On the global existence for the Muskat problem

Journal of the European Mathematical Society

Volumn 15, Issue 1, 2013, Pages 201-227

  Constantin, Peter ^a   Córdoba, Diego ^b   Gancedo, Francisco ^c   Strain, Robert M ^d  

^a PRINCETON UNIVERSITY   (United States)

^b INSTITUTO DE CIENCIAS MATEMÁTICAS   (Spain)

^c UNIVERSITY OF CHICAGO   (United States)

^d UNIVERSITY OF PENNSYLVANIA   (United States)

Author keywords

Fluid interface; Global existence; Incompressible flows; Porous media

Indexed keywords

EID: 84870705750     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/360     Document Type: Article

References (22)

1. Ambrose, D.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Euro. J. Appl. Math. 15, 597-607 (2004) Zbl 1076.76027 MR 2128613
2. Bailly, J. H.: Local existence of classical solutions to first-order parabolic equations describing free boundaries. Nonlinear Anal. 32, 583-599 (1998) Zbl 0934.35211 MR 1612026
3. Bear, J.: Dynamics of Fluids in Porous Media. American Elsevier, New York (1972) Zbl 1191.76001
4. Bertozzi, A. L., Constantin, P.: Global regularity for vortex patches. Comm. Math. Phys. 152, 19-28 (1993) Zbl 0771.76014 MR 1207667
5. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229-243 (1998) Zbl 0923.76025 MR 0972259
6. Constantin, P., Foias, C.: Navier-Stokes Equation. Chicago Lectures in Math., Univ. of Chicago Press, Chicago, IL (1988) Zbl 0687.35071
7. Constantin, P., Pugh, M.: Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6, 393-415 (1993) Zbl 0808.35104 MR 1223740
8. Córdoba, A., Córdoba, D., Gancedo, F.: Interface evolution: the Hele-Shaw and Muskat problems. Ann. of Math. 173, 477-542 (2011) Zbl 1229.35204 MR 2753607
9. Córdoba, D., Gancedo, F.: Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Comm. Math. Phys. 273, 445-471 (2007) Zbl 1120.76064 MR 2318314
10. Córdoba, D., Gancedo, F.: A maximum principle for the Muskat problem for fluids with different densities. Comm. Math. Phys. 286, 681-696 (2009) Zbl 1173.35637 MR 2472040
11. Córdoba, D., Gancedo, F.: Absence of squirt singularities for the multi-phase Muskat problem. Comm. Math. Phys. 299, 561-575 (2010) Zbl 1198.35176 MR 2679821
12. Córdoba, D., Gancedo, F., Orive, R.: A note on the interface dynamics for convection in porous media. Phys. D, 237, 1488-1497 (2008) Zbl 1143.76574 MR 2454601
13. Escher, J., Simonett, G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations 2, 619-642 (1997) Zbl 1023.35527 MR 1441859
14. Escher, J., Matioc, B.-V.: On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results. Z. Anal. Anwend. 30, 193-218 (2011) Zbl 1223.35199 MR 2793001
15. Escher, J., Matioc, A.-V., Matioc, B.-V.: A generalized Rayleigh-Taylor condition for the Muskat problem. Nonlinearity 25, 73-92 (2012) Zbl 1243.35179 MR 2864377
16. Kim, I.: Uniqueness and existence results on the Hele-Shaw and the Stefan problems. Arch. Ration. Mech. Anal. 168, 299-328 (2003) Zbl 1044.76019 MR 1994745
17. Muskat, M., Wickoff, R. D.: The Flow of Homogeneous Fluids through Porous Media. McGraw-Hill, London (1937) JFM 63.1368.03
18. Saffman, P. G., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. London Ser. A 245, 312-329 (1958) Zbl 0086.41603 MR 0097227
19. Siegel, M., Caflisch, R., Howison, S.: Global existence, singular solutions, and ill-posedness for the Muskat problem. Comm. Pure Appl. Math. 57, 1374-1411 (2004) Zbl 1062.35089 MR 2070208
20. Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ (1993) Zbl 0821.42001 MR 1232192
21. Yi, F.: Local classical solution of Muskat free boundary problem, J. Partial Differential Equations 9, 84-96 (1996) Zbl 0847.35152 MR 1384001
22. Yi, F.: Global classical solution of Muskat free boundary problem, J. Math. Anal. Appl. 288, 442-461 (2003) Zbl 1038.35083 MR 2019452