Skip to main content

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 88))

Abstract

A large class of mathematical models for phase change problems in thermo-diffusion phenomena belong to the family of the so-called Stefan problems, after the Austrian mathematical-physicist Joseph Stefan, who, exactly one century ago, published a series of papers on some of these problems [St]. However, the first work on this type of problems seems to be due to Lamé and Clapeyron [LC] and since then they have interested a large number of mathematicians.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Baiocchi, C & Capelo, A. Disequazioni variazionali e quasi variazionali. Applicazioni a problemi di frontiera libera Vol. I, II, Qua-derni dell’U. Mat. Ital., Pitagora, Bologna, 1978; English transl. J. Wiley, Chichester-New York, 1984.

    Google Scholar 

  2. Barbu, V. Optimal Control of Variational Inequalities, Research Notes in Math. No. 100, Pitman, Boston, London, 1984

    Google Scholar 

  3. Bensoussan,A. & Lions J.L. des Inéquations Variationelles en Contrôle StochastiqueParis,1978; English transl., North-Holland, Amsterdam, 1982.

    Google Scholar 

  4. Bossavit, A. Damlamian, A. & Frémond, M. Free Boundary problems: applications and theoryVol. III, IV, Research Notes in Math. Nos. 120/121, Pitman, Boston, London, 1985.

    Google Scholar 

  5. Brauner, CM., Nicolaenko B. & Frémond, M. Homographic Approximations of Free Boundary Problems Characterized by Variational Inequalities, Sci. & Comp., Adv. in Math. Suppl. Studies (Academic Press), Vol. 10 (1986), 119–152.

    Google Scholar 

  6. Brézis, H. On some degenerate nonlinear parabolic equations in Browder FE (ed.) Nonlinear functional analysis, Part I, Symp. Pure Math. 18 (1970), 28–38

    Google Scholar 

  7. Brézis, H. Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equationsin Contributions to Non Linear Funct. Anal., Academic Press, New York, (1971), 101–156.

    Google Scholar 

  8. Brière, T.-Application des Méthodes Variationelles à la cristallisa-tion d’un metal par passage dans une gaine de refroidissementAnn. Fac. Sc. Toulouse, 2 (1980), 219–247.

    Google Scholar 

  9. Caffareiii, L.A. -The Regularity of Free Boundaries in Higer DimensionsActa Math. 139 (1977), 155–184.

    Article  Google Scholar 

  10. Caffareiii, L.A. -Some Aspects of the One-Phase Stefan ProblemIndiana Univ. Math. J. 27 (1978), 73–77.

    Article  Google Scholar 

  11. Caffareiii, L.A. -A Remark on the Hausdorff Measure of a Free Boundary and the Convergence of Coincidence SetsBoll. Unione Mat. Ital. 18(5) (1981), 1297–1299.

    Google Scholar 

  12. Caffareiii, L. & Evans, L. -Continuity of the temperature in the two-phase Stefan problemsArch. Rational Mech. Anal. 81, 3 (1983), 199–220.

    Google Scholar 

  13. Caffareiii, L.A. & Friedman, A. -Continuity of the Temperature in the Stefan ProblemIndiana Univ. Math. J. 28 (1979), 53–70.

    Article  Google Scholar 

  14. Cannon, J.R. & DiBenedetto, E. -On the existence of weak solutions to an n-dimensional Stefan problem with nonlinear boundary conditionsSIAM J. Math. Anal. 11 (1980), 632–645.

    Article  Google Scholar 

  15. Chipot, M. -Variational Inequalities and Flow in Porous MediaSpringer-Verlag, Berlin and New York, 1984.

    Book  Google Scholar 

  16. Chipot, M. & Rodrigues, J.F. -On the Steady-State Continuous Casting Stefan Problem with Non-Linear Cooling Quart. Appl. Math. 40 (1983), 476–491.

    Google Scholar 

  17. Crank, J. -Free and moving boundary problemsOxford, Univ. Press, Oxford, 1984.

    Google Scholar 

  18. Damlamian, A. -Some results on the multiple-phase Stefan problem Comm. P.D.E. 2 (1977), 1017–1044 (also These, Univ. Paris VI, 1976)

    Article  Google Scholar 

  19. Damlamian, A. & Kenmochi, N. -Asymptotic Behaviour of Solutions to a Multi-Phase Stefan ProblemJapan J. Appl. Math. 3 (1986), 15–36.

    Article  Google Scholar 

  20. Danilyuk, I.I. -On the Stefan problem Russian Math. Surveys, 40:5 (1985), 157–223.

    Article  Google Scholar 

  21. DiBenedetto, E. - Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl. 4,130 (1982), 131–176

    Article  Google Scholar 

  22. DiBenedetto, E. -A boundary modulus of continuity for a class of singular parabolic equationsJ. Dif. Eq. 63 (1986), 418–447.

    Article  Google Scholar 

  23. DiBenedetto, E. & Friedman, A. - Periodic Behaviour for the evolutionary dam problem and related free boundary problems, Comm. P.D.E. 11 (1986), 1297–1377.

    Article  Google Scholar 

  24. Duvaut, G. – Résolution d’un problème de Stefan, C.R. Ac. Sci. Paris276-A 1973, 1461–1463.

    Google Scholar 

  25. Duvaut, G. & Lions, J.L. -Les inéquations en mécanique et en physiqueDunod, Paris, 1972; English transl. Springer, Berlin, 1976.

    Google Scholar 

  26. Elliott, C.M. & Ockendon, J.R.Weak and Variational Methods for Moving Boundary Problems Research Notes in Maths59, Pitman, Boston, London, 1982.

    Google Scholar 

  27. Fasano A. Primicerio, M. (eds.) -Free Boundary Problems: Th. Appl. Research Notes in Math. 78/79, Pitman, Boston, 1983.

    Google Scholar 

  28. Frémond, M. -Diffusion Problems with Free Boundaries Autumum Course on application of Analysis to Mechanics, Sept-Nov. 1976 International Centre for Theoretical Physics, Trieste

    Google Scholar 

  29. Friedman, A. -The Stefan problem in several space variablesTrans. Amer. Math. Soc. 133 (1968), 51–87.

    Article  Google Scholar 

  30. Friedman, A. -Variational Principles and Free-Boundary ProblemsWiley, New York, 1982.

    Google Scholar 

  31. Friedman, A., Huang, S. Sz Yong, J. -Optimal periodic control for the two-phase Stefan problem SIAM J.Control and Optim. 26 (1988), 23–41

    Article  Google Scholar 

  32. Friedman, A. & Kinderlehrer, D. -A one phase Stefan problem Indiana Univ. Math. J. 24 (1975), 1005–1035

    Article  Google Scholar 

  33. Gustafson, B. & Mossino, J. -Quelques inégalités isopérimétriques pour le problème de Stefan CR. Acad. Sci. Paris, 305–1 (1987), 669–672

    Google Scholar 

  34. Hoffmann, K.H. & Niezgodka, M. - Control of Parabolic Systems Involving Free Boundaries in [FP], Vol. II (1983), 431–453

    Google Scholar 

  35. Hoofman, K.H & Sprekels, J. -Real time control in free boundary problems connected with the continous casting of steel in Optimal Control pf P.D.Es, Eds. Hoffman & Krubs, Birkhäuser 1984, 127 –143

    Google Scholar 

  36. Hoffman, K.H & Sprekels, J. -Free Boundary Problems, Theory and Applications Pitman Longman Res. Notes (1989) (Proceedings of Irsee/Bravia Conf., 1987: in print)

    Google Scholar 

  37. Holappa, L., Laitinen, E., Louhenkilpi, S. & Neittaanmäki, P. -Optimization of the secondary cooling in the continuous casting of steel billetsProc. 24th Annual Conf. of Metallurgists, Vancouver, 1985.

    Google Scholar 

  38. Kamenomostkaya, S. -On the Stefan problem (in Russian), Naučnye Dokl. Vysšei Školy 1 (1958), 60–62 (Mat. Sb. 53(95) (1961), 489–514.

    Google Scholar 

  39. Kinderlehrer, D. & Stampacchia, G. -An introduction to variational inequalities and their applications Acad. Press, New York, 1980.

    Google Scholar 

  40. Ladyzenskaya, O.A., Solonnikov, V.A. Ural’eva, N.N. -Linear and Quasilinear Equations of Parabolic Type A.M.S. Transl. Monog. 23, Providence, 1968.

    Google Scholar 

  41. Lamé,G.& Clapeyron, B.P Mémoire sur la solidification par re froidissement d’un globe solid, Ann. Chem. Phys. 47 (1831), 250-256.

    Google Scholar 

  42. Larreq, M., Birat, J.P., Saguez, C. & Henry, J. -Optiminization of casting and cooling conditions on steel continuous on steel continuous casters IRSID, ACI-83 RE-1004, Jul. 1983

    Google Scholar 

  43. Lions, J.L. -Sur le contrôle optimale de systèmes governés par des equations aux dérivées partielles Dunod, Paris, 1968.

    Google Scholar 

  44. Lions, J.L. -Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires Dunod, Paris, 1969.

    Google Scholar 

  45. Lions, J.L-Sur quelques questions d’ Analyse, d’e Mécanique et de Contrôle Optimal Presses Univ. Montreal, 1976

    Google Scholar 

  46. Loper, D.E. (Ed.) -Structure and Dynamics of Partial Solidified SystemsNATO ASI Series E: 125; M. Nijhoff Pub., Dordrecht, 1987.

    Google Scholar 

  47. Louro, B. & Rodrigues, J.F. -Remarks on the quasi-steady one phase Stefan problem Proc. Royal Soc. Edinburg 102-A (1986), 263–275

    Google Scholar 

  48. Louro, B. & Rodrigues, J.F. -An evolutionary Stefan problem with zero specific heat, Proceedings”Free Boundary Problems: Theory and Applications” Irsee/Bavaria, 1987 (see [HS2])

    Google Scholar 

  49. Maganes, E. (ed)-Free Boundary Problems Proc. Pravia Seminar Sept. – Oct. 1979, Instituto Nazionale di Alta Matematica Francesco Severi, Vol. I and II, Roma, 1980

    Google Scholar 

  50. Magenes, E. -Problemi di Stefan Bifase in piu Variabili SpazialiLe Matematiche (Catania) 36 (1983), 65–108

    Google Scholar 

  51. Magenes, E., Verdi, C. & Visintin, A. -Semigroup approach to the Stefan problem with nonlinear fluxRend. Arad. Naz. Lincei 75(2) (1983), 24–33

    Google Scholar 

  52. Meirmanov, A.M. -On the Classical Solution of the Multidimensional Stefan Problem for Quasilinear Parabolic EquationsMat. Sbornik 112 (1980), 170–192 (also Math. USSR-Sb., 40, 2 (1981), 157–179)

    Google Scholar 

  53. Meirmanov, A.M. -Stefan Problems (in Russian) Nauka, Novosibirsk, 1986

    Google Scholar 

  54. Neittanmaki, P. & LaitinenE. - Optimization of Cooling Conditions in Continuous Casting in Proceedings of the Summer School in Numerical Analysis at Jyväskylä, June 1985, University of Jyväskylä, Report 31, ed. P. Neittanmaki.

    Google Scholar 

  55. Niezgodka, M. -Stefan-like problems in [FP] (1983), 321–347

    Google Scholar 

  56. Niezgodka, N. & Pawlow, I. -A generalized Stefan problem in several space variablesAppl. Math. Optim. 9 (1983), 193–224

    Article  Google Scholar 

  57. Nirenberg, L. -An extended interpolation inequalityAnn. Scuola Norm. Sup. Pisa 20 (1966), 733–737

    Google Scholar 

  58. Nochetto, R.estimates for two-phase Stefan problems in several space variables, I: Linear boundary conditions Calcolo XXII (1985), 457–499; II: Non-linear flux conditions, 501–534

    Google Scholar 

  59. Nochetto, R.-A class of non-degenerate two-phase Stefan problems in several space variables Comm. P.D.E. 12 (1987), 21–45

    Article  Google Scholar 

  60. Ockendon, J.R. fe Hodgkins, A.R. -Moving boundary problems in heat flow and diffusionClarendon Press, Oxford, 1975

    Google Scholar 

  61. Oleinik, O.A. -A method of solution of the general Stefan problemSoviet Math. Dokl. 1 (1960), 1350–1354

    Google Scholar 

  62. Panagiotopoulos, P.D. -Inequality Problems in Mechanics and ApplicationsBirkhäuser, Boston-Basel, 1985

    Google Scholar 

  63. Pietra, P. & Verdi, C. -Convergence of the Approximate Free Boundary for the Multidimensional One-Phase Stefan Problem Comp. Mech. Int. J. 1 (1986)

    Google Scholar 

  64. Primicerio, M.-Problemi di diffusione a frontiera liberaBoll. U.M.I. (5) 18-;A (1981), 11–68

    Google Scholar 

  65. Primicerio, M. -Mushy regions in phase-change problems in Applied Nonlinear Functional Analysis (eds. R. Gorenflo, K. Hoffmann), Lang, Frankfurt/Main (1982)

    Google Scholar 

  66. Rodrigues, J.F.Sur un problème à frontière libre stationnaire tra-duisant la cristallisation d’un métal C.R.A.S. Paris 290-A (1980), 823–825

    Google Scholar 

  67. Rodrigues, J.F. -Aspects of the Variational Approach to the Continuous Casting Problemin [BDF], Vol. III (1985), 72–83

    Google Scholar 

  68. Rodrigues, J.F. -An evolutionary continuous casting problem of Stefan typeQuaterly Appl. Math. 44 (1986), 109–131

    Google Scholar 

  69. Rodrigues, J.F. -On the variational inequality approach to the one-phase Stefan problemActa Appl. Math. 8 (1987), 1–35

    Article  Google Scholar 

  70. Rodrigues, J.F. -Obstacle Problems in Mathematical Physics North-Holland, Amsterdam, 1987

    Google Scholar 

  71. Rodrigues, J.F.-Free boundary stability in the one-phase Stefan problem in Nonlinear Anal, and Appl., Ed. V. Lakshmikantham, Marcel Dekker, New York (1987), 527–531

    Google Scholar 

  72. Rodrigues, J.F. & Santos, L. -Asymptotic Convergences in a One-Phase Continuous-Casting Stefan Problem with High Extraction Velocity IMA J. Appl. Math. (1989), in press

    Google Scholar 

  73. Rodrigues, J.F. & Yi Fahuai -On a two-phase continuous casting Stefan problem with nonlinear flux to appear)

    Google Scholar 

  74. Rubinstein, L.I. -The Stefan problem Amer. Math. Soc. Transl. Monogr. 27, Providence, 1971

    Google Scholar 

  75. Rulla, J. -Weak Solutions to Stefan Problems with Prescribed Convection SIAM J. Math. Anal. 18 (1987), 1784–1800

    Article  Google Scholar 

  76. Saguez, C-Contrôle optimal d’inèquations variationnelles avec observations de domains Rapport no. 286,1.R.I.A. (1978); see also C.R. Acad. Sc. Paris, 287-A (1978), 957–959

    Google Scholar 

  77. Saguez, C. –Contrôle optimal de systèmes à frontière libre Thèse d’Etat, Université Technologie de Compiègne, 1980

    Google Scholar 

  78. Saguez, C. & Larrec, Q. –Contrôle de systèmes à frontière libre, Applications à la coulèe continue d’acier 4ème Colloque Int. sur les Meth. Cal., I.R.I.A., Versailles, 1979

    Google Scholar 

  79. Smith, T.J. (Ed.) -Modelling the Flow and Solidification of Metals M. Nijhoff Pub., Dordrecht, 1987

    Google Scholar 

  80. Stefan, J.-Über einige Probleme der Theorie der Wärmeleitung, Sitzungsber Wien, Akad. Mat. Natur. 98 (1889), 473–484 see also pp. 614–634; 965–983; 1418–1442

    Google Scholar 

  81. Tarzia, D.A. -Sur le problème de Stefan à deux phaseC.R. Acad. Sci. Paris 288 (1970), 941–944

    Google Scholar 

  82. Tarzia, D.A -Una revisión sobre problemas de frontera móvil ylibre para la ecuación del calor. El problema de StefanMath. Notae 29 (1981/82),147–241

    Google Scholar 

  83. Visintin, A. -Sur le problème de Stefan avec flux non linéaireBoll. UMI C. 18 (1981), 63–86

    Google Scholar 

  84. Visintin, A. -General free boundary evolution problems in several space dimensions J. Math. Anal. Appl. 95 (1983), 117–143

    Article  Google Scholar 

  85. Wilson, D.G.Solomon, A.D. & Boggs, P.T. (eds.) - Moving boundary problems, Academic Press, New York, London, 1978

    Google Scholar 

  86. Yi Fauai An evolutionary continuous casting problem of two-phase and its periodic behaviour (to appear in Journal P.D.E.)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Birkhäuser Verlag Basel

About this chapter

Cite this chapter

Rodrigues, J.F. (1989). The Stefan Problem Revisited. In: Rodrigues, J.F. (eds) Mathematical Models for Phase Change Problems. International Series of Numerical Mathematics, vol 88. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9148-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9148-6_8

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9926-0

  • Online ISBN: 978-3-0348-9148-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics