# American Institute of Mathematical Sciences

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 & 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. doi: 10.1007/BF02412236. Google Scholar [2] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff, (1976). Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jmaa.2005.06.033. Google Scholar [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. Google Scholar [7] H. Brezis, "Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert,", North-Holland Math. Studies, (1973). Google Scholar [8] E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004). Google Scholar [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. doi: 10.1002/mma.1089. Google Scholar [10] M. Frémond, "Non-smooth Thermomechanics,", Springer-Verlag, (2002). Google Scholar [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. doi: 10.1016/S0022-247X(02)00127-0. Google Scholar [12] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires,", Dunod-Gauthier Villars, (1969). Google Scholar [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. doi: 10.1142/S0218202501001112. Google Scholar [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. Google Scholar [15] F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions,, Z. Anal. Anwendungen, 21 (2002), 335. Google Scholar [16] T. Roubiček, "Nonlinear Partial Differential Equations with Applications,", International Series of Numerical Mathematics, (2005). Google Scholar [17] G. Schimperna, F. Luterotti and U. Stefanelli, Local solution to Frémond's full model for irreversible phase transitions,, in, (2002), 323. Google Scholar [18] 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 [19] A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996). Google Scholar [20] J. B. Zelďovich and Y. P. Raizer, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena,", Academic Press, (1966). Google Scholar

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##### 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. doi: 10.1007/BF02412236. Google Scholar [2] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff, (1976). Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jmaa.2005.06.033. Google Scholar [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. Google Scholar [7] H. Brezis, "Opérateurs Maximaux Monotones et Sémi-groupes de Contractions dans les Espaces de Hilbert,", North-Holland Math. Studies, (1973). Google Scholar [8] E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004). Google Scholar [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. doi: 10.1002/mma.1089. Google Scholar [10] M. Frémond, "Non-smooth Thermomechanics,", Springer-Verlag, (2002). Google Scholar [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. doi: 10.1016/S0022-247X(02)00127-0. Google Scholar [12] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires,", Dunod-Gauthier Villars, (1969). Google Scholar [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. doi: 10.1142/S0218202501001112. Google Scholar [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. Google Scholar [15] F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions,, Z. Anal. Anwendungen, 21 (2002), 335. Google Scholar [16] T. Roubiček, "Nonlinear Partial Differential Equations with Applications,", International Series of Numerical Mathematics, (2005). Google Scholar [17] G. Schimperna, F. Luterotti and U. Stefanelli, Local solution to Frémond's full model for irreversible phase transitions,, in, (2002), 323. Google Scholar [18] 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 [19] A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996). Google Scholar [20] J. B. Zelďovich and Y. P. Raizer, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena,", Academic Press, (1966). Google Scholar
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