×

Some mathematical models in phase transition. (English) Zbl 1275.35048

Summary: 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.

MSC:

35B41 Attractors
35K55 Nonlinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Aizicovici, <em>Long-time convergence of solutions to a phase-field system</em>,, Math. Methods Appl. Sci., 24, 277 (2001) · Zbl 0984.35026
[2] N. D. Alikakos, <em>\(L^p\) bounds of solutions of reaction-diffusion equations</em>,, Commun. Partial Diff. Eqns., 4, 827 (1979) · Zbl 0421.35009
[3] S. M. Allen, <em>A microscopic theory of antiphase boundary motion and its application to antiphase domain coarsening</em>,, J. Chem. Phys., 28, 258 (1957)
[4] A. V. Babin, “Attractors of Evolution Equations,”, Studies in Mathematics and its Applications, 25 (1992) · Zbl 0778.58002
[5] P. W. Bates, <em>Inertial manifolds and inertial sets for the phase-field equations</em>,, J. Dyn. Diff. Eqns., 4, 375 (1992) · Zbl 0758.35040
[6] D. Brochet, <em>Finite dimensional exponential attractors for the phase-field model</em>,, Appl. Anal., 49, 197 (1993) · Zbl 0790.35052
[7] D. Brochet, <em>Universal attractor and inertial sets for the phase field model</em>,, Appl. Math. Letters, 4, 59 (1991) · Zbl 0773.35028
[8] M. Brokate, “Hysteresis and Phase Transitions,”, Applied Mathematical Sciences, 121 (1996) · Zbl 0951.74002
[9] J. W. Cahn, <em>Free energy of a nonuniform system. I. Interfacial free energy</em>,, J. Chem. Phys., 2, 258 (1958) · Zbl 1431.35066
[10] G. Caginalp, <em>The role of microscopic anisotropy in the macroscopic behavior of a phase boundary</em>,, Ann. Physics, 172, 136 (1986) · Zbl 0639.58038
[11] G. Caginalp, <em>An analysis of a phase field model of a free boundary</em>,, Arch. Rational Mech. Anal., 92, 205 (1986) · Zbl 0608.35080
[12] G. Caginalp, <em>Stefan and Hele-shaw type models as asymptotic limits of the phase-field equations</em>,, Phys. Review A, 39, 5887 (1989) · Zbl 1027.80505
[13] G. Caginalp, <em>Phase field equations in the singular limit of sharp interface problems</em>,, in, 43, 1 (1992) · Zbl 0760.76094
[14] G. Caginalp, <em>Convergence of the phase field model to its sharp interface limits</em>,, European J. Appl. Math., 9, 417 (1998) · Zbl 0930.35024
[15] G. Caginalp, <em>Numerical tests of a phase field model with second order accuracy</em>,, SIAM J. Appl. Math., 68, 1518 (2008) · Zbl 1151.80006
[16] G. Caginalp, <em>Efficient computation of a sharp interface spreading via phase field methods</em>,, Appl. Math. Letters, 2, 117 (1989) · Zbl 0705.65101
[17] B. Chalmers, “Principles of Solidification,”, R. E. Krieger Publishing (1977)
[18] X. Chen, <em>A rapidly converging phase field model</em>,, Discrete Cont. Dyn. Systems, 15, 1107 (2006)
[19] L. Cherfils, <em>Some results on the asymptotic behavior of the Caginalp system with singular potentials</em>,, Adv. Math. Sci. Appl., 17, 107 (2007) · Zbl 1145.35042
[20] L. Cherfils, <em>On the Caginalp system with dynamic boundary conditions and singular potentials</em>,, Appl. Math., 54, 89 (2009) · Zbl 1212.35012
[21] R. Chill, <em>Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions</em>,, Math. Nachr., 279, 1448 (2006) · Zbl 1107.35058
[22] C. I. Christov, <em>Heat conduction paradox involving second-sound propagation in moving media</em>,, Phys. Review Letters, 94 (2005)
[23] C. M. Elliott, <em>Global existence and stability of solutions to the phase field equations</em>,, in, 46 (1990) · Zbl 0733.35062
[24] H. P. Fischer, <em>Novel surface modes in spinodal decomposition</em>,, Phys. Review Letters, 79, 893 (1997)
[25] H. P. Fischer, <em>Diverging time and length scales of spinodal decomposition modes in thin flows</em>,, Europhys. Letters, 42, 49 (1998)
[26] C. G. Gal, <em>A Cahn-Hilliard model in bounded domains with permeable walls</em>,, Math. Methods Appl. Sci., 29, 2009 (2006) · Zbl 1109.35057
[27] C. G. Gal, <em>On the asymptotic behavior of the Caginalp system with dynamic boundary conditions</em>,, Commun. Pure Appl. Anal., 8, 689 (2009) · Zbl 1171.35337
[28] C. G. Gal, <em>Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions</em>,, in, 117 (2008) · Zbl 1178.35076
[29] S. Gatti, <em>Asymptotic behavior of a phase-field system with dynamic boundary conditions</em>,, in, 149 (2006) · Zbl 1123.35310
[30] J. W. Gibbs, “Collected Works,”, Yale University Press (1948) · JFM 54.0036.02
[31] G. Gilardi, <em>On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions</em>,, Commun. Pure Appl. Anal., 8, 881 (2009) · Zbl 1172.35417
[32] M. Grasselli, <em>Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials</em>,, Math. Nachr., 280, 1475 (2007) · Zbl 1133.35017
[33] A. E. Green, <em>A re-examination of the basic postulates of thermomechanics</em>,, Proc. Royal Society London A, 432, 171 (1991) · Zbl 0726.73004
[34] M. Gurtin, <em>Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance</em>,, Physica D, 92, 178 (1996) · Zbl 0885.35121
[35] S. I. Hariharan, <em>Comparison of asymptotic solutions of a phase-field model to a sharp-interface model</em>,, SIAM J. Appl. Math., 62, 244 (2001) · Zbl 1007.35095
[36] B. R. Hunt, <em>Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces</em>,, Nonlinearity, 12, 1263 (1999) · Zbl 0932.28006
[37] B. R. Hunt, <em>Prevalence: A translation-invariant “almost every” for infinite-dimensional spaces</em>,, Bull. Amer. Math. Soc., 27, 217 (1992) · Zbl 0763.28009
[38] M. A. Jendoubi, <em>A simple unified approach to some convergence theorems of L. Simon</em>,, J. Funct. Anal., 153, 187 (1998) · Zbl 0895.35012
[39] G. Lamé, <em>Mémoire sur la solidification par refroidissement d’un globe solide</em>,, Ann. Chem. Phys., 47, 250 (1831)
[40] L. D. Landau, “Statistical Physics (Part 1),”, Third edition (1980)
[41] S. Łojasiewicz, “Ensembles Semi-Analytiques,”, IHES (1965)
[42] A. M. Meirmanov, <em>The classical solution of a multidimensional Stefan problem for quasilinear parabolic equations (in Russian)</em>,, Mat. Sb. (N.S.), 112, 170 (1980)
[43] A. Miranville, <em>A generalization of the Caginalp phase-field system based on the Cattaneo law</em>,, Nonlinear Anal. TMA, 71, 2278 (2009) · Zbl 1167.35304
[44] A. Miranville, <em>Some generalizations of the Caginalp phase-field system</em>,, Appl. Anal., 88, 877 (2009) · Zbl 1178.35194
[45] A. Miranville, <em>Attractors for dissipative partial differential equations in bounded and unbounded domains</em>,, in, 103 (2008) · Zbl 1221.37158
[46] A. Miranville, <em>The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions</em>,, Discrete Cont. Dyn. Systems, 28, 275 (2010) · Zbl 1203.35046
[47] O. A. Oleinik, <em>A method of solution of the general Stefan problem</em>,, Soviet Math. Dokl., 1, 1350 (1960) · Zbl 0131.09202
[48] O. Penrose, <em>Thermodynamically consistent models of phase-field type for the kinetics of phase transitions</em>,, Physica D, 43, 44 (1990) · Zbl 0709.76001
[49] O. Penrose, <em>On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model</em>,, Physica D, 69, 107 (1993) · Zbl 0799.76084
[50] J. C. Robinson, <em>Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces</em>,, Nonlinearity, 22, 711 (2009) · Zbl 1163.54008
[51] L. Rubinstein, <em>On the solution of Stefan’s problem</em>,, (in Russian) Izvestia Akad. Nauk SSSR, 11, 37 (1947)
[52] S. I. Serdyukov, <em>Extended irreversible thermodynamics and generalization of the dual-phase-lag model in heat transfer</em>,, J. Non-Equilib. Thermodyn., 28, 1 (2003) · Zbl 0974.82026
[53] L. Simon, <em>Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems</em>,, Ann. Math., 118, 525 (1983) · Zbl 0549.35071
[54] J. Sprekels, <em>Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions</em>,, J. Math. Anal. Appl., 176, 200 (1993) · Zbl 0804.35063
[55] J. Stefan, <em>Uber einige Probleme der Theorie der Warmeleitung</em>,, S.-B. Wien Akad. Mat. Natur., 98, 173 (1889)
[56] R. Temam, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics,”, Second edition (1997) · Zbl 0871.35001
[57] A. Visintin, <em>Introduction to Stefan-type problems</em>,, in, 377 (2008) · Zbl 1183.35279
[58] S. Zelik, <em>The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension</em>,, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24, 1 (2000)
[59] Z. Zhang, <em>Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions</em>,, Commun. Pure Appl. Anal., 4, 683 (2005) · Zbl 1082.35033
[60] S. Zheng, <em>Global existence for a thermodynamically consistent model of phase field type</em>,, Diff. Integral Eqns., 5, 241 (1992) · Zbl 0768.35050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.