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On a two-phase continuous casting Stefan problem with nonlinear flux

Published online by Cambridge University Press:  16 July 2009

José Francisco Rodrigues  and
Fahuai Yi
José Francisco Rodrigues
Affiliation:
C.M.A.F.–I.N.I.C. and University of Lisbon, Av. Prof Gama Pinto, 2, 1699 Lisboa Codex, Portugal
Fahuai Yi
Affiliation:
Department of Mathematics, Suzhou University, Suzhou, China
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Abstract

We prove the existence of a weak solution for a two-phase continuous casting Stefan problem with a general monotone nonlinear cooling condition. We establish a sufficient condition for stability, which yields uniqueness and comparison results for the evolutionary and the steady- state solutions. We also discuss the asymptotic behaviour as t←∞ of the corresponding temperatures and enthalpies.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 1 , Issue 3 , September 1990 , pp. 259 - 278
Copyright
Copyright © Cambridge University Press 1990

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References

BriéRe, T. 1980 Application des méthodes variationelles ´ la cristallisation d'un metal par passage dans une game de refroidissement. Ann. Fac. Sc. Toulouse 2, 219–247.CrossRefGoogle Scholar
Cannon, J. R. & Dibenedetto, E. 1980 On the existence of weak solutions to an n–dimensional Stefan problem with nonlinear boundary conditions. SIAM J. Math. Anal. 11, 632–645.CrossRefGoogle Scholar
Chipot, M. & Rodrigues, J. F. 1983 On the steady-state continuous casting Stefan problem with non-linear cooling. Quart. Appi. Math. 40, 476–491.CrossRefGoogle Scholar
Damlamian, A. 1977 Some results on the multiple-phase Stefan problem. Comm. P.D.E. 2, 1017–1044 (also Thése, Univ. Paris VI, 1976).CrossRefGoogle Scholar
Damlamian, A. & Kenmochi, N. 1986 Asymptotic behaviour of solutions to a multi-phase Stefan problem. Japan J. Appl. Math. 3, 15–36.CrossRefGoogle Scholar
Danilyuk, I. I.On the Stefan problem. Russian Math. Surveys 40 (5), 157–223.CrossRefGoogle Scholar
DiBenedetto, E. 1982 Continuity of weak solutions to certain singular parabolic equations. Ann. Mat. Pura Appl. (4), 130, 131–176.CrossRefGoogle Scholar
DiBenedetto, E. 1986 A boundary modulus of continuity for a class of singular parabolic equations. J. Diff Eq. 63, 418–447.CrossRefGoogle Scholar
DiBenedetto, E. & Friedman, A. 1986 Periodic behaviour for the evolutionary dam problem and related free boundary problems. Comm. P.D.E. 11, 1297–1377.CrossRefGoogle Scholar
Friedman, A. 1968 The Stefan problem in several space variables. Trans. Amer. Math. Soc. 133, 51–87.CrossRefGoogle Scholar
Kamenostkaya, S. 1958 On the Stefan problem (in Russian), Naučnye Doki. Vysšel Školy 1, 60–62Google Scholar
Kamenostkaya, S.On the Stefan problem (in Russian), (Mat. Sb). 53 (95) (1961), 489–514.Google Scholar
Ladyzenskaya, O. A., Solonnikov, V. A. & Ural'eva, N. N. 1968 Linear and Quasilinear Equations of Parabolic Type, A.M.S. Transl. Monog. 23, Providence.CrossRefGoogle Scholar
Lions, J. L. 1969 Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod: Paris.Google Scholar
Meirmanov, A. M. 1986 Stefan Problems (in Russian), Nauka: Novosibirsk.Google Scholar
Niezgodka, N. & Pawlow, I. 1983 A generalized Stefan problem in several space variables. Appl. Math. Optim. 9, 193–224.CrossRefGoogle Scholar
Oleinik, O. A. 1960 A method of solution of the general Stefan problem. Soy. Math. Dokl. 1, 1350–1354.Google Scholar
Rodrigues, J. F. 1985 Aspects of the Variational Approach to the Continuous Casting Problem, pp. 72–83. In Research Notes in Match (Pitman) 120.Google Scholar
Rodrigues, J. F. 1986 An evolutionary continuous casting problem of Stefan type. Quart. Appl. Math. 44, 109–131.CrossRefGoogle Scholar
Rodrigues, J. F. 1989 The Stefan problem revisited. In Mathematical models for Phase Change Problems (ed. Rodrigues, J. F.), pp. 129–190. ISNM vol. 88, Birkhäuser Verlag: Basel.CrossRefGoogle Scholar
Rulla, J. 1987 Weak solutions to Stefan problems with prescribed convection. SIAM J. Math. Anal. 18, 1784–1800.CrossRefGoogle Scholar
Stampacchia, G. 1960 Problemi al contorno elliptici con datti discontinui dotati di soluzioni holderiane. Annali Math. Pura Appl. 51, 1–32.CrossRefGoogle Scholar
Visintin, A. 1981 Sur le probléme de Stefan avec flux non linéaire. Boll. UMI C. 18, 63–86.Google Scholar
Yi, F. 1989 An evolutionary continuous casting problem of two-phase and its periodic behaviour. J. J. Part. Diff Eq. 2 (3), 7–22.Google Scholar
Zaltzman, B. 1989 Multidimensional two-phase quasistationary Stefan problem in an unbounded region. Pre-print of Novosibirsk State University.Google Scholar