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September  2014, 19(7): 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

Singular limit of an integrodifferential system related to the entropy balance

Elena Bonetti 1, , Pierluigi Colli 1, and  Gianni Gilardi 1,

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia

Received  January 2013 Revised  May 2013 Published  August 2014

A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Convergence results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.
Keywords: thermal memory, phase field model, asymptotics on the memory term., nonlinear partial differential equations, Entropy equation.
Mathematics Subject Classification: Primary: 35K55, 35B40; Secondary: 35Q79, 80A2.
Citation: Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
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.  Google Scholar

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Podio-Guidugli, A virtual power format for thermomechanics, Contin. Mech. Thermodyn., 20 (2009), 479-487. doi: 10.1007/s00161-009-0093-5.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Podio-Guidugli, A virtual power format for thermomechanics, Contin. Mech. Thermodyn., 20 (2009), 479-487. doi: 10.1007/s00161-009-0093-5.  Google Scholar

[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.  Google Scholar

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