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Singular limit of an integrodifferential system related to the entropy balance
1. | Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia |
References:
[1] |
G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[2] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976. |
[3] |
E. Bonetti, P. Colli and M. Frémond, A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci., 13 (2003), 1565-1588.
doi: 10.1142/S0218202503003033. |
[4] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026.
doi: 10.3934/dcdsb.2006.6.1001. |
[5] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979.
doi: 10.1016/j.na.2006.02.035. |
[6] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295.
doi: 10.1016/j.jde.2009.02.007. |
[7] |
E. Bonetti, M. Frémond and E. Rocca, A new dual approach for a class of phase transitions with memory: Existence and long-time behaviour of solutions, J. Math. Pures Appl., 88 (2007), 455-481.
doi: 10.1016/j.matpur.2007.09.005. |
[8] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud., 5, North-Holland, Amsterdam, 1973. |
[9] |
G. Canevari and P. Colli, Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982.
doi: 10.3934/cpaa.2012.11.1959. |
[10] |
G. Canevari and P. Colli, Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162.
doi: 10.3233/ASY-2012-1142. |
[11] |
M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04800-9. |
[12] |
G. Gilardi and E. Rocca, Convergence of phase field to phase relaxation governed by the entropy balance with memory, Math. Methods Appl. Sci., 29 (2006), 2149-2179.
doi: 10.1002/mma.765. |
[13] |
A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermo-mechanics, Proc. Roy. Soc. Lond. A, 432 (1991), 171-194.
doi: 10.1098/rspa.1991.0012. |
[14] |
G. Gripenberg, S-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl., 34, Cambridge University Press, Cambridge, 1990.
doi: 10.1017/CBO9780511662805. |
[15] |
M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.
doi: 10.1007/BF00281373. |
[16] |
P. Podio-Guidugli, A virtual power format for thermomechanics, Contin. Mech. Thermodyn., 20 (2009), 479-487.
doi: 10.1007/s00161-009-0093-5. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
show all references
References:
[1] |
G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory. Theory and Applications, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[2] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976. |
[3] |
E. Bonetti, P. Colli and M. Frémond, A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci., 13 (2003), 1565-1588.
doi: 10.1142/S0218202503003033. |
[4] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1001-1026.
doi: 10.3934/dcdsb.2006.6.1001. |
[5] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Global solution to a singular integrodifferential system related to the entropy balance, Nonlinear Anal., 66 (2007), 1949-1979.
doi: 10.1016/j.na.2006.02.035. |
[6] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295.
doi: 10.1016/j.jde.2009.02.007. |
[7] |
E. Bonetti, M. Frémond and E. Rocca, A new dual approach for a class of phase transitions with memory: Existence and long-time behaviour of solutions, J. Math. Pures Appl., 88 (2007), 455-481.
doi: 10.1016/j.matpur.2007.09.005. |
[8] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud., 5, North-Holland, Amsterdam, 1973. |
[9] |
G. Canevari and P. Colli, Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982.
doi: 10.3934/cpaa.2012.11.1959. |
[10] |
G. Canevari and P. Colli, Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162.
doi: 10.3233/ASY-2012-1142. |
[11] |
M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04800-9. |
[12] |
G. Gilardi and E. Rocca, Convergence of phase field to phase relaxation governed by the entropy balance with memory, Math. Methods Appl. Sci., 29 (2006), 2149-2179.
doi: 10.1002/mma.765. |
[13] |
A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermo-mechanics, Proc. Roy. Soc. Lond. A, 432 (1991), 171-194.
doi: 10.1098/rspa.1991.0012. |
[14] |
G. Gripenberg, S-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl., 34, Cambridge University Press, Cambridge, 1990.
doi: 10.1017/CBO9780511662805. |
[15] |
M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.
doi: 10.1007/BF00281373. |
[16] |
P. Podio-Guidugli, A virtual power format for thermomechanics, Contin. Mech. Thermodyn., 20 (2009), 479-487.
doi: 10.1007/s00161-009-0093-5. |
[17] |
J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
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