Continuity for bounded solutions of multiphase Stefan problem
DiBenedetto, Emmanuele ; Vespri, Vincenzo
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 5 (1994), p. 297-302 / Harvested from Biblioteca Digitale Italiana di Matematica
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We establish the continuity of bounded local solutions of the equation β⁢ut=Δ⁢u. Here β is any coercive maximal monotone graph in R×R, bounded for bounded values of its argument. The multiphase Stefan problem and the Buckley-Leverett model of two immiscible fluids in a porous medium give rise to such singular equations.

In questa Nota si dimostra la continuità delle soluzioni locali limitate dell'equazione β⁢ut=Δ⁢u, dove β è un qualsiasi grafo massimale monotono e coercivo in R×R, che si mantiene limitato per valori limitati del suo argomento. A questo contesto appartengono sia il problema di Stefan multifase che il modello di Buckley-Leverett di due fluidi immiscibili in un mezzo poroso.

Publié le : 1994-12-01
@article{RLIN_1994_9_5_4_297_0,
     author = {Emmanuele DiBenedetto and Vincenzo Vespri},
     title = {Continuity for bounded solutions of multiphase Stefan problem},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {5},
     year = {1994},
     pages = {297-302},
     zbl = {0824.35142},
     mrnumber = {1320581},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1994_9_5_4_297_0}
}

DiBenedetto, Emmanuele; Vespri, Vincenzo. Continuity for bounded solutions of multiphase Stefan problem. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 5 (1994) pp. 297-302. http://gdmltest.u-ga.fr/item/RLIN_1994_9_5_4_297_0/

[1] Alt, H. W. - Dibenedetto, E., Non steady flow of water and oil through inhomogeneous porous media. Ann. Sc. Norm. Sup. Pisa, s. IV, vol. XII, 3, 1985, 335-392. | MR 837254 | Zbl 0608.76082

[2] Caffarelli, L. - Evans, L. C., Continuity of the temperature in the two phase Stefan problem. Arch. Rat. Mech. Anal., 81, 1983, 199-220. | MR 683353 | Zbl 0516.35080

[3] Chavent, G. - Jaffrè, J., Mathematical Models and Finite Elements Methods for Reservoir Simulation. North-Holland 1986. | Zbl 0603.76101

[4] Dibenedetto, E., Continuity of weak solutions to certain singular parabolic equations. Ann. Mat. Pura Appl., (4), CXXI, 1982, 131-176. | MR 663969 | Zbl 0503.35018

[5] Dibenedetto, E., The flow of two immiscible fluids through a porous medium: Regularity of the saturation. In: J. L. Ericksen - D. Kinderlherer (eds.), Theory and Applications of Liquid Crystals. IMA, vol. 5, Springer-Verlag, New York 1987, 123-141. | MR 900832 | Zbl 0694.35099

[6] Dibenedetto, E. - Vespri, V., On the singular equation β⁢ut=Δ⁢u. To appear. | MR 1365831 | Zbl 0849.35060

[7] Friedman, A., Variational Principles and Free Boundary Problems. Wiley-Interscience, New York1982. | MR 679313 | Zbl 0564.49002

[8] Kruzkov, S. N. - Sukorjanski, S. M., Boundary value problems for systems of equations of two phase porous flow type: statement of the problems, questions of solvability, justification of approximate methods. Mat. Sbornik, 44, 1977, 62-80. | Zbl 0398.35039

[9] Ladyzhenskaja, O. A. - Solonnikov, V. A. - Ural'Tzeva, N. N., Linear and Quasilinear Equations of Parabolic Type. AMS Transl. Math. Mono, 23, Providence RI 1968. | MR 241822 | Zbl 0174.15403

[10] Lions, J. L., Quelques méthodes de résolutions des problèmes aux limites non linéaires. Dunod Gauthiers-Villars, Paris1969. | MR 259693 | Zbl 0189.40603

[11] Sachs, P., The initial and boundary value problem for a class of degenerate parabolic equations. Comm. Part. Diff. Equ., 8, 1983, 693-734. | Zbl 0529.35038

[12] Ziemer, W. P., Interior and boundary continuity of weak solutions of degenerate parabolic equations. Trans. Amer. Math. Soc., 271 (2), 1982, 733-748. | MR 654859 | Zbl 0506.35053