Global existence for the confined muskat problem

SIAM Journal on Mathematical Analysis

Volumn 46, Issue 2, 2014, Pages 1651-1680

  Granero Belinchón, Rafael ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Darcy's law; Inhomogeneus Muskat problem; Well posedness

Indexed keywords

MATHEMATICAL MODELS;

DARCY'S LAW; FINITE DEPTH; GLOBAL EXISTENCE; INHOMOGENEUS MUSKAT PROBLEM; LIPSCHITZ CONTINUOUS; UNIFORMLY BOUNDED; WELLPOSEDNESS;

MATHEMATICAL TECHNIQUES;

EID: 84902663105     PISSN: 00361410     EISSN: 10957111     Source Type: Journal    
DOI: 10.1137/130912529     Document Type: Article

References (24)

1. D. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math., 15 (2004), pp. 597-607. (Pubitemid 40482778)
2. J. Bear, Dynamics of Fluids in Porous Media, Dover, New York, 1988.
3. L.C. Berselli, D. Córdoba, and R. Granero-Belinchón, Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound., to appear.
4. A. Castro, D. Cordoba, C. Fefferman, and F. Gancedo, Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal., 208 (2013), pp. 805-909.
5. A. Castro, D. Cordoba, C. Fefferman, F. Gancedo, and M. Lopez-Fernandez, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175 (2012), pp. 909-948.
6. M. Cerminara and A. Fasano, Modelling the dynamics of a geothermal reservoir fed by gravity driven flow through overstanding saturated rocks, J. Volcan. Geotherm. Res., 233 (2012), pp. 37-54.
7. C. Cheng, D. Coutand, and S. Shkoller, Global existence and decay for solutions of the Hele-Shaw flow with injection, preprint, arXiv:1208.6213 [math.AP], 2012.
8. P. Constantin, D. Cordoba, F. Gancedo, and R. Strain, On the global existence for the Muskat problem, J. Eur. Math. Soc. (JEMS), 15 (2013), pp. 201-227.
9. P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), pp. 393-415.
10. A. Cordoba, D. Córdoba, and F. Gancedo, Interface evolution: The Hele-Shaw and Muskat problems, Ann. of Math. (2), 173 (2011), pp. 477-542.
11. D. Córdoba and F. Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Comm. Math. Phys., 273 (2007), pp. 445-471.
12. D. Córdoba and F. Gancedo, A maximum principle for the Muskat problem for fluids with different densities, Comm. Math. Phys., 286 (2009), pp. 681-696.
13. D. Córdoba, R. Granero-Belinchón, and R. Orive, The confined Muskat problem: Differences with the deep water regime, Comm. Math. Sci., 12 (2014), pp. 423-455.
14. J. Escher, A. Matioc, and B. Matioc, A generalized Rayleigh-Taylor condition for the Muskat problem, Nonlinearity, 25 (2011), pp. 73-92.
15. J. Escher and B. Matioc, On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results, Z. Anal. Anwendungen, 30 (2011), pp. 193-218.
16. J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), pp. 619-642.
17. J. Gómez-Serrano and R. Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: A computer-assisted proof, Nonlinearity, to appear.
18. H. Hele-Shaw, Flow of water, Nature, 58 (1898), p. 520.
19. H. Kawarada and H. Koshigoe, Unsteady flow in porous media with a free surface, Japan J. Indust. Appl. Math., 8 (1991), pp. 41-84.
20. H. Knüpfer and N. Masmoudi, Darcy's Flow with Prescribed Contact Angle-Well-Posedness and Lubrication Approximation, preprint, arXiv:1204.2278 [math.AP], 2012.
21. A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.
22. M. Muskat, The flow of homogeneous fluids through porous media, Soil Sci., 46 (1938), p. 169.
23. D. Nield and A. Bejan, Convection in Porous Media, Springer Verlag, New York, 2006.
24. M. Siegel, R. Caflisch, and S. Howison, Global existence, singular solutions, and illposedness for the Muskat problem, Comm. Pure Appl. Math., 57 (2004), pp. 1374-1411. (Pubitemid 39234321)