Well-posedness of the Muskat problem with H2 initial data

Advances in Mathematics

Volumn 286, Issue , 2016, Pages 32-104

  Cheng, C H Arthur ^a   Granero Belinchón, Rafael ^b   Shkoller, Steve ^b  

^a NATIONAL CENTRAL UNIVERSITY   (Taiwan)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Free boundary problems; Hele Shaw; Moving interfaces; Muskat; Regularity

Indexed keywords

EID: 84942354084     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2015.08.026     Document Type: Article

References (49)

1. Ambrose D. The zero surface tension limit of two-dimensional interfacial Darcy flow. J. Math. Fluid Mech. 2014, 16:105-143.
2. Bear J. Dynamics of Fluids in Porous Media 1988, Dover Publications.
3. Berselli L., Córdoba D., Granero-Belinchón R. Local solvability and turning for the inhomogeneous Muskat problem. Interfaces Free Bound. 2014, 16(2):175-213.
4. Castro A., Cordoba D., Fefferman C., Gancedo F. Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal. 2013, 208(3):805-909.
5. Castro A., Cordoba D., Fefferman C., Gancedo F., López-Fernández M. Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. of Math. 2012, 175:909-948.
6. Castro Á., Córdoba D., Fefferman C., Gancedo F., Orive R. Splash singularities for the one-phase Muskat problem in stable regimesincompressible flow in porous media with fractional diffusion arxiv:1311.7653.
7. Castro A., Córdoba D., Gancedo F. Some recent results on the Muskat problem. Journ. Equ. Dériv. Partielles 2010, 6:1-14.
8. Cerminara M., Fasano A. Modelling the dynamics of a geothermal reservoir fed by gravity driven flow through overstanding saturated rocks. J. Volcanol. Geotherm. Res. 2012, 233:37-54.
9. Chen X. The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Ration. Mech. Anal. 1993, 123(2):117-151.
10. Cheng C.H.A., Coutand D., Shkoller S. Global existence and decay for solutions of the Hele-Shaw flow with injection. Interfaces Free Bound. 2014, 16(3):297-338.
11. C.H.A. Cheng, S. Shkoller, Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains, arXiv e-prints, Aug. 2014.
12. Constantin P., Cordoba D., Gancedo F., Rodríguez-Piazza L., Strain R. On the Muskat problem: global in time results in 2d and 3d arXiv preprint. arxiv:1310.0953.
13. Constantin P., Cordoba D., Gancedo F., Strain R. On the global existence for the Muskat problem. J. Eur. Math. Soc. (JEMS) 2013, 15:201-227.
14. Constantin P., Pugh M. Global solutions for small data to the Hele-Shaw problem. Nonlinearity 1993, 6:393-415.
15. Cordoba A., Córdoba D., Gancedo F. Interface evolution: the Hele-Shaw and Muskat problems. Ann. of Math. 2011, 173:1:477-542.
16. Córdoba D., Gancedo F. Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Comm. Math. Phys. 2007, 273(2):445-471.
17. Córdoba D., Gancedo F. A maximum principle for the Muskat problem for fluids with different densities. Comm. Math. Phys. 2009, 286(2):681-696.
18. Córdoba D., Gancedo F. Absence of squirt singularities for the multi-phase Muskat problem. Comm. Math. Phys. 2009, 1-15.
19. Córdoba D., Granero-Belinchón R., Orive R. On the confined Muskat problem: differences with the deep water regime. Commun. Math. Sci. 2014, 12(3):423-455.
20. Cordoba D., Pernás-Castaño T. Non-splat singularity for the one-phase Muskat problem arxiv:1409.2483.
21. Coutand D., Hole J., Shkoller S. Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit. SIAM J. Math. Anal. 2013, 45(6):3690-3767.
22. Coutand D., Shkoller S. The interaction between quasilinear elastodynamics and the Navier-Stokes equations. Arch. Ration. Mech. Anal. 2006, 179(3):303-352.
23. Coutand D., Shkoller S. Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc. 2007, 20(3):829-930.
24. Coutand D., Shkoller S. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete Contin. Dyn. Syst. Ser. S 2010, 3(3):429-449.
25. Coutand D., Shkoller S. Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. 2012, 206(2):515-616.
26. Coutand D., Shkoller S. On the finite-time splash and splat singularities for the 3-D free-surface Euler equations. Comm. Math. Phys. 2014, 325(1):143-183.
27. D. Coutand, S. Shkoller, On the impossibility of finite-time splash singularities for vortex sheets, arXiv e-prints, July 2014.
28. De Parseval Y., Pillai K., Advani S. A simple model for the variation of permeability due to partial saturation in dual scale porous media. Transp. Porous Media 1997, 27(3):243-264.
29. Elliott C.M., Ockendon J.R. Weak and Variational Methods for Moving Boundary Problems 1982, vol. 59. Pitman, Boston.
30. Escher J., Matioc A.-V., Matioc B.-V. A generalized Rayleigh-Taylor condition for the Muskat problem. Nonlinearity 2012, 25(1):73-92.
31. Escher J., Matioc B.-V. On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results. Z. Anal. Anwend. 2011, 30(2):193-218.
32. Escher J., Simonett G. Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations 1997, 2(4):619-642.
33. Escher J., Simonett G. A center manifold analysis for the Mullins-Sekerka model. J. Differential Equations 1998, 143(2):267-292.
34. C. Fefferman, A.D. Ionescu, V. Lie, On the absence of "splash" singularities in the case of two-fluid interfaces, arXiv e-prints, Dec. 2013.
35. Friedman A. Free boundary problems arising in tumor models. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 2004, 9(3-4).
36. Gancedo F., Strain R. Absence of splash singularities for SQG sharp fronts and the Muskat problem arXiv preprint. arxiv:1309.4023.
37. Gómez-Serrano J., Granero-Belinchón R. On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof. Nonlinearity 2014, 27(6):1471-1498.
38. Granero-Belinchón R. Global existence for the confined Muskat problem. SIAM J. Math. Anal. 2014, 46(2):1651-1680.
39. Hadžić M., Shkoller S. Global stability and decay for the classical Stefan problem. Comm. Pure Appl. Math. 2014.
40. M. Hadžić, S. Shkoller, Well-posedness for the classical Stefan problem and the zero surface tension limit, arXiv e-prints, Dec. 2011.
41. Hele-Shaw H.S. On the motion of a viscous fluid between two parallel plates. Trans. R. Inst. Nav. Archit. 1898, 40:218.
42. Kato T., Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 1988, 41(7):891-907.
43. Kenig C.E., Ponce G., Vega L. Well-posedness and scattering results for the generalized Korteweg-De Vries equation via the contraction principle. Comm. Pure Appl. Math. 1993, 46(4):527-620.
44. Kinderlehrer D., Nirenberg L. Regularity in free boundary problems. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1977, 4(2):373-391.
45. Kinderlehrer D., Nirenberg L. The smoothness of the free boundary in the one phase Stefan problem. Comm. Pure Appl. Math. 1978, 31(3):257-282.
46. Meirmanov A. The Stefan Problem. de Gruyter Expositions in Mathematics 1992, vol. 3.
47. Muskat M. The flow of homogeneous fluids through porous media. Soil Sci. 1938, 46(2):169.
48. Nield D., Bejan A. Convection in Porous Media 2006, Springer-Verlag.
49. Pozrikidis C. Numerical simulation of blood and interstitial flow through a solid tumor. J. Math. Biol. 2010, 60(1):75-94.