Continuity for bounded solutions of multiphase Stefan problem

Emmanuele DiBenedetto; Vincenzo Vespri

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 4, page 297-302
  • ISSN: 1120-6330

Abstract

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We establish the continuity of bounded local solutions of the equation β ⁢ u t = 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.

How to cite

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DiBenedetto, Emmanuele, and 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 5.4 (1994): 297-302. <http://eudml.org/doc/244315>.

@article{DiBenedetto1994,
abstract = {We establish the continuity of bounded local solutions of the equation \( \beta (u)\_\{t\} = \Delta u \). Here \( \beta \) is any coercive maximal monotone graph in \( \mathbb\{R\} \times \mathbb\{R\} \), 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.},
author = {DiBenedetto, Emmanuele, Vespri, Vincenzo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Singular parabolic equations; Regularity; Stefan problem; Maximal monotone graphs; singular parabolic equations; regularity; maximal monotone graphs},
language = {eng},
month = {12},
number = {4},
pages = {297-302},
publisher = {Accademia Nazionale dei Lincei},
title = {Continuity for bounded solutions of multiphase Stefan problem},
url = {http://eudml.org/doc/244315},
volume = {5},
year = {1994},
}

TY - JOUR
AU - DiBenedetto, Emmanuele
AU - Vespri, Vincenzo
TI - Continuity for bounded solutions of multiphase Stefan problem
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/12//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 4
SP - 297
EP - 302
AB - We establish the continuity of bounded local solutions of the equation \( \beta (u)_{t} = \Delta u \). Here \( \beta \) is any coercive maximal monotone graph in \( \mathbb{R} \times \mathbb{R} \), 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.
LA - eng
KW - Singular parabolic equations; Regularity; Stefan problem; Maximal monotone graphs; singular parabolic equations; regularity; maximal monotone graphs
UR - http://eudml.org/doc/244315
ER -

References

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  2. CAFFARELLI, L. - EVANS, L. C., Continuity of the temperature in the two phase Stefan problem. Arch. Rat. Mech. Anal., 81, 1983, 199-220. Zbl0516.35080MR683353DOI10.1007/BF00250800
  3. CHAVENT, G. - JAFFRÈ, J., Mathematical Models and Finite Elements Methods for Reservoir Simulation. North-Holland1986. Zbl0603.76101
  4. DIBENEDETTO, E., Continuity of weak solutions to certain singular parabolic equations. Ann. Mat. Pura Appl., (4), CXXI, 1982, 131-176. Zbl0503.35018MR663969DOI10.1007/BF01761493
  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 York1987, 123-141. Zbl0694.35099MR900832DOI10.1007/978-1-4613-8743-5_7
  6. DIBENEDETTO, E. - VESPRI, V., On the singular equation β ⁢ u t = Δ ⁢ u . To appear. Zbl0849.35060MR1365831DOI10.1007/BF00382749
  7. FRIEDMAN, A., Variational Principles and Free Boundary Problems. Wiley-Interscience, New York1982. Zbl0564.49002MR679313
  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. Zbl0398.35039
  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. Zbl0174.15403MR241822
  10. LIONS, J. L., Quelques méthodes de résolutions des problèmes aux limites non linéaires. Dunod Gauthiers-Villars, Paris1969. Zbl0189.40603MR259693
  11. SACHS, P., The initial and boundary value problem for a class of degenerate parabolic equations. Comm. Part. Diff. Equ., 8, 1983, 693-734. Zbl0529.35038
  12. ZIEMER, W. P., Interior and boundary continuity of weak solutions of degenerate parabolic equations. Trans. Amer. Math. Soc., 271 (2), 1982, 733-748. Zbl0506.35053MR654859DOI10.2307/1998907

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