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Asymptotic behavior of the Caginalp phase-field system with coupled dynamic boundary conditions
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 |
References:
[1] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[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), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[3] |
H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Letters, 79 (1997), 893-896.
doi: 10.1103/PhysRevLett.79.893. |
[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), 49-54.
doi: 10.1209/epl/i1998-00550-y. |
[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), 3028-3037.
doi: 10.1063/1.475690. |
[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), 535–-556.
doi: 10.1007/s00030-008-7029-9. |
[7] |
C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation," GAKUTO Internat. Ser. Math. Sci. Appl., 29, Gakkotōsho, Tokyo, 2008, 117–-139. |
[8] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems," Lecture Notes in Pure and Applied Mathematics, 251, Chapman & Hall/CRC, Boca Raton, FL, (2006), 149-170. |
[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), 67-98.
doi: 10.3934/dcds.2010.28.67. |
[10] |
O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, AMS, Providence, RI, 1967. |
[11] |
A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().
|
[12] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[13] |
A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310.
doi: 10.3934/dcds.2010.28.275. |
[14] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[15] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems," Springer-Verlag, New York, 1986. |
show all references
References:
[1] |
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.
doi: 10.1007/BF00254827. |
[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), 211-238.
doi: 10.3934/dcds.2004.10.211. |
[3] |
H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Letters, 79 (1997), 893-896.
doi: 10.1103/PhysRevLett.79.893. |
[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), 49-54.
doi: 10.1209/epl/i1998-00550-y. |
[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), 3028-3037.
doi: 10.1063/1.475690. |
[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), 535–-556.
doi: 10.1007/s00030-008-7029-9. |
[7] |
C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions, in "Nonlinear Phenomena with Energy Dissipation," GAKUTO Internat. Ser. Math. Sci. Appl., 29, Gakkotōsho, Tokyo, 2008, 117–-139. |
[8] |
S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in "Differential Equations: Inverse and Direct Problems," Lecture Notes in Pure and Applied Mathematics, 251, Chapman & Hall/CRC, Boca Raton, FL, (2006), 149-170. |
[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), 67-98.
doi: 10.3934/dcds.2010.28.67. |
[10] |
O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, AMS, Providence, RI, 1967. |
[11] |
A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations,, Electron. J. Differential Equations, 2002 ().
|
[12] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Meth. Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[13] |
A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dynam. Systems, 28 (2010), 275-310.
doi: 10.3934/dcds.2010.28.275. |
[14] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," 2nd edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[15] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. Part I. Fixed-Point Theorems," Springer-Verlag, New York, 1986. |
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