April  2013, 6(2): 331-351. doi: 10.3934/dcdss.2013.6.331

A well-posedness result for irreversible phase transitions with a nonlinear heat flux law

1. 

Dipartimento di Matematica, Università di Brescia, via Branze 38, 25123 Brescia

Received  October 2011 Revised  March 2012 Published  November 2012

In this paper, we deal with a PDE system describing a phase transition problem characterized by irreversible evolution and ruled by a nonlinear heat flux law. Its derivation comes from the modelling approach proposed by M. Frémond. Our main result consists in showing the global-in-time existence and the uniqueness of the solution of the related initial and boundary value problem.
Citation: Giovanna Bonfanti, Fabio Luterotti. A well-posedness result for irreversible phase transitions with a nonlinear heat flux law. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 331-351. doi: 10.3934/dcdss.2013.6.331
References:
[1]

C. Baiocchi, Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl. (IV), 76 (1967), 233-304. doi: 10.1007/BF02412236.

[2]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.

[3]

G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24.

[4]

G. Bonfanti, M. Frémond and F. Luterotti, Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear Anal. Real World Appl., 5 (2004), 123-140.

[5]

G. Bonfanti and F. Luterotti, Well-posedness results and asymptotic behaviour for a phase transition model taking into account microscopic accelerations, J. Math. Anal. Appl., 320 (2006), 95-107. doi: 10.1016/j.jmaa.2005.06.033.

[6]

G. Bonfanti and F. Luterotti, Global solution to a phase transition model with microscopic movements and accelerations in one space dimension, Comm. Pure Appl. Anal., 5 (2006), 763-777.

[7]

H. Brezis, "Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North-Holland, Amsterdam, 1973.

[8]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 2004.

[9]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089.

[10]

M. Frémond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 2002.

[11]

Ph. Laurençot, G. Schimperna and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0.

[12]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires," Dunod-Gauthier Villars, Paris, 1969.

[13]

F. Luterotti, G. Schimperna and U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci., 11 (2001), 809-825. doi: 10.1142/S0218202501001112.

[14]

F. Luterotti, G. Schimperna and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quarterly Appl. Math., 60 (2002), 301-316.

[15]

F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350.

[16]

T. Roubiček, "Nonlinear Partial Differential Equations with Applications," International Series of Numerical Mathematics, 153. Birkhäuser Verlag, Basel, 2005.

[17]

G. Schimperna, F. Luterotti and U. Stefanelli, Local solution to Frémond's full model for irreversible phase transitions, in "Mathematical Models and Methods for Smart Materials" (eds. Mauro Fabrizio, Barbara Lazzari and Angelo Morro), Proc. INdAM meeting in Cortona, June 2001, (2002), 323-328.

[18]

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.

[19]

A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996.

[20]

J. B. Zelďovich and Y. P. Raizer, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena," Academic Press, New York, 1966.

show all references

References:
[1]

C. Baiocchi, Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl. (IV), 76 (1967), 233-304. doi: 10.1007/BF02412236.

[2]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff, Leyden, 1976.

[3]

G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24.

[4]

G. Bonfanti, M. Frémond and F. Luterotti, Existence and uniqueness results to a phase transition model based on microscopic accelerations and movements, Nonlinear Anal. Real World Appl., 5 (2004), 123-140.

[5]

G. Bonfanti and F. Luterotti, Well-posedness results and asymptotic behaviour for a phase transition model taking into account microscopic accelerations, J. Math. Anal. Appl., 320 (2006), 95-107. doi: 10.1016/j.jmaa.2005.06.033.

[6]

G. Bonfanti and F. Luterotti, Global solution to a phase transition model with microscopic movements and accelerations in one space dimension, Comm. Pure Appl. Anal., 5 (2006), 763-777.

[7]

H. Brezis, "Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert," North-Holland Math. Studies, 5, North-Holland, Amsterdam, 1973.

[8]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 2004.

[9]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089.

[10]

M. Frémond, "Non-smooth Thermomechanics," Springer-Verlag, Berlin, 2002.

[11]

Ph. Laurençot, G. Schimperna and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442. doi: 10.1016/S0022-247X(02)00127-0.

[12]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires," Dunod-Gauthier Villars, Paris, 1969.

[13]

F. Luterotti, G. Schimperna and U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci., 11 (2001), 809-825. doi: 10.1142/S0218202501001112.

[14]

F. Luterotti, G. Schimperna and U. Stefanelli, Global solution to a phase field model with irreversible and constrained phase evolution, Quarterly Appl. Math., 60 (2002), 301-316.

[15]

F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350.

[16]

T. Roubiček, "Nonlinear Partial Differential Equations with Applications," International Series of Numerical Mathematics, 153. Birkhäuser Verlag, Basel, 2005.

[17]

G. Schimperna, F. Luterotti and U. Stefanelli, Local solution to Frémond's full model for irreversible phase transitions, in "Mathematical Models and Methods for Smart Materials" (eds. Mauro Fabrizio, Barbara Lazzari and Angelo Morro), Proc. INdAM meeting in Cortona, June 2001, (2002), 323-328.

[18]

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.

[19]

A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996.

[20]

J. B. Zelďovich and Y. P. Raizer, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena," Academic Press, New York, 1966.

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