×

Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. (English) Zbl 0991.35047

This paper is concerned with a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. Existence, uniqueness and regularity of a weak solution regarding the above system are proved.

MSC:

35K65 Degenerate parabolic equations
76S05 Flows in porous media; filtration; seepage
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Alt, H. W.; di Benedetto, E., Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12, 335-392 (1985) · Zbl 0608.76082
[3] Alt, H. W.; Luckhaus, S., Quasilinear elliptic-parabolic differential equations, Math. Z., 183, 311-341 (1983) · Zbl 0497.35049
[4] Antontsev, S. N., On the solvability of boundary value problems for degenerate two-phase porous flow equations, Dinamika Splosnoı Sredy Vyp., 10, 28-53 (1972)
[5] Antontsev, S. N.; Kazhikhov, A. V.; Monakhov, V. N., Boundary-Value Problems in the Mechanics of Nonuniform Fluids. Boundary-Value Problems in the Mechanics of Nonuniform Fluids, Studies in Mathematics and Its Applications (1990), North-Holland: North-Holland Amsterdam · Zbl 0696.76001
[6] Arbogast, T. J., The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow, Nonlinear Anal., 19, 1009-1031 (1992) · Zbl 0783.76090
[7] Arbogast, T. J.; Wheeler, M.; Zhang, N., A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal., 33, 1669-1687 (1996) · Zbl 0856.76033
[8] Aziz, K.; Settari, A., Petroleum Reservoir Simulation (1979), Applied Science: Applied Science London
[9] Bear, J., Dynamics of Fluids in Porous Media (1972), Dover: Dover New York · Zbl 1191.76001
[10] Chavent, G.; Jaffré, J., Mathematical Models and Finite Elements for Reservoir Simulation (1978), North-Holland: North-Holland Amsterdam
[11] Chen, Z.; Espedal, M.; Ewing, R., Continuous-time finite element analysis of multiphase flow in groundwater hydrology, Appl. Math., 40, 203-226 (1995) · Zbl 0847.76030
[12] Chen, Z.; Ewing, R., Fully-discrete finite element analysis of multiphase flow in groundwater hydrology, SIAM J. Numer. Anal., 34, 2228-2253 (1997) · Zbl 0901.76031
[13] Chen, Z.; Ewing, R., Mathematical analysis for reservoir models, SIAM J. Math. Anal., 30, 431-453 (1999) · Zbl 0922.35074
[14] Z. Chen, and, R. Ewing, Degenerate two-phase incompressible flow. II. Optimal error estimates, Numer. Math, in press.; Z. Chen, and, R. Ewing, Degenerate two-phase incompressible flow. II. Optimal error estimates, Numer. Math, in press. · Zbl 1097.76064
[15] Z. Chen, and, R. Ewing, Degenerate two-phase incompressible flow III: Local refinement and domain decomposition, Numer. Math, in press.; Z. Chen, and, R. Ewing, Degenerate two-phase incompressible flow III: Local refinement and domain decomposition, Numer. Math, in press. · Zbl 1032.76590
[16] Chen, Z.; Ewing, R.; Espedal, M., Multiphase flow simulation with various boundary conditions, (Peters, A., Numerical Methods in Water Resources (1994), Kluwer Academic: Kluwer Academic Netherlands), 925-932
[17] de Giorgi, E., Sulla differenziabilita e lanaliticita delle estremali degli integrali multipli regolari, Mem. Acc. Sci. Torino Cl. Sc. Fis. Mat. Nat., 3, 25-43 (1957) · Zbl 0084.31901
[18] di Benedetto, E., Degenerate Parabolic Equations (1993), Springer-Verlag: Springer-Verlag New York
[19] Friedman, A., Variational Principles and Free-Boundary Problems (1982), Wiley: Wiley New York · Zbl 0564.49002
[20] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0361.35003
[21] Kroener, D.; Luckhaus, S., Flow of oil and water in a porous medium, J. Differential Equations, 55, 276-288 (1984) · Zbl 0509.35048
[22] Kruzkov, S. N.; Sukorjanskiı, S. M., Boundary problems for systems of equations of two-phase porous flow type; statement of the problems, questions of solvability, justification of approximate methods, Math. USSR Sb., 33, 62-80 (1977) · Zbl 0398.35039
[23] Ladyzenskaja, O. A.; Solonnikov, V. A.; Uralćeva, N. N., Linear and Quasilinear Equations of Parabolic Type. Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs, 23 (1968), Amer. Math. Soc: Amer. Math. Soc Providence
[24] Ladyzenskaja, O. A.; Uralćeva, N. N., Linear and Quasilinear Elliptic Equations. Linear and Quasilinear Elliptic Equations, Translation of Mathematical Monographs, 46 (1968), Amer. Math. Soc: Amer. Math. Soc Providence
[25] Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation (1977), Elsevier: Elsevier New York · Zbl 0204.28001
[26] Porzio, M. M.; Vespri, V., Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103, 146-178 (1993) · Zbl 0796.35089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.