.2012.5.485

Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions

Monica Conti 1, , Stefania Gatti 2, and  Alain Miranville 3,

1. 

Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano
2. 

Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, I-41125 Modena, Italy
3. 

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

Received  May 2011 Revised  July 2011 Published  October 2011

This paper deals with the longtime behavior of the Caginalp phase-field system with coupled dynamic boundary conditions on both state variables. We prove that the system generates a dissipative semigroup in a suitable phase-space and possesses the finite-dimensional smooth global attractor and an exponential attractor.
Keywords: Caginalp system, exponential attractor., phase transition, global attractor, dynamic boundary conditions.
Mathematics Subject Classification: Primary: 35K55, 35J60; Secondary: 80A2.
Citation: Monica Conti, Stefania Gatti, Alain Miranville. Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 485-505. doi: 10.3934/dcdss.2012.5.485
References:
[1]

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

[2]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications,, Discrete Contin. Dynam. Systems, 10 (2004), 211.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[3]

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

[4]

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.  doi: 10.1209/epl/i1998-00550-y.  Google Scholar

[5]

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

[6]

C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008).  doi: 10.1007/s00030-008-7029-9.  Google Scholar

[7]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, 29 (2008).   Google Scholar

[8]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, 251 (2006), 149.   Google Scholar

[9]

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

[10]

O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967).   Google Scholar

[11]

A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().   Google Scholar

[12]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709.  doi: 10.1002/mma.590.  Google Scholar

[13]

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

[14]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2nd edition, 68 (1997).   Google Scholar

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems,", Springer-Verlag, (1986).   Google Scholar

show all references

References:
[1]

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

[2]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation. Partial differential equations and applications,, Discrete Contin. Dynam. Systems, 10 (2004), 211.  doi: 10.3934/dcds.2004.10.211.  Google Scholar

[3]

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

[4]

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.  doi: 10.1209/epl/i1998-00550-y.  Google Scholar

[5]

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

[6]

C. G. Gal, M. Grasselli and A. Miranville, Robust exponential attractors for singularly perturbed phase-field equations with dynamic boundary conditions,, NoDEA Nonlinear Differential Equations Appl., 15 (2008).  doi: 10.1007/s00030-008-7029-9.  Google Scholar

[7]

C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, 29 (2008).   Google Scholar

[8]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, 251 (2006), 149.   Google Scholar

[9]

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

[10]

O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, (1967).   Google Scholar

[11]

A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().   Google Scholar

[12]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Meth. Appl. Sci., 28 (2005), 709.  doi: 10.1002/mma.590.  Google Scholar

[13]

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

[14]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", 2nd edition, 68 (1997).   Google Scholar

[15]

E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems,", Springer-Verlag, (1986).   Google Scholar

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