An evolutionary continuous casting problem of Stefan type

Rodrigues, José-Francisco

doi:10.1090/qam/840448

Author: José-Francisco Rodrigues
Journal: Quart. Appl. Math. 44 (1986), 109-131
MSC: Primary 49A29; Secondary 35R35, 80A20
DOI: https://doi.org/10.1090/qam/840448
MathSciNet review: 840448
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A simplified model for the solidification of an ingot being cast continuously by withdrawing it from a mould at constant rate is studied via the variational inequalities approach. Existence, uniqueness and regularity results are given for this one-phase Stefan type problem, together with a detailed analysis for the free boundary (the solid-liquid interface) particularly with respect to the asymptotic behavior as time $t \uparrow \infty$ where the steady state is attained.

Hans Wilhelm Alt, The fluid flow through porous media. Regularity of the free surface, Manuscripta Math. 21 (1977), no. 3, 255–272. MR 449170, DOI https://doi.org/10.1007/BF01167879
Claudio Baiocchi and António Capelo, Variational and quasivariational inequalities, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. Applications to free boundary problems; Translated from the Italian by Lakshmi Jayakar. MR 745619

H. Brézis, Nouveaux théorèmes de régularité pour les problèmes unilatéraux, Rencontre entre Physicien Théoricien et Mathématicien, Strasbourg, Vol. 12 (1971)

Thierry Brière, Application des méthodes variationnelles à la cristallisation d’un métal par passage dans une gaine de refroidissement, Ann. Fac. Sci. Toulouse Math. (5) 2 (1980), no. 3-4, 219–247 (French, with English summary). MR 614013
Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), no. 3-4, 155–184. MR 454350, DOI https://doi.org/10.1007/BF02392236
Luis A. Caffarelli, A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets, Boll. Un. Mat. Ital. A (5) 18 (1981), no. 1, 109–113 (English, with Italian summary). MR 607212
Michel Chipot and José-Francisco Rodrigues, On the steady-state continuous casting Stefan problem with nonlinear cooling, Quart. Appl. Math. 40 (1982/83), no. 4, 476–491. MR 693880, DOI https://doi.org/10.1090/S0033-569X-1983-0693880-7

G. Duvaut, Résolution d’un problème de Stefan, C. R. Acad. Sc. Paris, 276A, 1461–1463 (1973)

Avner Friedman, Variational principles and free-boundary problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. A Wiley-Interscience Publication. MR 679313
Avner Friedman and David Kinderlehrer, A one phase Stefan problem, Indiana Univ. Math. J. 24 (1974/75), no. 11, 1005–1035. MR 385326, DOI https://doi.org/10.1512/iumj.1975.24.24086
Avner Friedman and Alessandro Torelli, Correction to the paper: “A free boundary problem connected with nonsteady filtration in porous media” [Nonlinear Anal. 1 (1976/77), no. 5, 503–545; MR 58 #29277], Nonlinear Anal. 2 (1978), no. 4, 513–518. MR 512490, DOI https://doi.org/10.1016/0362-546X%2878%2990060-3
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696

D. Kröner, The regularity of the free boundary for a nonsteady filtration problem in the case of a compressible flow, Preprint no. 661, SFB72, Universität Bonn (1984), (to appear in Boll. Un. Mat. Ital.)

O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires; Dunod, Paris (1969)

Jacques-Louis Lions, Sur quelques questions d’analyse, de mécanique et de contrôle optimal, Les Presses de l’Université de Montréal, Montreal, Que., 1976 (French). Collection de la Chaire Aisenstadt. MR 0493633
M. K. V. Murthy and G. Stampacchia, A variational inequality with mixed boundary conditions, Israel J. Math. 13 (1972), 188–224 (1973). MR 333430, DOI https://doi.org/10.1007/BF02760237
L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 733–737. MR 208360
José-Francisco Rodrigues, Sur un problème à frontière libre stationnaire traduisant la cristallisation d’un métal, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 17, A823–A825 (French, with English summary). MR 580575
José-Francisco Rodrigues, Stabilité de la frontière libre dans le problème de la digue, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 16, 565–568 (French, with English summary). MR 679945
José-Francisco Rodrigues, Some remarks on the asymptotic behaviour of strong solutions to monotone parabolic variational inequalities, Rend. Mat. (7) 4 (1984), no. 3, 457–470. MR 870823
J.-F. Rodrigues, Aspects of the variational approach to a continuous casting problem, Free boundary problems: applications and theory, Vol. III (Maubuisson, 1984) Res. Notes in Math., vol. 120, Pitman, Boston, MA, 1985, pp. 72–83. MR 863163
Alessandro Torelli, On a free boundary value problem connected with a non steady filtration phenomenon, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 1, 33–59. MR 443564
Alessandro Torelli, Existence and uniqueness of the solution for a non steady free boundary problem, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 2, 423–466 (English, with Italian summary). MR 0603295

A. Torelli, On a free boundary problem connected with a nonsteady filtration phenomenon of compressible fluid, Pub. no. 148, Lab Anal. Num. del C. N. R., Pavia (1977)