A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure

Journal of Differential Equations

Volumn 248, Issue 6, 2010, Pages 1561-1577

  Mikelić, Andro ^a,b  

^a UNIVERSITÉ DE LYON   (France)

^b UNIVERSITÉ LYON 1   (France)

Author keywords

Degenerate nonlinear parabolic PDE; Pseudoparabolic equations; Unsaturated flows

Indexed keywords

EID: 75249094418     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.11.022     Document Type: Article

References (30)

1. Barenblatt G.I., Entov V.M., and Ryzhik V.M. Theory of Fluid Flows Through Natural Rocks (1990), Kluwer, Dordrecht, Boston, Leiden
2. Barenblatt G.I., Bertsch M., Dalpasso R., and Ughi M. A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat-equation arising in the theory of heat and mass-exchange in stably stratified turbulent shear-flow. SIAM J. Math. Anal. 24 6 (November 1993) 1414-1439
3. Bear J. Dynamics of Fluids in Porous Media (1988), Dover Publications Inc., New York
4. Beliaev A.Yu. Homogenization of two-phase flows in porous media with hysteresis in the capillary relation. European J. Appl. Math. 14 1 (February 2003) 61-84
5. Beliaev A.Yu., and Hassanizadeh S.M. A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media. Transp. Porous Media 43 3 (June 2001) 487-510
6. Bourgeat A., and Panfilov M. Effective two-phase flow through highly heterogeneous porous media. Capillary non-equilibrium effects. Comput. Geosci. 2 3 (1998) 191-215
7. Cuesta C., and Hulshof J. A model problem for groundwater flow with dynamic capillary pressure: stability of travelling waves. Nonlinear Anal. 52 4 (February 2003) 1199-1218
8. Cuesta C., van Duijn C.J., and Hulshof J. Infiltration in porous media with dynamic capillary pressure: travelling waves. European J. Appl. Math. 11 (August 2000) 381-397
9. DiBenedetto E., and Pierre M. On the maximum principle for pseudoparabolic equations. Indiana Univ. Math. J. 30 6 (1981) 821-854
10. Düll W.P. Some qualitative properties of solutions to a pseudoparabolic equation modeling solvent uptake in polymeric solids. Comm. Partial Differential Equations 31 8 (August 2006) 1117-1138
11. van Duijn C.J., Peletier L.A., and Pop I.S. A new class of entropy solutions of the Buckley-Leverett equation. SIAM J. Math. Anal. 39 2 (2007) 507-536
12. Evans L.C. Partial Differential Equations. Grad. Stud. Math. (1975), American Mathematical Society, Providence, RI
13. Evans L.C. A survey of entropy methods for partial differential equations. Bull. Amer. Math. Soc. 41 4 (2004) 409-438
14. Gray W.G., and Hassanizadeh S.M. Unsaturated flow theory including interfacial phenomena. Water Resour. Res. 27 (1991) 1855-1863
15. Gray W.G., and Hassanizadeh S.M. Paradoxes and realities in unsaturated flow theory. Water Resour. Res. 27 (1991) 1847-1854
16. Hassanizadeh S.M., and Gray W.G. Thermodynamic basis of capillary-pressure in porous-media. Water Resour. Res. 29 10 (October 1993) 3389-3405
17. Hassanizadeh S.M., and Gray W.G. Toward an improved description of the physics of two-phase flow. Adv. Water Resour. 16 (1993) 53-67
18. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. vol. 840 (1993), Springer Verlag, Berlin
19. Hulshof J., and King J.R. Analysis of a Darcy flow model with a dynamic pressure saturation relation. SIAM J. Appl. Math. 59 1 (October 1998) 318-346
20. King J.R., and Cuesta C.M. Small- and waiting-time behavior of a Darcy flow model with a dynamic pressure saturation relation. SIAM J. Appl. Math. 66 5 (2006) 1482-1511
21. Mikelić A., and Robert R. On the equations describing a relaxation toward a statistical equilibrium state in the two-dimensional perfect fluid dynamics. SIAM J. Math. Anal. 29 5 (September 1998) 1238-1255
22. Mikelić A., and Bruining J. Analysis of model equations for stress-enhanced diffusion in coal layers. Part I: Existence of a weak solution. SIAM J. Math. Anal. 40 4 (2008) 1671-1691
23. Novick-Cohen A., and Pego R.L. Stable patterns in a viscous diffusion equation. Trans. Amer. Math. Soc. 324 (1991) 331-351
24. Padron V. Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations. Comm. Partial Differential Equations 23 3 (1998) 457-486
25. Ptashnyk M. Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities. Nonlinear Anal. 66 12 (2007) 2653-2675
26. Showalter R.E., and Ting T.W. Pseudoparabolic partial differential equations. SIAM J. Math. Anal. 1 (1970) 1-26
27. Smiles D.E., Vachaud G., and Vauclin M.A. Test of the uniqueness of the soil moisture characteristic during transient, non-hysteretic flow of water in a rigid soil. Soil Sci. Soc. Amer. Proc. 35 (1971) 535-539
28. F. Stauffer, Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media, in: IAHR Sympos. on Scale Effects in Porous Media, Thessaloniki, Greece, August 29-September 1, 1978
29. Topp G.C., Klute A., and Peters D.B. Comparison of water content-pressure head data obtained by equilibrium, steady-state, and unsteady-state methods. Soil Sci. Soc. Amer. Proc. 31 (1967) 312-314
30. Vachaud G., Vauclin M., and Wakil M.A. Study of the uniqueness of the soil moisture characteristic during desorption by vertical drainage. Soil Sci. Soc. Amer. Proc. 36 (1972) 531-532