Variational Formulation of Phase Transitions with Glass Formation

Augusto Visintin

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 1, page 75-111
  • ISSN: 0392-4041

Abstract

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In the framework of the theory of nonequilibrium thermodynamics, phase transitions with glass formation in binary alloys are here modelled as a multi-non-linear system of PDEs. A weak formulation is provided for an initial- and boundary-value problem, and existence of a solution is studied. This model is then reformulated as a minimization problem, on the basis of a theory that was pioneered by Fitzpatrick [MR 1009594]. This provides a tool for the analysis of compactness and structural stability of the dependence of the solution(s) on data and operators, via De Giorgi's notion of γ -convergence. This latter issue is here dealt with in some simpler settings.

How to cite

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Visintin, Augusto. "Variational Formulation of Phase Transitions with Glass Formation." Bollettino dell'Unione Matematica Italiana 6.1 (2013): 75-111. <http://eudml.org/doc/294034>.

@article{Visintin2013,
abstract = {In the framework of the theory of nonequilibrium thermodynamics, phase transitions with glass formation in binary alloys are here modelled as a multi-non-linear system of PDEs. A weak formulation is provided for an initial- and boundary-value problem, and existence of a solution is studied. This model is then reformulated as a minimization problem, on the basis of a theory that was pioneered by Fitzpatrick [MR 1009594]. This provides a tool for the analysis of compactness and structural stability of the dependence of the solution(s) on data and operators, via De Giorgi's notion of $\gamma$-convergence. This latter issue is here dealt with in some simpler settings.},
author = {Visintin, Augusto},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {75-111},
publisher = {Unione Matematica Italiana},
title = {Variational Formulation of Phase Transitions with Glass Formation},
url = {http://eudml.org/doc/294034},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Visintin, Augusto
TI - Variational Formulation of Phase Transitions with Glass Formation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/2//
PB - Unione Matematica Italiana
VL - 6
IS - 1
SP - 75
EP - 111
AB - In the framework of the theory of nonequilibrium thermodynamics, phase transitions with glass formation in binary alloys are here modelled as a multi-non-linear system of PDEs. A weak formulation is provided for an initial- and boundary-value problem, and existence of a solution is studied. This model is then reformulated as a minimization problem, on the basis of a theory that was pioneered by Fitzpatrick [MR 1009594]. This provides a tool for the analysis of compactness and structural stability of the dependence of the solution(s) on data and operators, via De Giorgi's notion of $\gamma$-convergence. This latter issue is here dealt with in some simpler settings.
LA - eng
UR - http://eudml.org/doc/294034
ER -

References

top
  1. ALEXIADES, V. - CANNON, J. R., Free boundary problems in solidification of alloys. S.I.A.M. J. Math. Anal.11 (1980), 254-264. Zbl0436.35079MR559867DOI10.1137/0511025
  2. ALEXIADES, V. - SOLOMON, A. D., Mathematical Modeling of Melting and Freezing Processes. Hemisphere Publishing, Washington DC1993. 
  3. ALEXIADES, V. - WILSON, D. G. - SOLOMON, A. D., Macroscopic global modeling of binary alloy solidification processes. Quart. Appl. Math.43 (1985), 143-158. Zbl0582.35115MR793522DOI10.1090/qam/793522
  4. ALT, H. W. - LUCKHAUS, S., Quasilinear elliptic-parabolic differential equations. Math. Z.183 (1983), 311-341. Zbl0497.35049MR706391DOI10.1007/BF01176474
  5. ANSINI, N. - DAL MASO, G. - ZEPPIERI, C. I., New results on Gamma-limits of integral functionals. Preprint SISSA, Trieste, 2012. MR3165285DOI10.1016/j.anihpc.2013.02.005
  6. ATTOUCH, H., Variational Convergence for Functions and Operators. Pitman, Boston1984. Zbl0561.49012MR773850
  7. BARBU, V., Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, Berlin2010. Zbl1197.35002MR2582280DOI10.1007/978-1-4419-5542-5
  8. BRAIDES, A., Γ -Convergence for Beginners. Oxford University Press, Oxford2002. Zbl1198.49001MR1968440DOI10.1093/acprof:oso/9780198507840.001.0001
  9. BRAIDES, A. - DEFRANCESCHI, A., Homogenization of Multiple Integrals. Oxford University Press, Oxford1998. Zbl0911.49010MR1684713
  10. BREZIS, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam1973. Zbl0252.47055MR348562
  11. BREZIS, H. - EKELAND, I., Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps and II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971-974, and ibid. 1197-1198. Zbl0332.49032MR637214
  12. BROKATE, M. - SPREKELS, J., Hysteresis and Phase Transitions. Springer, Heidelberg1996. Zbl0951.74002MR1411908DOI10.1007/978-1-4612-4048-8
  13. BURACHIK, R. S. - SVAITER, B. F., Maximal monotone operators, convex functions, and a special family of enlargements. Set-Valued Analysis10 (2002), 297-316. Zbl1033.47036MR1934748DOI10.1023/A:1020639314056
  14. BURACHIK, R. S. - SVAITER, B. F., Maximal monotonicity, conjugation and the duality product. Proc. Amer. Math. Soc.131 (2003), 2379-2383. Zbl1019.47038MR1974634DOI10.1090/S0002-9939-03-07053-9
  15. CALLEN, H. B., Thermodynamics and an Introduction to Thermostatistics. Wiley, New York1985. Zbl0989.80500
  16. CHALMERS, B., Principles of Solidification. Wiley, New York1964. 
  17. CHIADÒ PIAT, V. - DAL MASO, G. - DEFRANCESCHI, A., G-convergence of monotone operators, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 7 (1990), 123-160. Zbl0731.35033MR1065871DOI10.1016/S0294-1449(16)30298-0
  18. CHRISTIAN, J. W., The Theory of Transformations in Metals and Alloys. Part 1: Equilibrium and General Kinetic Theory. Pergamon Press, London2002. 
  19. COLLI, P. L. - VISINTIN, A., On a class of doubly nonlinear evolution problems. Communications in P.D.E.s, 15 (1990), 737-756. Zbl0707.34053MR1070845DOI10.1080/03605309908820706
  20. DAL MASO, G., An Introduction to Γ -Convergence. Birkhäuser, Boston1993. Zbl0816.49001MR1201152DOI10.1007/978-1-4612-0327-8
  21. DE GIORGI, E. - FRANZONI, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), 842-850. MR448194
  22. DE GROOT, S. R., Thermodynamics of Irreversible Processes. Amsterdam, North-Holland1961. Zbl0045.27104MR43729
  23. DE GROOT, S. R. - MAZUR, P., Non-equilibrium Thermodynamics. Amsterdam, North-Holland1962. Zbl1375.82003MR140332
  24. DIBENEDETTO, E. - SHOWALTER, R. E., Implicit degenerate evolution equations and applications. S.I.A.M. J. Math. Anal.12 (1981), 731-751. Zbl0477.47037MR625829DOI10.1137/0512062
  25. DONNELLY, J. D. P., A model for non-equilibrium thermodynamic processes involving phase changes. J. Inst. Math. Appl.24 (1979), 425-438. Zbl0426.35060MR556152
  26. EKELAND, I. - TEMAM, R., Analyse Convexe et Problèmes Variationnelles. DunodGauthier-Villars, Paris1974. MR463993
  27. ECKART, C., The thermodynamics of irreversible processes I: The simple fluid. Physical Reviews, 58 (1940). The thermodynamics of irreversible processes II. Fluid mixtures. Physical Reviews58 (1940). Zbl66.1077.01
  28. ELLIOTT, C. M. - OCKENDON, J. R., Weak and Variational Methods for Moving Boundary Problems. Pitman, Boston, 1982. Zbl0476.35080MR650455
  29. FENCHEL, W., Convex Cones, Sets, and Functions. Princeton Univ., 1953. Zbl0053.12203
  30. FITZPATRICK, S., Representing monotone operators by convex functions. Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988. Zbl0669.47029MR1009594
  31. FLEMINGS, M. C., Solidification Processing. McGraw-Hill, New York1973. 
  32. FRANCFORT, G. - MURAT, F. - TARTAR, L., Homogenization of monotone operators in divergence form with x -dependent multivalued graphs. Ann. Mat. Pura Appl. (4) 118 (2009), 631-652. Zbl1180.35077MR2533960DOI10.1007/s10231-009-0094-9
  33. FRÉMOND, M., Phase Change in Mechanics. Springer, Berlin2012. 
  34. GHOUSSOUB, N., A variational theory for monotone vector fields. J. Fixed Point Theory Appl.4 (2008), 107-135. Zbl1177.35093MR2447965DOI10.1007/s11784-008-0083-4
  35. GHOUSSOUB, N., Self-Dual Partial Differential Systems and their Variational Principles. Springer, 2009. Zbl1357.49004MR2458698
  36. GHOUSSOUB, N. - TZOU, L., A variational principle for gradient flows. Math. Ann.330 (2004), 519-549. Zbl1062.35008MR2099192DOI10.1007/s00208-004-0558-6
  37. GUPTA, S. C., The Classical Stefan Problem. Basic Concepts, Modelling and Analysis. North-Holland Series. Elsevier, Amsterdam2003. MR2032973
  38. GURTIN, M. E., Thermomechanics of Evolving Phase Boundaries in the Plane. Clarendon Press, Oxford1993. Zbl0787.73004MR1402243
  39. HIRIART-URRUTY, J.-B. - LEMARECHAL, C., Convex Analysis and Optimization Algorithms. Springer, Berlin1993. 
  40. KONDEPUDI, D. - PRIGOGINE, I., Modern Thermodynamics. Wiley, New York1998. 
  41. KURZ, W. - FISHER, D. J., Fundamentals of Solidification. Trans Tech, Aedermannsdorf1989. 
  42. LIONS, J. L. - MAGENES, E., Non-Homogeneous Boundary Value Problems and Applications. Vols. I, II. Springer, Berlin1972 (French edition: Dunod, Paris1968). MR350177
  43. LUCKHAUS, S., Solidification of alloys and the Gibbs-Thomson law. Preprint, 1994. 
  44. LUCKHAUS, S. - VISINTIN, A., Phase transition in a multicomponent system. Manuscripta Math.43 (1983), 261-288. Zbl0525.35012MR707047DOI10.1007/BF01165833
  45. MARTINEZ-LEGAZ, J.-E. - SVAITER, B. F., Monotone operators representable by l.s.c. convex functions. Set-Valued Anal.13 (2005), 21-46. Zbl1083.47036MR2128696DOI10.1007/s11228-004-4170-4
  46. MARTINEZ-LEGAZ, J.-E. - SVAITER, B. F., Minimal convex functions bounded below by the duality product. Proc. Amer. Math. Soc.136 (2008), 873-878. Zbl1133.47040MR2361859DOI10.1090/S0002-9939-07-09176-9
  47. MARTINEZ-LEGAZ, J.-E. - THÉRA, M., A convex representation of maximal monotone operators. J. Nonlinear Convex Anal.2 (2001), 243-247. Zbl0999.47037MR1848704
  48. MARQUES ALVES, M. - SVAITER, B.F., Brndsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces. J. Convex Analysis15 (2008), 693-706. Zbl1161.47034MR2489609
  49. MÜLLER, I., A History of Thermodynamics. Springer, Berlin2007. 
  50. MÜLLER, I. - WEISS, W., Entropy and Energy. A Universal Competition. Springer, Berlin2005. MR2282821
  51. MÜLLER, I. - WEISS, W., A history of thermodynamics of irreversible processes. (in preparation). 
  52. NAYROLES, B., Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B282 (1976), A1035-A1038. Zbl0345.73037MR418609
  53. PENROSE, O. - FIFE, P.C., Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D43 (1990), 44-62. Zbl0709.76001MR1060043DOI10.1016/0167-2789(90)90015-H
  54. 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. Zbl0799.76084MR1245658DOI10.1016/0167-2789(93)90183-2
  55. PRIGOGINE, I., Thermodynamics of Irreversible Processes. Wiley-Interscience, New York1967. Zbl0115.23101MR135908
  56. ROCKAFELLAR, R. T., Convex Analysis. Princeton University Press, Princeton1969. MR274683
  57. STEFANELLI, U., The Brezis-Ekeland principle for doubly nonlinear equations. S.I.A.M. J. Control Optim.8 (2008), 1615-1642. Zbl1194.35214MR2425653DOI10.1137/070684574
  58. SOLOMON, A. D. - ALEXIADES, V. - WILSON, D. G., A numerical simulation of a binary alloy solidification process. S.I.A.M. J. Sci. Statist. Comput.6 (1985), 911-922. Zbl0588.65084MR801180DOI10.1137/0906061
  59. 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. 
  60. SVAITER, B. F., Fixed points in the family of convex representations of a maximal monotone operator. Proc. Amer. Math. Soc.131 (2003), 3851-3859. Zbl1053.47046MR1999934DOI10.1090/S0002-9939-03-07083-7
  61. TARTAR, L., Nonlocal effects induced by homogenization. In: Partial Differential Equations and the Calculus of Variations, Vol. II (F. Colombini, A. Marino, L. Modica and S. Spagnolo, Eds.) Birkhäuser (Boston1989), 925-938. MR1034036
  62. TARTAR, L., Memory effects and homogenization. Arch. Rational Mech. Anal.111 (1990), 121-133. Zbl0725.45012MR1057651DOI10.1007/BF00375404
  63. TARTAR, L., The General Theory of Homogenization. A Personalized Introduction. Springer, Berlin; UMI, Bologna, 2009. Zbl1188.35004MR2582099DOI10.1007/978-3-642-05195-1
  64. TARZIA, D. A., A bibliography on moving-free boundary problems for the heat-diffusion equation. The Stefan and related problems. Universidad Austral, Departamento de Matematica, Rosario, 2000. Zbl0963.35207MR1802028
  65. VISINTIN, A., Models of Phase Transitions. Birkhäuser, Boston1996. Zbl0882.35004MR1423808DOI10.1007/978-1-4612-4078-5
  66. VISINTIN, A., Introduction to Stefan-type problems. In: Handbook of Differential Equations: Evolutionary Differential Equations vol. IV (C. Dafermos and M. Pokorny, eds.) North-Holland, Amsterdam (2008), chap. 8, 377-484. Zbl1183.35279MR1500159
  67. VISINTIN, A., Extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl.18 (2008), 633-650. Zbl1191.47067MR2489147
  68. VISINTIN, A., Phase transitions and glass formation. S.I.A.M. J. Math. Anal.41 (2009), 1725-1756. Zbl1203.35055MR2564192DOI10.1137/080733607
  69. VISINTIN, A., Scale-transformations in the homogenization of nonlinear magnetic processes. Archive Rat. Mech. Anal.198 (2010), 569-611. Zbl1233.78043MR2721590DOI10.1007/s00205-010-0296-8
  70. VISINTIN, A., Structural stability of doubly nonlinear flows. Boll. Un. Mat. Ital. IV (2011), 363-391. Zbl1235.35032MR2906767
  71. VISINTIN, A., On the structural stability of monotone flows. Boll. Un. Mat. Ital. IV (2011), 471-479. Zbl1243.49015MR2906771
  72. VISINTIN, A., Structural stability of rate-independent nonpotential flows. Discrete and Continuous Dynamical Systems6 (2013), 257-275. doi:10.3934/dcdss.2013.6.257 Zbl1262.35141MR2983478DOI10.3934/dcdss.2013.6.257
  73. VISINTIN, A., Variational formulation and structural stability of monotone equations. Calc. Var. Partial Differential Equations (in press). Zbl1304.47073MR3044140DOI10.1007/s00526-012-0519-y
  74. WOODRUFF, D. P., The Solid-Liquid Interface. Cambridge Univ. Press, Cambridge1973. 
  75. WOODS, L. C., The Thermodynamics of Fluid Systems. Clarendon Press, Oxford1975. 
  76. ZEIDLER, E., Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators. Springer, New York1990. Zbl0684.47029MR1033498DOI10.1007/978-1-4612-0985-0

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