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

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

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.
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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