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A vanishing diffusion limit in a nonstandard system of phase field equations
1. | Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy |
2. | Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1 |
3. | Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin |
References:
[1] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. |
[2] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes De Contractions Dans les Espaces de Hilbert, (French) North-Holland Mathematics Studies, No. 5, Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. |
[3] |
P. Colli, G. Gilardi, P. Krejčí and J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, to appear in Math. Methods Appl. Sci., (2014).
doi: 10.1002/mma.2892. |
[4] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behaviour for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.
doi: 10.1137/110828526. |
[5] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368. |
[6] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Differential Equations, 254 (2013), 4217-4244.
doi: 10.1016/j.jde.2013.02.014. |
[7] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity, Boll. Unione Mat. Ital. (9), 5 (2012), 495-513. |
[8] |
P. Colli and J. Sprekels, On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type, Ann. Mat. Pura Appl. (4), 169 (1995), 269-289.
doi: 10.1007/BF01759357. |
[9] |
P. Colli and J. Sprekels, Global solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law, Adv. Math. Sci. Appl., 9 (1999), 383-391. |
[10] |
G. Gilardi, P. Krejčí and J. Sprekels, Hysteresis in phase-field models with thermal memory, Math. Methods Appl. Sci., 23 (2000), 909-922.
doi: 10.1002/1099-1476(20000710)23:10<909::AID-MMA142>3.0.CO;2-E. |
[11] |
P. Krejčí and J. Sprekels, Hysteresis operators in phase-field models of Penrose-Fife type, Appl. Math., 43 (1998), 207-222.
doi: 10.1023/A:1023276524286. |
[12] |
P. Krejčí and J. Sprekels, A hysteresis approach to phase-field models, Nonlinear Anal., 39 (2000), 569-586.
doi: 10.1016/S0362-546X(98)00222-3. |
[13] |
P. Krejčí and J. Sprekels, Phase-field models with hysteresis, J. Math. Anal. Appl., 252 (2000), 198-219.
doi: 10.1006/jmaa.2000.6974. |
[14] |
P. Krejčí, J. Sprekels and S. Zheng, Asymptotic behaviour for a phase-field system with hysteresis, J. Differential Equations, 175 (2001), 88-107.
doi: 10.1006/jdeq.2001.3950. |
[15] |
P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.
doi: 10.1007/s11587-006-0008-8. |
[16] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
show all references
References:
[1] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. |
[2] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes De Contractions Dans les Espaces de Hilbert, (French) North-Holland Mathematics Studies, No. 5, Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. |
[3] |
P. Colli, G. Gilardi, P. Krejčí and J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, to appear in Math. Methods Appl. Sci., (2014).
doi: 10.1002/mma.2892. |
[4] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behaviour for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.
doi: 10.1137/110828526. |
[5] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368. |
[6] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Differential Equations, 254 (2013), 4217-4244.
doi: 10.1016/j.jde.2013.02.014. |
[7] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity, Boll. Unione Mat. Ital. (9), 5 (2012), 495-513. |
[8] |
P. Colli and J. Sprekels, On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type, Ann. Mat. Pura Appl. (4), 169 (1995), 269-289.
doi: 10.1007/BF01759357. |
[9] |
P. Colli and J. Sprekels, Global solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law, Adv. Math. Sci. Appl., 9 (1999), 383-391. |
[10] |
G. Gilardi, P. Krejčí and J. Sprekels, Hysteresis in phase-field models with thermal memory, Math. Methods Appl. Sci., 23 (2000), 909-922.
doi: 10.1002/1099-1476(20000710)23:10<909::AID-MMA142>3.0.CO;2-E. |
[11] |
P. Krejčí and J. Sprekels, Hysteresis operators in phase-field models of Penrose-Fife type, Appl. Math., 43 (1998), 207-222.
doi: 10.1023/A:1023276524286. |
[12] |
P. Krejčí and J. Sprekels, A hysteresis approach to phase-field models, Nonlinear Anal., 39 (2000), 569-586.
doi: 10.1016/S0362-546X(98)00222-3. |
[13] |
P. Krejčí and J. Sprekels, Phase-field models with hysteresis, J. Math. Anal. Appl., 252 (2000), 198-219.
doi: 10.1006/jmaa.2000.6974. |
[14] |
P. Krejčí, J. Sprekels and S. Zheng, Asymptotic behaviour for a phase-field system with hysteresis, J. Differential Equations, 175 (2001), 88-107.
doi: 10.1006/jdeq.2001.3950. |
[15] |
P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.
doi: 10.1007/s11587-006-0008-8. |
[16] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura. Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
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