# American Institute of Mathematical Sciences

2012, 5(1): 93-113. doi: 10.3934/dcdss.2012.5.93

## Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero

 1 CMI, Université de Provence, 39 rue Frédéric Joliot-Curie 13453 Marseille cedex 13 2 CNRS and Laboratoire de Mathématiques, Université de Paris-Sud 11, F-91405 Orsay Cedex 3 Université Paris-Est Marne-La-Vallée, 5 bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France

Received  June 2009 Revised  December 2009 Published  February 2011

In this paper we consider a two-phase flow problem in porous media and study its singular limit as the viscosity of the air tends to zero; more precisely, we prove the convergence of subsequences to solutions of a generalized Richards model.
Citation: Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93
##### References:
 [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic equations,, Math.-Z., 183 (1983), 311. doi: 10.1007/BF01176474. [2] H. Brezis, "Analyse Fonctionnelle Théorie et Applications,'', Masson, (1993). [3] Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution,, J. Differential Equations, 171 (2001), 203. doi: 10.1006/jdeq.2000.3848. [4] F. Z. Daïm, R. Eymard and D. Hilhorst, Existence of a solution for two phase flow in porous media: the case that the porosity depends on the pressure,, J. Math. Anal. Appl., 326 (2007), 332. doi: 10.1016/j.jmaa.2006.02.082. [5] C. J. Van Duijn and L. A. Peletier, Nonstationary filtration in partially satured porous media,, Arch. Rational Mech. Anal., 78 (1982), 173. doi: 10.1007/BF00250838. [6] R. Eymard, M. Gutnic and D. Hilhorst, The finit volume method for an elliptic-parabolic equation,, Acta Mathematica Universitatis Comenianae, 67 (1998), 181. [7] J. Hulshof and N. Wolanski, Monotone flows in n-dimensional partially saturated porous media: Lipschitz-continuity of the interface,, Arch. Rational Mech. Anal., 102 (1988), 287. doi: 10.1007/BF00251532. [8] O. A. Ladyhenskaya and N. N. Ural'ceva, "Linear and Quasilinear Elliptic Equations,'', American Mathematical Society, (1964). [9] O. A. Ladyhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', American Mathematical Society, (1968). [10] F. Otto, $L^1$-concentration and uniqueness for quasilinear elliptic-parabolic equations,, J. Differential Equations, 131 (1996), 20. doi: 10.1006/jdeq.1996.0155. [11] I. S. Pop, Error estimates for a time discretization method for the Richard's equation,, Computational Geosciences, 6 (2002), 141. doi: 10.1023/A:1019936917350. [12] F. A. Radu, I. S. Pop and P. Knabner, Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards' equation,, SIAM Journal on Numerical Analysis, 42 (2004), 1452. doi: 10.1137/S0036142902405229.

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##### References:
 [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic equations,, Math.-Z., 183 (1983), 311. doi: 10.1007/BF01176474. [2] H. Brezis, "Analyse Fonctionnelle Théorie et Applications,'', Masson, (1993). [3] Z. Chen, Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution,, J. Differential Equations, 171 (2001), 203. doi: 10.1006/jdeq.2000.3848. [4] F. Z. Daïm, R. Eymard and D. Hilhorst, Existence of a solution for two phase flow in porous media: the case that the porosity depends on the pressure,, J. Math. Anal. Appl., 326 (2007), 332. doi: 10.1016/j.jmaa.2006.02.082. [5] C. J. Van Duijn and L. A. Peletier, Nonstationary filtration in partially satured porous media,, Arch. Rational Mech. Anal., 78 (1982), 173. doi: 10.1007/BF00250838. [6] R. Eymard, M. Gutnic and D. Hilhorst, The finit volume method for an elliptic-parabolic equation,, Acta Mathematica Universitatis Comenianae, 67 (1998), 181. [7] J. Hulshof and N. Wolanski, Monotone flows in n-dimensional partially saturated porous media: Lipschitz-continuity of the interface,, Arch. Rational Mech. Anal., 102 (1988), 287. doi: 10.1007/BF00251532. [8] O. A. Ladyhenskaya and N. N. Ural'ceva, "Linear and Quasilinear Elliptic Equations,'', American Mathematical Society, (1964). [9] O. A. Ladyhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'', American Mathematical Society, (1968). [10] F. Otto, $L^1$-concentration and uniqueness for quasilinear elliptic-parabolic equations,, J. Differential Equations, 131 (1996), 20. doi: 10.1006/jdeq.1996.0155. [11] I. S. Pop, Error estimates for a time discretization method for the Richard's equation,, Computational Geosciences, 6 (2002), 141. doi: 10.1023/A:1019936917350. [12] F. A. Radu, I. S. Pop and P. Knabner, Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards' equation,, SIAM Journal on Numerical Analysis, 42 (2004), 1452. doi: 10.1137/S0036142902405229.
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