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.

show all references

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.

[1]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217

[2]

Yūki Naito, Takasi Senba. Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1861-1880. doi: 10.3934/cpaa.2013.12.1861

[3]

Giuseppe Maria Coclite, Helge Holden, Kenneth H. Karlsen. Wellposedness for a parabolic-elliptic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 659-682. doi: 10.3934/dcds.2005.13.659

[4]

Yūki Naito, Takasi Senba. Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model. Conference Publications, 2011, 2011 (Special) : 1111-1118. doi: 10.3934/proc.2011.2011.1111

[5]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two--phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006

[6]

Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81

[7]

Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691

[8]

Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211

[9]

Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281

[10]

Youcef Amirat, Laurent Chupin, Rachid Touzani. Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2445-2464. doi: 10.3934/cpaa.2014.13.2445

[11]

Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335

[12]

Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23

[13]

Bilal Saad, Mazen Saad. Numerical analysis of a non equilibrium two-component two-compressible flow in porous media. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 317-346. doi: 10.3934/dcdss.2014.7.317

[14]

Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268

[15]

Clément Cancès. On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Networks & Heterogeneous Media, 2010, 5 (3) : 635-647. doi: 10.3934/nhm.2010.5.635

[16]

Christian Heinemann, Christiane Kraus. Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2565-2590. doi: 10.3934/dcds.2015.35.2565

[17]

Shifeng Geng, Zhen Wang. Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant. Communications on Pure & Applied Analysis, 2012, 11 (2) : 475-500. doi: 10.3934/cpaa.2012.11.475

[18]

Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059

[19]

Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307

[20]

Kristian Bredies. Weak solutions of linear degenerate parabolic equations and an application in image processing. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1203-1229. doi: 10.3934/cpaa.2009.8.1203

2016 Impact Factor: 0.781

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]