American Institute of Mathematical Sciences

April  2014, 7(2): 271-306. doi: 10.3934/dcdss.2014.7.271

Some mathematical models in phase transition

 1 Laboratoire de Mathématiques et Applications, Université de Poitiers, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex

Received  March 2013 Revised  July 2013 Published  September 2013

Our aim in these notes is to discuss the well-posedness and the asymptotic behavior, in terms of finite-dimensional attractors, of models in phase transition. In particular, we focus on the Caginalp phase field model.
Citation: Alain Miranville. Some mathematical models in phase transition. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 271-306. doi: 10.3934/dcdss.2014.7.271
References:
 [1] S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system,, Math. Methods Appl. Sci., 24 (2001), 277. doi: 10.1002/mma.215. Google Scholar [2] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Commun. Partial Diff. Eqns., 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar [3] S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening,, J. Chem. Phys., 28 (1957), 258. doi: 10.1016/0001-6160(79)90196-2. Google Scholar [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992). Google Scholar [5] P. W. Bates and S. Zheng, Inertial manifolds and inertial sets for the phase-field equations,, J. Dyn. Diff. Eqns., 4 (1992), 375. doi: 10.1007/BF01049391. Google Scholar [6] D. Brochet, X. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model,, Appl. Anal., 49 (1993), 197. doi: 10.1080/00036819108840173. Google Scholar [7] D. Brochet and D. Hilhorst, Universal attractor and inertial sets for the phase field model,, Appl. Math. Letters, 4 (1991), 59. doi: 10.1016/0893-9659(91)90076-8. Google Scholar [8] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Applied Mathematical Sciences, 121 (1996). doi: 10.1007/978-1-4612-4048-8. Google Scholar [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 2 (1958), 258. doi: 10.1063/1.1744102. Google Scholar [10] G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary,, Ann. Physics, 172 (1986), 136. doi: 10.1016/0003-4916(86)90022-9. Google Scholar [11] G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar [12] G. Caginalp, Stefan and Hele-shaw type models as asymptotic limits of the phase-field equations,, Phys. Review A, 39 (1989), 5887. doi: 10.1103/PhysRevA.39.5887. Google Scholar [13] G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems,, in, 43 (1992), 1. doi: 10.1007/978-1-4613-9211-8_1. Google Scholar [14] G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits,, European J. Appl. Math., 9 (1998), 417. doi: 10.1017/S0956792598003520. Google Scholar [15] G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy,, SIAM J. Appl. Math., 68 (2008), 1518. doi: 10.1137/070680965. Google Scholar [16] G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface spreading via phase field methods,, Appl. Math. Letters, 2 (1989), 117. doi: 10.1016/0893-9659(89)90002-5. Google Scholar [17] B. Chalmers, "Principles of Solidification,", R. E. Krieger Publishing, (1977). doi: 10.1007/978-1-4684-1854-5_5. Google Scholar [18] X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model,, Discrete Cont. Dyn. Systems, 15 (2006), 1107. Google Scholar [19] L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials,, Adv. Math. Sci. Appl., 17 (2007), 107. Google Scholar [20] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials,, Appl. Math., 54 (2009), 89. doi: 10.1007/s10492-009-0008-6. Google Scholar [21] R. Chill, E. Fašangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Nachr., 279 (2006), 1448. doi: 10.1002/mana.200410431. Google Scholar [22] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media,, Phys. Review Letters, 94 (2005). doi: 10.1103/PhysRevLett.94.154301. Google Scholar [23] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in, (1990), 46. Google Scholar [24] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, Phys. Review Letters, 79 (1997), 893. doi: 10.1103/PhysRevLett.79.893. Google Scholar [25] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, Europhys. Letters, 42 (1998), 49. Google Scholar [26] C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls,, Math. Methods Appl. Sci., 29 (2006), 2009. doi: 10.1002/mma.757. Google Scholar [27] C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 689. doi: 10.3934/cpaa.2009.8.689. Google Scholar [28] C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, (2008), 117. Google Scholar [29] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, (2006), 149. doi: 10.1201/9781420011135.ch9. Google Scholar [30] J. W. Gibbs, "Collected Works,", Yale University Press, (1948). Google Scholar [31] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 881. doi: 10.3934/cpaa.2009.8.881. Google Scholar [32] M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials,, Math. Nachr., 280 (2007), 1475. doi: 10.1002/mana.200510560. Google Scholar [33] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. Royal Society London A, 432 (1991), 171. doi: 10.1098/rspa.1991.0012. Google Scholar [34] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5. Google Scholar [35] S. I. Hariharan and G. W. Young, Comparison of asymptotic solutions of a phase-field model to a sharp-interface model,, SIAM J. Appl. Math., 62 (2001), 244. doi: 10.1137/S0036139900374908. Google Scholar [36] B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263. doi: 10.1088/0951-7715/12/5/303. Google Scholar [37] B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: A translation-invariant "almost every'' for infinite-dimensional spaces,, Bull. Amer. Math. Soc., 27 (1992), 217. doi: 10.1090/S0273-0979-1992-00328-2. Google Scholar [38] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187. doi: 10.1006/jfan.1997.3174. Google Scholar [39] G. Lamé and B. P. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe solide,, Ann. Chem. Phys., 47 (1831), 250. Google Scholar [40] L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1),", Third edition, (1980). Google Scholar [41] S. Łojasiewicz, "Ensembles Semi-Analytiques,", IHES, (1965). Google Scholar [42] A. M. Meirmanov, The classical solution of a multidimensional Stefan problem for quasilinear parabolic equations (in Russian),, Mat. Sb. (N.S.), 112 (1980), 170. Google Scholar [43] A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law,, Nonlinear Anal. TMA, 71 (2009), 2278. doi: 10.1016/j.na.2009.01.061. Google Scholar [44] A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system,, Appl. Anal., 88 (2009), 877. doi: 10.1080/00036810903042182. Google Scholar [45] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [46] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, Discrete Cont. Dyn. Systems, 28 (2010), 275. doi: 10.3934/dcds.2010.28.275. Google Scholar [47] O. A. Oleinik, A method of solution of the general Stefan problem,, Soviet Math. Dokl., 1 (1960), 1350. Google Scholar [48] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Physica D, 43 (1990), 44. doi: 10.1016/0167-2789(90)90015-H. Google Scholar [49] O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model,, Physica D, 69 (1993), 107. doi: 10.1016/0167-2789(93)90183-2. Google Scholar [50] J. C. Robinson, Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces,, Nonlinearity, 22 (2009), 711. doi: 10.1088/0951-7715/22/4/001. Google Scholar [51] L. Rubinstein, On the solution of Stefan's problem,, (in Russian) Izvestia Akad. Nauk SSSR, 11 (1947), 37. Google Scholar [52] S. I. Serdyukov, N. M. Voskresenskii, V. K. Bel'nov and I. I. Karpov, Extended irreversible thermodynamics and generalization of the dual-phase-lag model in heat transfer,, J. Non-Equilib. Thermodyn., 28 (2003), 1. doi: 10.1515/JNETDY.2003.013. Google Scholar [53] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,, Ann. Math., 118 (1983), 525. doi: 10.2307/2006981. Google Scholar [54] J. Sprekels and S. Zheng, Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions,, J. Math. Anal. Appl., 176 (1993), 200. doi: 10.1006/jmaa.1993.1209. Google Scholar [55] J. Stefan, Uber einige Probleme der Theorie der Warmeleitung,, S.-B. Wien Akad. Mat. Natur., 98 (1889), 173. Google Scholar [56] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, (1997). Google Scholar [57] A. Visintin, Introduction to Stefan-type problems,, in, (2008), 377. doi: 10.1016/S1874-5717(08)00008-X. Google Scholar [58] S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1. Google Scholar [59] Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions,, Commun. Pure Appl. Anal., 4 (2005), 683. doi: 10.3934/cpaa.2005.4.683. Google Scholar [60] S. Zheng, Global existence for a thermodynamically consistent model of phase field type,, Diff. Integral Eqns., 5 (1992), 241. Google Scholar

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References:
 [1] S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system,, Math. Methods Appl. Sci., 24 (2001), 277. doi: 10.1002/mma.215. Google Scholar [2] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Commun. Partial Diff. Eqns., 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar [3] S. M. Allen and J. W. Cahn, A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening,, J. Chem. Phys., 28 (1957), 258. doi: 10.1016/0001-6160(79)90196-2. Google Scholar [4] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", Studies in Mathematics and its Applications, 25 (1992). Google Scholar [5] P. W. Bates and S. Zheng, Inertial manifolds and inertial sets for the phase-field equations,, J. Dyn. Diff. Eqns., 4 (1992), 375. doi: 10.1007/BF01049391. Google Scholar [6] D. Brochet, X. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model,, Appl. Anal., 49 (1993), 197. doi: 10.1080/00036819108840173. Google Scholar [7] D. Brochet and D. Hilhorst, Universal attractor and inertial sets for the phase field model,, Appl. Math. Letters, 4 (1991), 59. doi: 10.1016/0893-9659(91)90076-8. Google Scholar [8] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Applied Mathematical Sciences, 121 (1996). doi: 10.1007/978-1-4612-4048-8. Google Scholar [9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 2 (1958), 258. doi: 10.1063/1.1744102. Google Scholar [10] G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary,, Ann. Physics, 172 (1986), 136. doi: 10.1016/0003-4916(86)90022-9. Google Scholar [11] G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar [12] G. Caginalp, Stefan and Hele-shaw type models as asymptotic limits of the phase-field equations,, Phys. Review A, 39 (1989), 5887. doi: 10.1103/PhysRevA.39.5887. Google Scholar [13] G. Caginalp and X. Chen, Phase field equations in the singular limit of sharp interface problems,, in, 43 (1992), 1. doi: 10.1007/978-1-4613-9211-8_1. Google Scholar [14] G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits,, European J. Appl. Math., 9 (1998), 417. doi: 10.1017/S0956792598003520. Google Scholar [15] G. Caginalp, X. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy,, SIAM J. Appl. Math., 68 (2008), 1518. doi: 10.1137/070680965. Google Scholar [16] G. Caginalp and E. A. Socolovsky, Efficient computation of a sharp interface spreading via phase field methods,, Appl. Math. Letters, 2 (1989), 117. doi: 10.1016/0893-9659(89)90002-5. Google Scholar [17] B. Chalmers, "Principles of Solidification,", R. E. Krieger Publishing, (1977). doi: 10.1007/978-1-4684-1854-5_5. Google Scholar [18] X. Chen, G. Caginalp and C. Eck, A rapidly converging phase field model,, Discrete Cont. Dyn. Systems, 15 (2006), 1107. Google Scholar [19] L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials,, Adv. Math. Sci. Appl., 17 (2007), 107. Google Scholar [20] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials,, Appl. Math., 54 (2009), 89. doi: 10.1007/s10492-009-0008-6. Google Scholar [21] R. Chill, E. Fašangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions,, Math. Nachr., 279 (2006), 1448. doi: 10.1002/mana.200410431. Google Scholar [22] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media,, Phys. Review Letters, 94 (2005). doi: 10.1103/PhysRevLett.94.154301. Google Scholar [23] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase field equations,, in, (1990), 46. Google Scholar [24] H. P. Fischer, P. Maass and W. Dieterich, Novel surface modes in spinodal decomposition,, Phys. Review Letters, 79 (1997), 893. doi: 10.1103/PhysRevLett.79.893. Google Scholar [25] H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows,, Europhys. Letters, 42 (1998), 49. Google Scholar [26] C. G. Gal, A Cahn-Hilliard model in bounded domains with permeable walls,, Math. Methods Appl. Sci., 29 (2006), 2009. doi: 10.1002/mma.757. Google Scholar [27] C. G. Gal and M. Grasselli, On the asymptotic behavior of the Caginalp system with dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 689. doi: 10.3934/cpaa.2009.8.689. Google Scholar [28] C. G. Gal, M. Grasselli and A. Miranville, Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions,, in, (2008), 117. Google Scholar [29] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions,, in, (2006), 149. doi: 10.1201/9781420011135.ch9. Google Scholar [30] J. W. Gibbs, "Collected Works,", Yale University Press, (1948). Google Scholar [31] G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions,, Commun. Pure Appl. Anal., 8 (2009), 881. doi: 10.3934/cpaa.2009.8.881. Google Scholar [32] M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials,, Math. Nachr., 280 (2007), 1475. doi: 10.1002/mana.200510560. Google Scholar [33] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. Royal Society London A, 432 (1991), 171. doi: 10.1098/rspa.1991.0012. Google Scholar [34] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5. Google Scholar [35] S. I. Hariharan and G. W. Young, Comparison of asymptotic solutions of a phase-field model to a sharp-interface model,, SIAM J. Appl. Math., 62 (2001), 244. doi: 10.1137/S0036139900374908. Google Scholar [36] B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263. doi: 10.1088/0951-7715/12/5/303. Google Scholar [37] B. R. Hunt, T. Sauer and J. A. Yorke, Prevalence: A translation-invariant "almost every'' for infinite-dimensional spaces,, Bull. Amer. Math. Soc., 27 (1992), 217. doi: 10.1090/S0273-0979-1992-00328-2. Google Scholar [38] M. A. Jendoubi, A simple unified approach to some convergence theorems of L. Simon,, J. Funct. Anal., 153 (1998), 187. doi: 10.1006/jfan.1997.3174. Google Scholar [39] G. Lamé and B. P. Clapeyron, Mémoire sur la solidification par refroidissement d'un globe solide,, Ann. Chem. Phys., 47 (1831), 250. Google Scholar [40] L. D. Landau and E. M. Lifschitz, "Statistical Physics (Part 1),", Third edition, (1980). Google Scholar [41] S. Łojasiewicz, "Ensembles Semi-Analytiques,", IHES, (1965). Google Scholar [42] A. M. Meirmanov, The classical solution of a multidimensional Stefan problem for quasilinear parabolic equations (in Russian),, Mat. Sb. (N.S.), 112 (1980), 170. Google Scholar [43] A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law,, Nonlinear Anal. TMA, 71 (2009), 2278. doi: 10.1016/j.na.2009.01.061. Google Scholar [44] A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system,, Appl. Anal., 88 (2009), 877. doi: 10.1080/00036810903042182. Google Scholar [45] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [46] A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions,, Discrete Cont. Dyn. Systems, 28 (2010), 275. doi: 10.3934/dcds.2010.28.275. Google Scholar [47] O. A. Oleinik, A method of solution of the general Stefan problem,, Soviet Math. Dokl., 1 (1960), 1350. Google Scholar [48] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Physica D, 43 (1990), 44. doi: 10.1016/0167-2789(90)90015-H. Google Scholar [49] O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent'' phase-field model,, Physica D, 69 (1993), 107. doi: 10.1016/0167-2789(93)90183-2. Google Scholar [50] J. C. 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