Introduction to the models of phase transitions

Visintin, A.

Visintin, A.

Bollettino dell'Unione Matematica Italiana, Tome 1-A (1998), p. 1-47 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Résumé

Le transizioni di fase si presentano in svariati processi fisici: un esempio tipico è la transizione solido-liquido. Il classico modello matematico, noto come problema di Stefan, tiene conto solo dello scambio del calore latente e della diffusione termica nelle fasi. Si tratta di un problema di frontiera libera, poiché l'evoluzione dell'interfaccia solido liquido è una delle incognite. In questo articolo si rivedono le formulazioni forte e debole di tale problema, e quindi si considerano alcune generalizzazioni fisicamente motivate. In particolare si presenta un modello su scala mesoscopica per la tensione superficiale, il superraffreddamento e la nucleazione.

Publié le : 1998-02-01

MR 1619027 | Zbl 0903.35097

@article{BUMI_1998_8_1B_1_1_0,
     author = {A. Visintin},
     title = {Introduction to the models of phase transitions},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {1-A},
     year = {1998},
     pages = {1-47},
     zbl = {0903.35097},
     mrnumber = {1619027},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_1998_8_1B_1_1_0}
}

Visintin, A. Introduction to the models of phase transitions. Bollettino dell'Unione Matematica Italiana, Tome 1-A (1998) pp. 1-47. http://gdmltest.u-ga.fr/item/BUMI_1998_8_1B_1_1_0/

Bibliographie

[1] Abraham, F. F., Homogeneous Nucleation Theory, Academic Press, New York (1974).

[2] Alexiades, V.-Solomon, A. D., Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing, Washington DC (1993).

[3] Allen, S. M.-Cahn, J. W., A microscopic theory for antiphase motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.

[4] Almgren, F., Existence and Regularity Almost Everywhere of Elliptic Variational Problems with Constraints, Memoirs A.M.S., 165 (1976). | MR 420406 | Zbl 0327.49043

[5] Almgren, F.-Taylor, J. E.-Wang, L., Curvature-driven flows: a variational approach, S.I.A.M. J. Contr. Optimiz., 31 (1993), 387-437. | MR 1205983 | Zbl 0783.35002

[6] Almgren, F.-Wang, L., Mathematical existence of crystal growth with Gibbs-Thomson curvature effects, J. Geom. Anal. (to appear). | MR 1758583 | Zbl 0981.74041

[7] Alt, H. W.-Caffarelli, L., Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144. | MR 618549 | Zbl 0449.35105

[8] Alt, H. W.-Pawlow, I., A mathematical model of dynamics of non-isothermal phase separation, Physica D, 59 (1992), 389-416 | MR 1192751 | Zbl 0763.58031

[9] Athanassopoulos, I.-Caffarelli, L.A.-Salsa, S., Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems, Ann. Math., 143 (1996), 413-434 | MR 1394964 | Zbl 0853.35049

[10] Athanassopoulos, I.-Caffarelli, L. A.-Salsa, S., Regularity of the free boundary in phase transition problems, Acta Math., 176 (1996). | Zbl 0891.35164

[11] Athanassopoulos, I.-Caffarelli, L. A.-Salsa, S., Degenerate phase transition problems of parabolic type: smoothness of the front, Preprint, Princeton (1996). | Zbl 0924.35197

[12] Atthey, D. R., A finite difference scheme for melting problems, J. Inst. Math. Appl., 13 (1974), 353-366. | MR 351295

[13] Baiocchi, C., Su un problema di frontiera libera connesso a questioni di idraulica, Ann. Mat. Pura Appl., 92 (1972), 107-127. | MR 408443 | Zbl 0258.76069

[14] Baiocchi, C.-Capelo, A., Variational and Quasivariational Inequalities, Applications to Free Boundary Problems, Wiley, Chichester (1983). | MR 745619 | Zbl 0551.49007

[15] Barles, G., Remark on a flame propagation model, Rapport I.N.R.I.A., 464 (1985).

[16] Bénilan, Ph., Equations d'évolution dans un espace de Banach quelconque et applications, Thèse, Orsay (1972).

[17] Berger, A. E.-Brézis, H.-Rogers, J. W., A numerical method for solving the problem $u_{t} - Δ f (u) = 0$ , R.A.I.R.O., Analyse Numérique, 13 (1979), 297-312. | MR 555381 | Zbl 0426.65052

[18] Berger, A. E.-Rogers, J. W., Some properties of the nonlinear semigroup for the problem $u_{t} - Δ f (u) = 0$ , Nonlinear Analysis, T.M.A., 8 (1984), 909-939. | MR 753767 | Zbl 0557.35129

[19] Bermudez, A.-Saguez, C., Mathematical formulation and numerical solution of an alloy solidification problem, in: Free Boundary Problems: Theory and Applications (A. Fasano - M. Primicerio, eds.), Pitman, Boston (1983), pp. 237-247. | Zbl 0506.65043

[20] Beurling, A., On free boundary problems for the Laplace equation, Advanced Study Seminars, 1 (1957), 248-263. | Zbl 0099.08302

[21] Blowey, J. F.-Elliott, C. M., The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy, Euro J. App. Math., 2 (1991), 233-280. | MR 1123143 | Zbl 0797.35172

[22] Bossavit, A., Stefan models for eddy currents in steel, in: Free Boundary Problems: Theory and Applications (A. Fasano - M. Primicerio, eds.), Pitman, Boston (1983), pp. 349-364. | Zbl 0506.65044

[23] Bossavit, A., Free boundaries in induction heating, Control Cybernetics, 14 (1985), 69-96. | Zbl 0615.35086

[24] Bossavit, A., Mixed methods for a vectorial Stefan problem, in: Free Boundary Problems: Theory and Applications (K.-H. Hoffmann - J. Sprekels, eds.), Longman, Harlow (1990), pp. 25-37. | MR 1077030 | Zbl 0724.65118

[25] Bossavit, A., Électromagnétisme, en vue de la modélisation, Springer, Paris (1993). | MR 1616583 | Zbl 0787.65090

[26] Bossavit, A.-Damlamian, A., Homogenization of the Stefan problem and application to composite magnetic media, I.M.A. J. Appl. Math., 27 (1981), 319-334. | MR 633807 | Zbl 0468.35053

[27] Bossavit, A.-Damlamian, A.-Fre ?Mond (Eds.), M., Free Boundary Problems: Theory and Applications, Pitman, Boston (1985).

[28] Bossavit, A.-Vérité, J. C., A mixed FEM-BIEM method to solve eddy currents problems, I.E.E.E. Trans. Magn.MAG-18 (1982), 431-435.

[29] Brakke, K. A., The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton (1978). | MR 485012 | Zbl 0386.53047

[30] Brézis, H., On some degenerate nonlinear parabolic equations, in: Nonlinear Functional Analysis (F. E. Browder, ed.). Proc. Symp. Pure Math., XVIII. A.M.S., Providence (1970), pp. 28-38. | MR 273468 | Zbl 0231.47034

[31] Brézis, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam (1973). | MR 348562 | Zbl 0252.47055

[32] Brice, J. C., The Growth of Crystals from Liquids, North-Holland, Amsterdam (1973).

[33] Brokate, M.-Sprekels, J., Hysteresis and Phase Transitions, Springer, Heidelberg (1996). | MR 1411908 | Zbl 0951.74002

[34] G. Buttazzo - A. Visintin (eds.), Motion by Mean Curvature and Related Topics, De Gruyter, Berlin (1994). | MR 1277387 | Zbl 0794.00019

[35] Caffarelli, L. A., The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184. | MR 454350 | Zbl 0386.35046

[36] Caffarelli, L. A., Some aspects of the one-phase Stefan problem, Indiana Univ. Math. J., 27 (1978), 73-77. | MR 466965 | Zbl 0393.35064

[37] Caffarelli, L. A.-Evans, L. C., Continuity of the temperature in the two-phase Stefan problem, Arch. Rational Mech. Anal., 81 (1983), 199-220. | MR 683353 | Zbl 0516.35080

[38] Caffarelli, L. A.-Friedman, A., Continuity of the temperature in the Stefan prob- lem, Indiana Univ. Math. J., 28 (1979), 53-70. | MR 523623 | Zbl 0406.35032

[39] Caginalp, G., An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. | MR 816623 | Zbl 0608.35080

[40] Caginalp, G., Stefan and Hele-Shaw type models as asymptotic limits of phase field equations,, Pys. Rev. A, 39 (1989), 5887-5896. | MR 998924 | Zbl 1027.80505

[41] Caginalp, G., The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, I.M.A. J. Appl. Math., 44 (1990), 77-94. | MR 1044256 | Zbl 0712.35114

[42] Caginalp, G.-Xie, W., Phase field and sharp-interface alloy models, Phys. Rev. E, 48 (1993), 1897-1909 | MR 1377919

[43] Cahn, J. W., Theory of crystal growth and interface motion in crystalline materials, Acta Metall., 8 (1960), 554-562.

[44] Cahn, J. W., On spinodal decomposition, Acta Metall., 9 (1961), 795-801.

[45] Cahn, J. W.-Hilliard, J. E., Free energy of a nonuniform system. I. Interfacial free energy. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 28 (1957), 258-267; 31 (1959), 688-699.

[46] Cahn, J. W.-Taylor, J. E., Surface motion by surface diffusion, Acta Metall. Mater., 42 (1994), 1045-1063.

[47] Cannon, J. R., The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications, Vol. 23, Addison Wesley, Menlo Park (1984). | MR 747979 | Zbl 0567.35001

[48] Cannon, J. R.-Dibenedetto, E., The steady state Stefan problem with convection, with mixed temperature and non-linear heat flux boundary conditions, in: Free Boundary Problems (E. Magenes, ed.), Istituto di Alta Matematica, Roma (1980), pp. 231-266. | MR 630724 | Zbl 0493.76088

[49] Cannon, J. R.-Hill, C. D., On the infinite differentiability of the free boundary in a Stefan problem, J. Math. Anal. Appl., 22 (1968), 385-387. | MR 225013 | Zbl 0167.10504

[50] J. M. Chadam - H. Rasmussen (eds.), Emerging Applications in Free Boundary Problems, Longman, Harlow (1993). | MR 1216352

[51] J. M. Chadam - H. Rasmussen (eds.) Free Boundary Problems Involving Solids, Longman, Harlow (1993). | MR 1216392

[52] J. M. Chadam - H. Rasmussen (eds.), Free Boundary Problems in Fluid Flow with Applications, Longman, Harlow (1993). | MR 1216373

[53] Chalmers, B., Principles of Solidification, Wiley, New York (1964).

[54] Chen, X.-Reitich, F., Local existence and uniqueness of solutions of the Stefan prob- lem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 154 (1992), 350-362. | MR 1151039 | Zbl 0761.35113

[55] Chen, Y. G.-Giga, Y.-Goto, S., Uniqueness and existence of viscosity solutions of generalized solutions of mean curvature flow equation, J. Diff. Geom., 33 (1991), 749-786. | MR 1100211 | Zbl 0696.35087

[56] Christian, J. W., The Theory of Transformations in Metals and Alloys. Part 1: Equilibrium and General Kinetic Theory, Pergamon Press, London (1975).

[57] Colli, P.-Gilardi, G.-Grasselli, M., Weak solution to hyperbolic Stefan problems with memory, No.D.E.A. (to appear). | MR 1433315 | Zbl 0878.35121

[58] Colli, P.-Gilardi, G.-Grasselli, M., Global smooth solution to the standard phase-field model with memory, Adv. Diff. Eqs. (to appear). | MR 1441852 | Zbl 1023.45500

[59] Colli, P.-Gilardi, G.-Grasselli, M., Asymptotic analysis of a phase field model with memory for vanishing time relaxation, Preprint, Dipartimento di Matematica dell'Università di Torino (1995). | MR 1679579 | Zbl 0927.35022

[60] Colli, P.-Grasselli, M., Hyperbolic phase change problems in heat conduction with memory, Proc. Roy. Soc. Edingurgh A, 123 (1993), 571-592. | MR 1226618 | Zbl 0788.35151

[61] Colli, P.-Sprekels, J., On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type, Ann. Mat. Pura Appl., 169 (1995), 269-289. | MR 1378478 | Zbl 0852.35030

[62] Colli, P.-Sprekels, J., Stefan problems and the Penrose-Fife phase field model, Adv. Math. Sci. Appl. (to appear). | MR 1476282 | Zbl 0892.35158

[63] Crank, J., Free and Moving Boundary Problems, Clarendon Press, Oxford (1984). | MR 776227 | Zbl 0547.35001

[64] Crank, J.-Ockendon, J. R., Proceedings of an I.M.A. Conference on Crystal Growth, I.M.A. J. of Appl. Math., 35 (1985) 115-264.

[65] Crowley, A. B., Numerical solution of alloy solidification problem revisited, in: Free Boundary Problems: Theory and Applications (A. Bossavit - A. Damlamian - M. Frémond, eds.), Pitman, Boston (1985), pp. 122-131. | Zbl 0593.35100

[66] Crowley, A. B.-Ockendon, J. R., On the numerical solution of an alloy solidification problem, Int. J. Heat Mass Transfer, 22 (1979), 941-947.

[67] Cryer, C. W., A Bibliography of Free Boundary Problems, M.R.C. Rep. No. 1793, Math. Res. Cent., Univ. of Wisconsin (1977).

[68] Damlamian, A., Some results on the multi-phase Stefan problem, Comm. in P.D.E.s, 2 (1977), 1017-1044. | MR 487015 | Zbl 0399.35054

[69] Damlamian, A., Homogenization for eddy currents, Delft Progress Report, 6 (1981), 268-275. | Zbl 0483.35004

[70] Damlamian, A., Asymptotic behavior of solutions to a multi-phase Stefan problem, in: Free Boundary Problems: Theory and Applications (K.-H. Hoffmann and J. Sprekels, eds.), Longman, Harlow (1990), pp. 811-817. | Zbl 0619.35053

[71] Damlamian, A.-Kenmochi, N., Asymptotic behavior of solutions to a multi-phase Stefan problem, Japan J. Appl. Math., 3 (1986), 15-36. | MR 899211 | Zbl 0619.35053

[72] Damlamian, A.-Kenmochi, N., Periodicity and almost periodicity of the solutions to a multi-phase Stefan problem, Nonlinear Analysis, T.M.A., 12 (1988), 921-934. | MR 960635 | Zbl 0673.35103

[73] A. Damlamian - J. Spruck - A. Visintin (eds.), Curvature Flows and Related Topics, Gakkotosho, Tokyo (1995). | MR 1365295

[74] Danilyuk, I. I., On the Stefan problem, Russian Math. Surveys, 40 (1985), 157-223. | MR 810813 | Zbl 0604.35080

[75] J. I. Diaz - M. A. Herrero - A. Liñán - J. L. Vázquez (eds.), Free Boundary Problems: Theory and Applications, Longman, Harlow (1995). | MR 1342322

[76] Dibenedetto, E., Regularity results for the n-dimensional two-phase Stefan problem, Boll. Un. Mat. Ital. Suppl. (1980), 129-152. | MR 677695 | Zbl 0458.35098

[77] Dibenedetto, E., Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl., 121 (1982), 131-176. | MR 663969 | Zbl 0503.35018

[78] Dibenedetto, E.-Friedman, A., The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. A.M.S., 282 (1984), 183-204. | MR 728709 | Zbl 0621.35102

[79] Dibenedetto, E.-Friedman, A., Conduction-convection problems with change of phase, J. Diff. Eqs., 62 (1986), 129-185. | MR 833415 | Zbl 0593.35085

[80] Dibenedetto, E.-Vespri, V., On the singular equation $β {(u)}_{t} = Δ u$ , Arch. Rat. Mech. Anal., 132 (1995), 247-309. | MR 1365831 | Zbl 0849.35060

[81] Dobrushin, R.-Kotecký, R.-Shlosman, S., Wulff Construction. A Global Shape from Local Interaction, A.M.S., Providence (1992). | MR 1181197 | Zbl 0917.60103

[82] Donnelly, J. D. P., A model for non-equilibrium thermodynamic processes involving phase changes, J. Inst. Math. Appl., 24 (1979), 425-438. | MR 556152 | Zbl 0426.35060

[83] Doremus, R. H., Rates of Phase Transformations, Academic Press, Orlando (1985).

[84] Duvaut, G., Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degrés), C.R. Acad. Sci. Paris, Série I, 276-A (1973), 1461-1463. | MR 328346 | Zbl 0258.35037

[85] Duvaut, G., The solution of a two-phase Stefan by a variational inequality, in: Moving Boundary Problems in Heat Flow and Diffusion (J. R. Ockendon - W. R. Hodgkins, eds.), Clarendon Press, Oxford (1975), pp. 173-181.

[86] Elliott, C. M., On a variational inequality formulation of an electrochemical machining moving boundary problem and its approximation by the finite element method, J. Inst. Maths. Applics., 25 (1980), 121-131. | MR 571974 | Zbl 0428.49003

[87] Elliott, C. M., The Cahn-Hilliard model for the kinetics of phase separation, in: Mathematical Models for Phase Change Problems (J.-F. Rodrigues, ed.), Birkhäuser, Basel (1989), pp. 35-73. | MR 1038064 | Zbl 0692.73003

[88] Elliott, C. M.-Janovsky, V., A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburg, 88A (1981), 93-107. | MR 611303 | Zbl 0455.76043

[89] Elliott, C. M.-Ockendon, J. R., Weak and Variational Methods for Moving Boundary Problems, Pitman, Boston (1982). | MR 650455 | Zbl 0476.35080

[90] Elliott, C. M.-Zheng, S., On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal., 96 (1986), 339-357. | MR 855754 | Zbl 0624.35048

[91] Evans, G. W., A note on the existence of a solution to a Stefan problem, Quart. Appl. Math., IX (1951), 185-193. | Zbl 0043.41101

[92] Evans, L. C.-Soner, M.-Souganidis, P. E., Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. | MR 1177477 | Zbl 0801.35045

[93] Evans, L. C.-Spruck, J., Motion of level sets by mean curvature I, J. Diff. Geom., 33 (1991), 635-681. | MR 1100206 | Zbl 0726.53029

[94] Fasano, A., Las Zonas Pastosas en el Problema de Stefan, Cuad. Inst. Mat. Beppo Levi, No. 13, Rosario (1987). | MR 914370 | Zbl 0642.35081

[95] Fasano, A.-Primicerio, M., General free boundary problems for the heat equation, J. Math. Anal. Appl., I, 57 (1977), 694-723; II, 58 (1977), 202-231; III, 59 (1977), 1-14. | Zbl 0355.35037

[96] Fasano, A.-Primicerio, M., Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J. Math. Anal. Appl., 72 (1979), 247-273. | MR 552335 | Zbl 0421.35080

[97] A. Fasano - M. Primicerio (eds.), Free Boundary Problems: Theory and Applications, Pitman, Boston (1983). | MR 714939 | Zbl 0504.00012

[98] Fasano, A.-Primicerio, M., Freezing in porous media: a review of mathematical models, Proc. German-Italian Symp. (V. Boffi - H. Neunzert, eds.), Teubner (1984), pp. 288-311. | MR 788551 | Zbl 0578.76102

[99] Fasano, A.-Primicerio, M., Phase-change with volumetric heat sources: Stefan's scheme vs. enthalpy formulation, Boll. Un. Mat. Ital. Suppl., 4 (1985), 131-149. | MR 784301 | Zbl 0578.35087

[100] Fasano, A.-Primicerio, M., Mushy regions with variable temperature in melting processes, Boll. Un. Mat. Ital., 4-B (1985), 601-626. | MR 805431 | Zbl 0591.35088

[101] Fasano, A.-Primicerio, M., A parabolic-hyperbolic free boundary problem, S.I.A.M. J. Math. Anal., 17 (1986), 67-73. | MR 819213 | Zbl 0594.35092

[102] Fasano, A.-Primicerio, M.-Kamin, S., Regularity of weak solutions of one-dimensional two-phase Stefan problems, Ann. Mat. Pura Appl., 115 (1977), 341-348. | MR 477460 | Zbl 0378.35008

[103] Fix, G., Numerical methods for alloy solidification, in: Moving Boundary Problems (D. G. Wilson - A. D. Solomon - P. T. Boggs, eds.) Academic Press, New York (1978), pp. 109-128. | MR 484050 | Zbl 0444.65081

[104] Fix, G., Numerical simulation of free boundary problems using phase field methods, in: The Mathematics of Finite Element and Applications (J. R. Whiteman, ed.), Academic Press, London (1982). | Zbl 0505.65061

[105] Fix, G., Phase field methods for free boundary problems, in: Free Boundary Problems, Theory and Applications (A. Fasano - M. Primicerio, eds.), Pitman, Boston (1983), pp. 580-589. | Zbl 0518.35086

[106] Flemings, M. C., Solidification Processing, McGraw-Hill, New York (1973).

[107] Frémond, M., Variational formulation of the Stefan problem, coupled Stefan problem, frost propagation in porous media, in: Proc. Conf. Computational Methods in Nonlinear Mechanics (J. T. Oden, ed.), University of Texas, Austin (1974), pp. 341-349. | MR 398299 | Zbl 0316.76063

[108] Frémond, M.-Visintin, A., Dissipation dans le changement de phase. Surfusion. Changement de phase irréversible, C.R. Acad. Sci. Paris, Série II, 301 (1985), 1265-1268. | MR 880589 | Zbl 0582.73007

[109] Friedman, A., Free boundary problems for parabolic equations. I, II, III, J. Math. Mech., 8 (1959), 499-517; 9 (1960), 19-66; 9 (1960), 327-345. | Zbl 0199.42301

[110] Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs (1964). | MR 181836 | Zbl 0144.34903

[111] Friedman, A., The Stefan problem in several space variables, Trans. Amer. Math. Soc., 133 (1968), 51-87. | MR 227625 | Zbl 0162.41903

[112] Friedman, A., One dimensional Stefan problems with non-monotone free boundary, Trans. Amer. Math. Soc., 133 (1968), 89-114. | MR 227626 | Zbl 0162.42001

[113] Friedman, A., Analyticity of the free boundary for the Stefan problem, Arch. Rational Mech. Anal., 61 (1976), 97-125. | MR 407452 | Zbl 0329.35034

[114] Friedman, A., Variational Principles and Free Boundary Problems, Wiley, New York (1982). | MR 679313 | Zbl 0564.49002

[115] A. Friedman (ed.), Mathematics in Industrial Problems, Parts 1 - 6, Springer, New York (1988-1993). | MR 968664 | Zbl 0731.00004

[116] Friedman, A.-Kinderlehrer, D., A one phase Stefan problem, Indiana Univ. Math. J., 25 (1975), 1005-1035. | MR 385326 | Zbl 0334.49002

[117] A. Friedman - J. Spruck (eds.), Variational and Free Boundary Problems, Springer, New York (1993). | MR 1320769 | Zbl 0782.00065

[118] Gage, M., An isoperimetric inequality with applications to curve shortening, Duke Math. J., 50 (1983), 1225-1229. | MR 726325 | Zbl 0534.52008

[119] Götz, I.G.-Primicerio, M., Spherically symmetrical Stefan problem with the Gibbs-Thomdon law at the moving boundary, Euro. J. Appl. Math., 7 (1996), 249-275. | MR 1401170 | Zbl 0879.35163

[120] Götz, I. G.-Zaltzman, B. B., Nonincrease of mushy region in a nonhomogeneous Stefan problem, Quart. Appl. Math., XLIX (1991), 741-746. | MR 1134749 | Zbl 0756.35119

[121] Grayson, M., The heat equation shrinks embedded plane curves to points, J. Diff. Geom., 26 (1987), 285-314. | MR 906392 | Zbl 0667.53001

[122] Gurtin, M. E., On a theory of phase transitions with interfacial energy, Arch. Rational Mech. Anal., 87 (1985), 187-212. | MR 768066

[123] Gurtin, M. E., On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rat. Mech. Anal., 96 (1986), 199-241. | MR 855304 | Zbl 0654.73008

[124] Gurtin, M. E., Toward a nonequilibrium thermodynamics of two phase materials, Arch. Rat. Mech. Anal., 100 (1988), 275-312. | MR 918798 | Zbl 0673.73007

[125] Gurtin, M. E., Multiphase thermomechanics with interfacial structure. 1. Heat conduction and the capillary balance law, Arch. Rational Mech. Anal., 104 (1988), 195-221. | MR 1017288 | Zbl 0723.73016

[126] Gurtin, M. E., On diffusion in two-phase systems: the sharp interface versus the transition layer, in: P.D.E.s and Continuum Models of Phase Transitions (M. Rascle - D. Serre - M. Slemrod, eds.), Springer, Heidelberg (1989), pp. 99-112. | MR 1036065 | Zbl 0991.80500

[127] Gurtin, M. E., On thermomechanical laws for the motion of a phase interface, Zeit. Angew. Math. Phys., 42 (1991), 370-388. | MR 1115197 | Zbl 0754.73020

[128] Gurtin, M. E., Thermomechanics of Evolving Phase Boundaries in the Plane, Clarendon Press, Oxford (1993). | MR 1402243 | Zbl 0787.73004

[129] Gurtin, M. E., The dynamics of solid-solid phase transitions. 1. Coherent interfaces, Arch. Rational Mech. Anal., 123 (1993), 305-335. | MR 1233641 | Zbl 0788.73017

[130] Gurtin, M. E., Thermodynamics and supercritical Stefan equations with nucleations, Quart. Appl. Math., LII (1994), 133-155. | MR 1262324 | Zbl 0809.35171

[131] M. E. Gurtin - G. Mcfadden (eds.), On the Evolution of Phase Boundaries, Springer, New York (1991). | MR 1226911 | Zbl 0745.00047

[132] Gurtin, M. E.-Soner, H. M., Some remarks on the Stefan problem with surface structure, Quart. Appl. Math., L (1992), 291-303. | MR 1162277 | Zbl 0763.35110

[133] Hanzawa, E.-I., Classical solution of the Stefan problem, Tohoku Math., 33 (1981), 297-335. | MR 633045 | Zbl 0571.35109

[134] Hill, J. M., One-Dimensional Stefan Problem: An Introduction, Longman, Harlow (1987). | MR 895137 | Zbl 0698.35002

[135] Hohenberg, P. C.-Halperin, B. I., Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479.

[136] K.-H. Hoffmann - J. Sprekels (eds.), Free Boundary Problems: Theory and Applications, Longman, Harlow (1990). | Zbl 0703.00015

[137] K.-H. Hoffmann - J. Sprekels (eds.), Free Boundary Value Problems, Birkhäuser, Boston (1990). | MR 1111018 | Zbl 0702.00021

[138] Hömberg, D., A mathematical model for the phase transitions in eutectoid carbon steel, I.M.A. J. Appl. Math., 54 (1995), 31-57. | MR 1319948 | Zbl 0830.65126

[139] Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., 20 (1984), 237-266. | MR 772132 | Zbl 0556.53001

[140] Ilmanen, T., Elliptic regularization and partial regularity for motion by mean curvature, Memoirs A.M.S., 520 (1994). | MR 1196160 | Zbl 0798.35066

[141] Jiang, L. S., The two-phase Stefan problem. I, II, Chinese Math.4 (1963), 686-702; 5 (1964), 36-53. | Zbl 0149.31601

[142] Kamenomostskaya, S., On the Stefan problem, Math. Sbornik, 53 (1961) 489-514 (Russian). | Zbl 0102.09301

[143] Kenmochi, N., Systems of nonlinear P.D.E.s arising from dynamical phase transitions, in: Modelling and Analysis of Phase Transition and Hysteresis Phenomena (A. Visintin, ed.), Springer, Heidelberg (1994), pp. 39-86. | MR 1321831 | Zbl 0824.35144

[144] Kenmochi, N.-Niezgódka, M., Viscosity approach to modelling non-isothermal diffusive phase separation, Japan J. Indust. Appl. Math., 13 (1996), 135-16. | MR 1377464 | Zbl 0865.35062

[145] Kinderlehrer, D.-Nirenberg, L., Regularity in free boundary value problems, Ann. Scuola Norm. Sup. Pisa, 4 (1977), 373-391. | MR 440187 | Zbl 0352.35023

[146] Kinderlehrer, D.-Nirenberg, L., The smoothness of the free boundary in the one-phase Stefan problem, Comm. Pure Appl. Math., 31 (1978), 257-282. | MR 480348 | Zbl 0391.35060

[147] Kinderlehrer, D.-Stampacchia, G., An Introduction to Variational Inequalities and their Applications, Academic Press, New York (1980). | MR 567696 | Zbl 0457.35001

[148] Kolodner, I. I., Free boundary problem for the heat equation with applications to problems with change of phase, Comm. Pure Appl. Math., 10 (1957), 220-231. | Zbl 0070.43803

[149] Kurz, W.-Fisher, D. J., Fundamentals of Solidification, Trans Tech, Aedermannsdorf (1989).

[150] Lamé, G.-Clayperon, B. P., Mémoire sur la solidification par refroidissement d'un globe solide, Ann. Chem. Phys., 47 (1831), 250-256.

[151] Langer, J. S., Instabilities and pattern formation in crystal growth, Rev. Mod. Phys., 52 (1980), 1-28.

[152] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris (1969). | MR 259693 | Zbl 0189.40603

[153] Luckhaus, S., Solutions of the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, Euro. J. Appl. Math., 1 (1990), 101-111. | MR 1117346 | Zbl 0734.35159

[154] Luckhaus, S., Solidification of alloys and the Gibbs-Thomson law, Preprint (1994).

[155] Luckhaus, S.-Modica, L., The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal., 107 (1989), 71-83. | MR 1000224 | Zbl 0681.49012

[156] Luckhaus, S.-Sturzenhecker, T., Implicit time discretization for the mean curvature flow equation, Calc. Var., 3 (1995), 253-271. | MR 1386964 | Zbl 0821.35003

[157] Luckhaus, S.-Visintin, A., Phase transition in a multicomponent system, Manuscripta Math., 43 (1983), 261-288. | MR 707047 | Zbl 0525.35012

[158] E. Magenes (ed.), Free Boundary Problems, Istituto di Alta Matematica, Roma (1980).

[159] Magenes, E., Problemi di Stefan bifase in più variabili spaziali, Le Matematiche, 36 (1981), 65-108. | MR 736797 | Zbl 0545.35096

[160] Massari, U.-Pepe, L., Su di una impostazione parametrica del problema dei capillari, Ann. Univ. Ferrara, 20 (1974), 21-31. | MR 388973 | Zbl 0365.76035

[161] Mcgeough, J. A., Principles of Electrochemical Machining, Chapman and Hall, London (1974).

[162] Mcgeough, J. A.-Rasmussen, H., On the derivation of the quasi-steady model in electrochemical machining, J. Inst. Maths. Applics., 13 (1974), 13-21.

[163] Meirmanov, A. M., On the classical solvability of the Stefan problem, Soviet Math. Dokl., 20 (1979), 1426-1429. | Zbl 0498.35087

[164] Meirmanov, A. M., On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations, Math. USSR.-Sbornik, 40 (1981), 157-178. | Zbl 0467.35053

[165] Meirmanov, A. M., An example of nonexistence of a classical solution of the Stefan problem, Soviet Math. Dokl., 23 (1981), 564-566. | MR 620870 | Zbl 0545.35097

[166] Meirmanov, A. M., The Stefan Problem, De Gruyter, Berlin (1992) (Russian edition: Nauka, Novosibirsk (1986)). | MR 1154310 | Zbl 0751.35052

[167] Meirmanov, A. M., The Stefan problem with surface tension in the three dimensional case with spherical symmetry: non-existence of the classical solution, Euro J. Appl. Math., 5 (1994), 1-20. | MR 1270785 | Zbl 0812.35165

[168] Modica, L., Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré. Anal. Nonlin., 4 (1987), 487-512. | MR 921549 | Zbl 0642.49009

[169] Modica, L., Gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142. | MR 866718 | Zbl 0616.76004

[170] Mullins, W. W.-Sekerka, R. F., Morphological stability of a particle growing by diffusion and heat flow, J. Appl. Phys., 34 (1963), 323-329.

[171] Mullins, W. W.-Sekerka, R. F., Stability of a planar interface during solidification of a dilute alloy, J. Appl. Phys., 35 (1964), 441-451.

[172] Niezgódka, M., Stefan-like problems, in: Free Boundary Problems: Theory and Applications (A. Fasano - M. Primicerio, eds.), Pitman, Boston (1983), pp. 321-347. | MR 714922 | Zbl 0512.35075

[173] M. Niezgódka (ed.), Free Boundary Problems: Theory and Applications, Gakkotosho, Tokyo (to appear). | MR 1462964

[174] Novick-Cohen, A.-Segel, L. A., Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10 (1984), 278-298. | MR 763473

[175] J. R. Ockendon - W. R. Hodgkins (eds.), Moving Boundary Problems in Heat Flow and Diffusion, Clarendon Press, Oxford (1975). | Zbl 0295.76064

[176] Oleĭnik, O. A., A method of solution of the general Stefan problem, Soviet Math. Dokl., 1 (1960), 1350-1353. | MR 125341 | Zbl 0131.09202

[177] Oleĭnik, O. A.-Primicerio, M.-Radkevich, E. V., Stefan-like problems, Meccanica, 28 (1993), 129-143. | Zbl 0786.76091

[178] Osher, S.-Sethian, J. A., Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. | MR 965860 | Zbl 0659.65132

[179] R. Pamplin (ed.), Crystal Growth, Pergamon Press, Oxford (1975).

[180] Penrose, O.-Fife, P. C., Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D, 43 (1990), 44-62. | MR 1060043 | Zbl 0709.76001

[181] Penrose, O.-Fife, P. C., On the relation between the standard phase-field model and a «thermodynamically consistent» phase-field model, Physica D, 69 (1993), 107-113. | MR 1245658 | Zbl 0799.76084

[182] Plotnikov, P. I.-Starovoitov, V. N., The Stefan problem with surface tension as a limit of phase field model, Diff. Eqs., 29 (1993), 395-404. | MR 1236334 | Zbl 0802.35165

[183] Prigogine, I., Thermodynamics of Irreversible Processes, Wiley-Interscience, New York (1967). | Zbl 0115.23101

[184] Primicerio, M., Problemi a contorno libero per l'equazione della diffusione, Rend. Sem. Mat. Univ. Politecn. Torino, 32 (1973-74), 183-206. | MR 477466 | Zbl 0307.76048

[185] Primicerio, M., Problemi di diffusione a frontiera libera, Boll. Un. Mat. Ital., 18-A (1981), 11-68. | Zbl 0468.35082

[186] Primicerio, M., Mushy regions in phase-change problems, in: Applied Functional Analysis (R. Gorenflo - K.-H. Hoffmann, eds.), Lang, Frankfurt (1983), pp. 251-269. | MR 685609 | Zbl 0518.35087

[187] Radkevitch, E., Gibbs-Thomson law and existence of the classical solution of the modified Stefan problem, Soviet Dokl. Acad. Sci., 43 (1991), 274-278. | Zbl 0782.35087

[188] Richardson, S., Hele-Shaw flow with a free boundary produced by the injections of a fluid into a narrow channel, J. Fluid. Mech., 56 (1972), 609-618. | Zbl 0256.76024

[189] Richardson, S., Some Hele-Shaw flow with time-dependent free boundaries, J. Fluid. Mech., 102 (1981), 263-278. | MR 612095 | Zbl 0451.76015

[190] Rodrigues, J.-F., An evolutionary continuous casting problem of Stefan type, Quart. Appl. Math., XLIV (1986), 109-131. | MR 840448 | Zbl 0618.35068

[191] Rodrigues, J.-F., Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam (1987). | MR 880369 | Zbl 0606.73017

[192] Rodrigues, J.-F., The variational inequality approach to the one-phase Stefan problem, Acta Appl. Math., 8 (1987), 1-35. | MR 871691 | Zbl 0653.35083

[193] Rodrigues (Ed.), J.-F., Mathematical Models for Phase Change Problems, Birkhäuser, Basel (1989). | MR 1038061 | Zbl 0676.00021

[194] Rodrigues, J.-F., Variational methods in the Stefan problem, in: Modelling and Analysis of Phase Transition and Hysteresis Phenomena (A. Visintin, ed.). Springer, Heidelberg (1994), pp. 147-212. | MR 1321833 | Zbl 0819.35154

[195] Romano, A., Thermomechanics of Phase Transitions in Classical Field Theory, World Scientific, Singapore (1993). | MR 1347688 | Zbl 0827.73001

[196] Rubinstein, L., On the determination of the position of the boundary which separates two phases in the one-dimensional problem of Stefan, Dokl. Acad. Nauk USSR, 58 (1947), 217-220. | MR 22979 | Zbl 0032.13101

[197] Rubinstein, L., The Stefan Problem, A.M.S., Providence (1971) (Russian edition: Zvaigzne, Riga (1967)). | MR 222436 | Zbl 0434.35086

[198] Rubinstein, L., On mathematical modelling of growth of an individual monocomponent crystal from melt in a non-uniform temperature field, Control and Cybernetics, 12 (1983), 5-18. | MR 720596 | Zbl 0512.93043

[199] Rubinstein, L., On mathematical models of solid-liquid zones in two phase mono- component system and in binary alloys, in: Free Boundary Problems: Theory and Applications (A. Fasano - M. Primicerio, eds.), Pitman, Boston (1983), pp. 275-282. | Zbl 0509.76095

[200] Rubinstein, L., Mathematical modelling of growth of an individual monocomponent crystal from its melt in a non-homogeneous temperature field, in: Free Boundary Problems: Theory and Applications (A. Bossavit - A. Damlamian - M. Frémond, eds.), Pitman, Boston (1985), pp. 166-178. | MR 863169 | Zbl 0593.35105

[201] Rubinstein, L.-Fasano, A.-Primicerio, M., Remarks on the analyticity of the free boundary for the one-dimensional Stefan problem, Ann. Mat. Pura Appl., 125 (1980), 295-311. | MR 605212 | Zbl 0456.35096

[202] Saffman, P. G.-Taylor, G. I., The penetration of fluid into a porous medium or Hele-Shaw cell, Proc. Roy. Soc. London Ser. A, 245 (1958), 312-329. | MR 97227 | Zbl 0086.41603

[203] Schaeffer, D., A new proof of infinite differentiability of the free boundary in the Stefan problem, J. Diff. Eqs., 20 (1976), 266-269. | MR 390499 | Zbl 0314.35044

[204] Sestini, G., Esistenza di una soluzione in problemi analoghi a quello di Stefan, Riv. Mat. Univ. Parma, 3 (1952), 3-23; 8 (1958), 1-209. | Zbl 0048.43406

[205] Sethian, J. A., Curvature and evolution of fronts, Comm. Math. Phys., 101 (1985), 487-499. | MR 815197 | Zbl 0619.76087

[206] Showalter, R. E., Mathematical formulation of the Stefan problem, Int. J. Eng. Sci., 20 (1982), 909-912. | MR 660566 | Zbl 0506.76103

[207] Showalter, R. E.-Walkington, N. J., A hyperbolic Stefan problem, Quart. Appl. Math., XLV (1987), 769-781. | MR 917025 | Zbl 0649.35086

[208] Skripov, V. P., Metastable Liquids, Wiley, Chichester (1974).

[209] Soner, M., Convergence of the phase-field equation to the Mullins-Sekerka problem with kinetic undercooling, Arch. Rational Mech. Anal., 131 (1995), 139-197. | MR 1346368 | Zbl 0829.73010

[210] Sprekels, J.-Zheng, S., Global smooth solutions to a thermodynamically consistent model of phase-field type in higher space dimensions, J. Math. Anal. Appl., 176 (1993), 200-223. | MR 1222165 | Zbl 0804.35063

[211] Stefan, J., Über einige Probleme der Theorie der Wärmeleitung, Sitzungber., Wien, Akad. Mat. Natur., 98 (1889), 473-484. Also ibid. pp. 614-634, 965-983, 1418-1442.

[212] Talamucci, F., Analysis of coupled heat-mass transport in freezing porous soils, Surveys for Mathematics in Industry, to appear. | Zbl 0901.76084

[213] Tarzia, D. A., Una revisión sobre problemas de frontera móvil y libre para la ecuación del calor. El problema de Stefan, Math. Notae, 29 (1981/82), 147-241. | Zbl 0847.00034

[214] Tarzia, D. A., The Two-Phase Stefan Problem and Some Related Conduction Problems, S.B.M.A.C., Gramado (1987). | Zbl 0774.35091

[215] Tarzia, D. A., A Bibliography on Moving-Free Boundary Problems for the Heat Diffusion Equation. The Stefan Problem, Prog. Naz. M.P.I. Ital., Firenze (1988). | MR 1007840 | Zbl 0694.35221

[216] Taylor, J. E., Mean curvature and weighted mean curvature, Acta Metall. Mater., 40 (1992), 1475-1485.

[217] Taylor, J. E.-Cahn, J. W., Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Stat. Phys., 77 (1994), 183-197. | MR 1300532 | Zbl 0844.35044

[218] Turnbull, D., Phase Changes, Solid State Phys., 3 (1956), 225-306.

[219] Ubbelohde, A. R., The Molten State of Matter, Wiley, Chichester (1978).

[220] Verdi, C.-Visintin, A., A mathematical model of the austenite-pearlite transformation in plain steel based on the Scheil additivity rule, Acta Metall., 35 (1987), 2711-2717.

[221] Visintin, A., Stefan problem with phase relaxation, I.M.A. J. Appl. Math., 34 (1985), 225-245. | MR 804824 | Zbl 0585.35053

[222] Visintin, A., Study of the eddy-current problem taking account of Hall's effect, App. Anal., 19 (1985), 217-226. | MR 799985 | Zbl 0563.35077

[223] Visintin, A., A new model for supercooling and superheating effects, I.M.A. J. Appl. Math., 36 (1986), 141-157. | MR 984464 | Zbl 0619.35116

[224] Visintin, A., Stefan problem with a kinetic condition at the free boundary, Ann. Mat. Pura Appl., 146 (1987), 97-122. | MR 916689 | Zbl 0643.35115

[225] Visintin, A., Coupled thermal and electromagnetic evolution in a ferromagnetic body, Z. Angew. Math. Mech., 67 (1987), 409-417. | MR 919395 | Zbl 0628.73118

[226] Visintin, A., Supercooling and superheating effects in heterogeneous systems, Quart. Appl. Math., XLV (1987), 239-263. | MR 895096 | Zbl 0636.35089

[227] Visintin, A., Mathematical models of solid-solid phase transformations in steel, I.M.A. J. Appl. Math., 39 (1987), 143-157. | MR 983739 | Zbl 0652.73050

[228] Visintin, A., The Stefan problem with surface tension, in: Mathematical Models of Phase Change Problems (J. F. Rodrigues, ed.). Birkhäuser, Basel (1989), pp. 191-213. | MR 1038070 | Zbl 0692.73002

[229] Visintin, A., Remarks on the Stefan problem with surface tension, in: Boundary Value Problems for Partial Differential Equations and Applications (C. Baiocchi - J. L. Lions, eds.). Dunod, Paris (1993). | MR 1260478 | Zbl 0803.35173

[230] Visintin (Ed.), A., Modelling and Analysis of Phase Transition and Hysteresis Phenomena, Springer, Heidelberg (1994).

[231] Visintin, A., Two-scale Stefan problem with surface tension, in: Nonlinear Analysis and Applications (N. Kenmochi - M. Niezgódka - P. Strzelecki, eds.), Gakkotosho, Tokyo (1996), pp. 405-424. | MR 1422948 | Zbl 0868.35144

[232] Visintin, A., Two-scale model of phase transitions, Physica D, 106 (1997), 66-80. | MR 1460449 | Zbl 0935.80005

[233] Visintin, A., Models of Phase Transitions, Birkhäuser, Boston (1996). | MR 1423808 | Zbl 0882.35004

[234] Visintin, A., Nucleation and mean curvature flow, Comm. in P.D.E.s (to appear). | MR 1608492 | Zbl 0901.53045

[235] Visintin, A., Stefan Problem with nucleation and mean curvature flow, in preparation.

[236] Visintin, A., Motion by mean curvature flow and nucleation, C. R. Acad. Sc. Paris, Serie I, 325 (1997), 55-60. | MR 1461397 | Zbl 0883.35136

[237] Wilson, D. G.-Solomon, A. D.-Alexiades, V., Progress with simple binary alloy solidification problems, in: Free Boundary Problems: Theory and Applications (A. Fasano - M. Primicerio, eds.), Pitman, Boston 1983, pp. 306-320. | MR 719507 | Zbl 0514.76105

[238] D. G. Wilson - A. D. Solomon - P. T. Boggs (eds.), Moving Boundary Problems, Academic Press, New York (1978). | MR 466887 | Zbl 0432.00011

[239] Wilson, D. G.-Solomon, A. D.-Trent, J. S., A Bibliography on Moving Boundary Problems with Key Word Index, Oak Ridge National Laboratory (1979).

[240] Woodruff, P. D., The Solid-Liquid Interface, Cambridge Univ. Press, Cambridge (1973).

[241] Ziemer, W. P., Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. A.M.S., 271 (1982), 733-748. | MR 654859 | Zbl 0506.35053