Interface evolution: The Hele-Shaw and Muskat problems

Annals of Mathematics

Volumn 173, Issue 1, 2011, Pages 477-542

  Córdoba, Antonio ^a   Córdoba, Diego ^b   Gancedo, Francisco ^c  

^a UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

^b INSTITUTO DE CIENCIAS MATEMÁTICAS   (Spain)

^c UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 78751624212     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.4007/annals.2011.173.1.10     Document Type: Article

References (19)

1. D. M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math. 15 (2004), 597-607.
2. G. R. Baker, D. I. Meiron, and S. A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech. 123 (1982), 477-501.
3. J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.
4. L. A. Caffarelli and A. Córdoba, Phase transitions: uniform regularity of the intermediate layers, J. Reine Angew. Math. 593 (2006), 209-235.
5. P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem, Nonlinearity 6 (1993), 393-415.
6. A. Córdoba and D. Córdoba, A pointwise estimate for fractionary derivatives with applications to partial differential equations, Proc. Natl. Acad. Sci. USA 100 (2003), 15316-15317.
7. A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), 511-528.
8. A. Córdoba and D. Córdoba, and F. Gancedo, Interface evolution: water waves in 2-D, Adv. Math. 223 (2010), 120-173.
9. D. Córdoba and F. Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Comm. Math. Phys. 273 (2007), 445-471.
10. D. Córdoba and F. Gancedo, A maximum principle for the Muskat problem for fluids with different densities, Comm. Math. Phys. 286 (2009), 681-696.
11. 1-domains, Studia Math. 66 (1979), 13-24.
12. R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65-124.
13. F. Gancedo, Existence for the α-patch model and the QG sharp front in Sobolev spaces, Adv. Math. 217 (2008), 2569-2598.
14. H. S. Hele-Shaw, The flow of water, Nature 58 (1898), 34-36.
15. T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys. 114 (1994), 312-338.
16. M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw Hill Book Co., New York, 1937.
17. P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London. Ser. A 245 (1958), 312-329. (2 plates).
18. M. Siegel, R. E. Caflisch, and S. Howison, Global existence, singular solutions, and ill-posedness for the Muskat problem, Comm. Pure Appl. Math. 57 (2004), 1374-1411.
19. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Series 43, Princeton Univ. Press, Princeton, NJ, 1993, Monographs in Harmonic Analysis, III.