On a conserved Penrose-Fife type system

Gianni Gilardi; Andrea Marson

Applications of Mathematics (2005)

  • Volume: 50, Issue: 5, page 465-499
  • ISSN: 0862-7940

Abstract

top
We deal with a class of Penrose-Fife type phase field models for phase transitions, where the phase dynamics is ruled by a Cahn-Hilliard type equation. Suitable assumptions on the behaviour of the heat flux as the absolute temperature tends to zero and to + ∞ are considered. An existence result is obtained by a double approximation procedure and compactness methods. Moreover, uniqueness and regularity results are proved as well.

How to cite

top

Gilardi, Gianni, and Marson, Andrea. "On a conserved Penrose-Fife type system." Applications of Mathematics 50.5 (2005): 465-499. <http://eudml.org/doc/33233>.

@article{Gilardi2005,
abstract = {We deal with a class of Penrose-Fife type phase field models for phase transitions, where the phase dynamics is ruled by a Cahn-Hilliard type equation. Suitable assumptions on the behaviour of the heat flux as the absolute temperature tends to zero and to $+\infty $ are considered. An existence result is obtained by a double approximation procedure and compactness methods. Moreover, uniqueness and regularity results are proved as well.},
author = {Gilardi, Gianni, Marson, Andrea},
journal = {Applications of Mathematics},
keywords = {Penrose-Fife model; Cahn-Hilliard equation; heat flux law; Penrose-Fife model; Cahn-Hilliard equation; heat flux law},
language = {eng},
number = {5},
pages = {465-499},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a conserved Penrose-Fife type system},
url = {http://eudml.org/doc/33233},
volume = {50},
year = {2005},
}

TY - JOUR
AU - Gilardi, Gianni
AU - Marson, Andrea
TI - On a conserved Penrose-Fife type system
JO - Applications of Mathematics
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 5
SP - 465
EP - 499
AB - We deal with a class of Penrose-Fife type phase field models for phase transitions, where the phase dynamics is ruled by a Cahn-Hilliard type equation. Suitable assumptions on the behaviour of the heat flux as the absolute temperature tends to zero and to $+\infty $ are considered. An existence result is obtained by a double approximation procedure and compactness methods. Moreover, uniqueness and regularity results are proved as well.
LA - eng
KW - Penrose-Fife model; Cahn-Hilliard equation; heat flux law; Penrose-Fife model; Cahn-Hilliard equation; heat flux law
UR - http://eudml.org/doc/33233
ER -

References

top
  1. A mathematical model of dynamics of non-isothermal phase separation, Physica D 59 (1992), 389–416. (1992) MR1192751
  2. Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. (1976) Zbl0328.47035MR0390843
  3. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. Vol.  5 North-Holland, Amsterdam, 1973. (1973) Zbl0252.47055MR0348562
  4. 10.1007/978-1-4612-4048-8_5, Springer-Verlag, New York, 1996. (1996) MR1411908DOI10.1007/978-1-4612-4048-8_5
  5. 10.1063/1.1744102, J.  Chem. Phys. 28 (1958), 258–267. (1958) DOI10.1063/1.1744102
  6. The conserved phase-field system with memory, Adv. Math. Sci. Appl. 11 (2001), 265–291. (2001) MR1841569
  7. On a Penrose-Fife phase field model with inhomogeneous Neumann boundary conditions for the temperature, Differ. Integral Equ. 17 (2004), 511–534. (2004) MR2054932
  8. Weak solutions to the Penrose-Fife phase field model for a class of admissible flux laws, Physica D 111 (1998), 311–334. (1998) MR1601442
  9. Global solution to the Penrose-Fife phase field model with special heat flux laws, In: Variations of Domains and Free-Boundary Problems in Solid Mechanics. Solid Mech. Appl.  66, P. Argoul, M. Frémond, Q. S. Nguyen (eds.), Kluwer Acad. Publ., Dordrecht, 1999, pp. 181–188. (1999) MR1672241
  10. 10.1007/BF01759357, Ann. Mat. Pura Appl. IV.  Ser. 169 (1995), 269–289. (1995) MR1378478DOI10.1007/BF01759357
  11. 10.1002/mma.423, Math. Methods Appl. Sci. 26 (2003), 1303–1325. (2003) MR2004103DOI10.1002/mma.423
  12. 10.1002/(SICI)1099-1476(19960910)19:13<1053::AID-MMA809>3.0.CO;2-S, Math. Methods Appl. Sci. 19 (1996), 1053–1072. (1996) MR1402815DOI10.1002/(SICI)1099-1476(19960910)19:13<1053::AID-MMA809>3.0.CO;2-S
  13. Global existence of smooth solution to the Penrose-Fife model for Ising ferromagnets, Adv. Math. Sci. Appl. 6 (1996), 227–241. (1996) MR1385769
  14. Non-isothermal phase transition models with Neumann boundary conditions, Nonlinear Anal. Theory Methods Appl. 53A (2003), 977-996. (2003) MR1978030
  15. Uniqueness of the solution to a nonlinear system arising in phase transition, Proceedings of the Conference Nonlinear Analysis and Applications (Warsaw, 1994). GAKUTO Intern. Ser. Math. Sci. Apl. Vol. 7, N.  Kenmochi (ed.), 1995, pp. 261–271. (1995) Zbl0873.35040MR1422940
  16. Weak solutions of nonlinear systems for non-isothermal phase transitions, Adv. Math. Sci. Appl. 9 (1999), 499–521. (1999) MR1690439
  17. Evolution equations of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal. Theory Methods Appl. 22 (1994), 1163–1180. (1994) MR1279139
  18. 10.1007/BF03167303, Japan J.  Ind. Appl. Math. 13 (1996), 135–169. (1996) MR1377464DOI10.1007/BF03167303
  19. 10.1006/jmaa.1994.1247, J.  Math. Anal. Appl. 185 (1994), 262–274. (1994) MR1283056DOI10.1006/jmaa.1994.1247
  20. Weak solutions to a Penrose-Fife model for phase transitions, Adv. Math. Sci. Appl. 5 (1995), 117–138. (1995) MR1325962
  21. 10.1006/jmaa.1997.5813, J.  Math. Anal. Appl. 219 (1998), 331–343. (1998) MR1606330DOI10.1006/jmaa.1997.5813
  22. Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969. (1969) Zbl0189.40603MR0259693
  23. Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967. (1967) MR0227584
  24. Statistical Mechanics and the kinetics of phase separation, In: Material Instabilities in Continuum Mechanics, J.  Ball (ed.), Oxford University Press, Oxford, 1988, pp. 373–394. (1988) MR0970534
  25. 10.1016/0167-2789(90)90015-H, Physica  D 43 (1990), 44–62. (1990) MR1060043DOI10.1016/0167-2789(90)90015-H
  26. 10.1016/0167-2789(93)90183-2, Physica  D 69 (1993), 107–113. (1993) MR1245658DOI10.1016/0167-2789(93)90183-2
  27. 10.1016/S0022-247X(03)00541-9, J. Math. Anal. Appl. 287 (2003 2002), 177–199. (2003 2002) MR2010264DOI10.1016/S0022-247X(03)00541-9
  28. The conserved Penrose-Fife system with Fourier heat flux law, Nonlinear Anal. Theory Methods Appl. 53A (2003), 1089–1100. (2003) MR1978036
  29. 10.1080/03605309308820946, Commun. Partial Differ. Equations 18 (1993), 701–727. (1993) MR1214877DOI10.1080/03605309308820946
  30. 10.1006/jmaa.1993.1209, J.  Math. Anal. Appl. 176 (1993), 200–223. (1993) MR1222165DOI10.1006/jmaa.1993.1209
  31. Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford, 1987. (1987) 

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.