Singular limit of an integrodifferential system related to the entropy balance

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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

Abstract
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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, (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, (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. 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. 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. 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. 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. 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., (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. 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. doi: 10.3233/ASY-2012-1142. Google Scholar
[11]	M. Frémond, Non-smooth Thermomechanics,, Springer-Verlag, (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. 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. 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., (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. doi: 10.1007/BF00281373. Google Scholar
[16]	P. Podio-Guidugli, A virtual power format for thermomechanics,, Contin. Mech. Thermodyn., 20 (2009), 479. 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. 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, (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, (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. 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. 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. 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. 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. 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., (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. 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. doi: 10.3233/ASY-2012-1142. Google Scholar
[11]	M. Frémond, Non-smooth Thermomechanics,, Springer-Verlag, (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. 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. 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., (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. doi: 10.1007/BF00281373. Google Scholar
[16]	P. Podio-Guidugli, A virtual power format for thermomechanics,, Contin. Mech. Thermodyn., 20 (2009), 479. 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. doi: 10.1007/BF01762360. Google Scholar

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