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 and 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.

[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.

[1]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[2]

Elena Bonetti, Pierluigi Colli, Mauro Fabrizio, Gianni Gilardi. Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1001-1026. doi: 10.3934/dcdsb.2006.6.1001

[3]

Jiaohui Xu, Tomás Caraballo, José Valero. Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021140

[4]

Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949

[5]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. Thermal control of the Souza-Auricchio model for shape memory alloys. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 369-386. doi: 10.3934/dcdss.2013.6.369

[6]

Kung-Ching Chang. In memory of professor Rouhuai Wang (1924-2001): A pioneering Chinese researcher in partial differential equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 571-575. doi: 10.3934/dcds.2016.36.571

[7]

Andrei Halanay, Luciano Pandolfi. Lack of controllability of thermal systems with memory. Evolution Equations and Control Theory, 2014, 3 (3) : 485-497. doi: 10.3934/eect.2014.3.485

[8]

Sergiu Aizicovici, Hana Petzeltová. Convergence to equilibria of solutions to a conserved Phase-Field system with memory. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 1-16. doi: 10.3934/dcdss.2009.2.1

[9]

S. Gatti, M. Grasselli, V. Pata, M. Squassina. Robust exponential attractors for a family of nonconserved phase-field systems with memory. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 1019-1029. doi: 10.3934/dcds.2005.12.1019

[10]

Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations and Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011

[11]

Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure and Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027

[12]

Monica Conti, Elsa M. Marchini, V. Pata. Global attractors for nonlinear viscoelastic equations with memory. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1893-1913. doi: 10.3934/cpaa.2016021

[13]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations and Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411

[14]

S. Gatti, Elena Sartori. Well-posedness results for phase field systems with memory effects in the order parameter dynamics. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 705-726. doi: 10.3934/dcds.2003.9.705

[15]

Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375

[16]

Monica Conti, Stefania Gatti, Alain Miranville. A singular cahn-hilliard-oono phase-field system with hereditary memory. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3033-3054. doi: 10.3934/dcds.2018132

[17]

Gianluca Mola. Global attractors for a three-dimensional conserved phase-field system with memory. Communications on Pure and Applied Analysis, 2008, 7 (2) : 317-353. doi: 10.3934/cpaa.2008.7.317

[18]

Eugenio Sinestrari. Wave equation with memory. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 881-896. doi: 10.3934/dcds.1999.5.881

[19]

Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220

[20]

Liping Luo, Zhenguo Luo, Yunhui Zeng. New results for oscillation of fractional partial differential equations with damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3223-3231. doi: 10.3934/dcdss.2020336

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]