Hysteresis operators in phase-field models of Penrose-fife type

Pavel Krejčí; Jürgen Sprekels

Applications of Mathematics (1998)

  • Volume: 43, Issue: 3, page 207-222
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Krejčí, Pavel, and Sprekels, Jürgen. "Hysteresis operators in phase-field models of Penrose-fife type." Applications of Mathematics 43.3 (1998): 207-222. <http://eudml.org/doc/33007>.

@article{Krejčí1998,
abstract = {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.},
author = {Krejčí, Pavel, Sprekels, Jürgen},
journal = {Applications of Mathematics},
keywords = {phase-field systems; phase transitions; hysteresis operators; well-posedness of parabolic systems; thermodynamic consistency; Penrose-Fife model; phase-field systems; phase transitions; hysteresis operators; well-posedness of parabolic systems; thermodynamic consistency; Penrose-Fife model},
language = {eng},
number = {3},
pages = {207-222},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hysteresis operators in phase-field models of Penrose-fife type},
url = {http://eudml.org/doc/33007},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Krejčí, Pavel
AU - Sprekels, Jürgen
TI - Hysteresis operators in phase-field models of Penrose-fife type
JO - Applications of Mathematics
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 43
IS - 3
SP - 207
EP - 222
AB - 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.
LA - eng
KW - phase-field systems; phase transitions; hysteresis operators; well-posedness of parabolic systems; thermodynamic consistency; Penrose-Fife model; phase-field systems; phase transitions; hysteresis operators; well-posedness of parabolic systems; thermodynamic consistency; Penrose-Fife model
UR - http://eudml.org/doc/33007
ER -

References

top
  1. Integral representation of functions and embedding theorems, Moscow, Nauka, 1975. (Russian) (1975) MR0430771
  2. 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. (1993) MR1246337
  3. 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. (1994) MR1277388
  4. 10.1007/978-1-4612-4048-8_5, Appl. Math. Sci. Vol.  121, Springer-Verlag, New York, 1996. (1996) MR1411908DOI10.1007/978-1-4612-4048-8_5
  5. 10.1007/BF00254827, Arch. Rational Mech. Anal. 92 (1986), 205–245. (1986) Zbl0608.35080MR0816623DOI10.1007/BF00254827
  6. 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. (1995) MR1378478
  7. Stefan problems and the Penrose-Fife phase-field model, Adv. Math. Sci. Appl. 7 (1997), 911–934. (1997) MR1476282
  8. Global solutions to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law. Preprint No. 351, WIAS Berlin, 1997. (1997) MR1690376
  9. 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. (1985) MR0880589
  10. Systems of nonlinear parabolic equations for phase change problems, Adv. Math. Sci. Appl. 3 (1993/94), 89–117. (1993/94) MR1287926
  11. A semidiscrete scheme for a Penrose-Fife system and some Stefan problems in R 3 , Adv. Math. Sci. Appl. 7 (1997), 491–523. (1997) MR1454679
  12. Existence and approximation results for phase-field systems of Penrose-Fife type and some Stefan problems, Ph.D. thesis, Humboldt University, Berlin, 1997. (1997) 
  13. Systems with hysteresis, Springer-Verlag, Heidelberg, 1989. (1989) MR0987431
  14. Hysteresis, convexity and dissipation in hyperbolic equations, Gakuto Int. Series Math. Sci. & Appl., Vol. 8, Gakkōtosho, Tokyo, 1996. (1996) MR2466538
  15. A hysteresis approach to phase-field models, Submitted. 
  16. Linear and quasilinear equations of parabolic type, American Mathematical Society, 1968. (1968) 
  17. 10.1006/jmaa.1994.1247, J. Math. Anal. Appl. 185 (1994), 262–274. (1994) MR1283056DOI10.1006/jmaa.1994.1247
  18. Weak solutions to a Penrose-Fife model for phase transitions, Adv. Math. Sci. Appl. 5 (1995), 117–138. (1995) MR1325962
  19. Mathematical models for hysteresis, Springer-Verlag, New York, 1991. (1991) MR1083150
  20. 10.1016/0167-2789(90)90015-H, Physica D 43 (1990), 44–62. (1990) MR1060043DOI10.1016/0167-2789(90)90015-H
  21. Maximum principle in differential equations, Prentice Hall, Englewood Cliffs, 1967. (1967) MR0219861
  22. 10.1006/jmaa.1993.1209, J. Math. Anal. Appl. 176 (1993), 200–223. (1993) MR1222165DOI10.1006/jmaa.1993.1209
  23. 10.1093/imamat/34.3.225, IMA J. Appl. Math. 34 (1985), 225–245. (1985) Zbl0585.35053MR0804824DOI10.1093/imamat/34.3.225
  24. 10.1093/imamat/35.2.233, IMA J. Appl. Math. 35 (1985), 233–256. (1985) Zbl0615.35090MR0839201DOI10.1093/imamat/35.2.233
  25. Differential models of hysteresis, Springer-Verlag, New York, 1994. (1994) Zbl0820.35004MR1329094

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.