Previous |  Up |  Next


phase-field systems; phase transitions; hysteresis operators; well-posedness of parabolic systems; thermodynamic consistency; Penrose-Fife model
Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.
[1] Besov, O. V., Il’in, V. P., Nikol’skii, S. M.: Integral representation of functions and embedding theorems. Moscow, Nauka, 1975. (Russian) MR 0430771
[2] Blowey, J. F., Elliott, C. M.: Curvature dependent phase boundary motion and double obstacle problems. Degenerate Diffusion, W.M. Ni, L.A. Peletier, and J.L. Vázquez (eds.), IMA Vol. Math. Appl. 47, Springer, New York, 1993, pp. 19–60. MR 1246337
[3] Blowey, J. F., Elliott, C. M.: A phase-field model with double obstacle potential. Motion by mean curvature and related topics, G. Buttazzo and A. Visintin (eds.), De Gruyter, Berlin, 1994, pp. 1–22. MR 1277388
[4] Brokate, M., Sprekels, J.: Hysteresis and phase transitions. Appl. Math. Sci. Vol.  121, Springer-Verlag, New York, 1996. DOI 10.1007/978-1-4612-4048-8_5 | MR 1411908
[5] Caginalp, G.: An analysis of a phase field model of a free boundary. Arch. Rational Mech. Anal. 92 (1986), 205–245. DOI 10.1007/BF00254827 | MR 0816623 | Zbl 0608.35080
[6] 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. (4) 169 (1995), 269–289. MR 1378478
[7] Colli, P., Sprekels, J.: Stefan problems and the Penrose-Fife phase-field model. Adv. Math. Sci. Appl. 7 (1997), 911–934. MR 1476282
[8] Colli, P., Sprekels, J.: Global solutions to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law. Preprint No. 351. WIAS Berlin, 1997. MR 1690376
[9] Frémond, M., Visintin, A.: Dissipation dans le changement de phase. Surfusion. Changement de phase irréversible. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 301 (1985), 1265–1268. MR 0880589
[10] Kenmochi, N., Niezgódka, M.: Systems of nonlinear parabolic equations for phase change problems. Adv. Math. Sci. Appl. 3 (1993/94), 89–117. MR 1287926
[11] Klein, O.: A semidiscrete scheme for a Penrose-Fife system and some Stefan problems in $R^3$. Adv. Math. Sci. Appl. 7 (1997), 491–523. MR 1454679
[12] Klein, O.: Existence and approximation results for phase-field systems of Penrose-Fife type and some Stefan problems. Ph.D. thesis, Humboldt University, Berlin, 1997.
[13] Krasnosel’skii, M. A., Pokrovskii, A. V.: Systems with hysteresis. Springer-Verlag, Heidelberg, 1989. MR 0987431
[14] Krejčí, P.: Hysteresis, convexity and dissipation in hyperbolic equations. Gakuto Int. Series Math. Sci. & Appl., Vol. 8, Gakkōtosho, Tokyo, 1996. MR 2466538
[15] Krejčí, P., Sprekels, J.: A hysteresis approach to phase-field models. Submitted.
[16] Ladyzhenskaya, O. A., Solonnikov, V. A., Ural’tseva, N. N.: Linear and quasilinear equations of parabolic type. American Mathematical Society, 1968.
[17] Laurençot, Ph.: Solutions to a Penrose-Fife model of phase-field type. J. Math. Anal. Appl. 185 (1994), 262–274. DOI 10.1006/jmaa.1994.1247 | MR 1283056
[18] Laurençot, Ph.: Weak solutions to a Penrose-Fife model for phase transitions. Adv. Math. Sci. Appl. 5 (1995), 117–138. MR 1325962
[19] Mayergoyz, I. D.: Mathematical models for hysteresis. Springer-Verlag, New York, 1991. MR 1083150
[20] Penrose, O., Fife, P.C.: Thermodynamically consistent models of phase field type for the kinetics of phase transitions. Physica D 43 (1990), 44–62. DOI 10.1016/0167-2789(90)90015-H | MR 1060043
[21] Protter, M. H., Weinberger, H. F.: Maximum principle in differential equations. Prentice Hall, Englewood Cliffs, 1967. MR 0219861
[22] 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. DOI 10.1006/jmaa.1993.1209 | MR 1222165
[23] Visintin, A.: Stefan problem with phase relaxation. IMA J. Appl. Math. 34 (1985), 225–245. DOI 10.1093/imamat/34.3.225 | MR 0804824 | Zbl 0585.35053
[24] Visintin, A.: Supercooling and superheating effects in phase transitions. IMA J. Appl. Math. 35 (1985), 233–256. DOI 10.1093/imamat/35.2.233 | MR 0839201 | Zbl 0615.35090
[25] Visintin, A.: Differential models of hysteresis. Springer-Verlag, New York, 1994. MR 1329094 | Zbl 0820.35004
Partner of
EuDML logo