Viscous displacement in porous media: the Muskat problem in 2D

Matioc, Bogdan–Vasile

doi:10.1090/tran/7287

Viscous displacement in porous media: the Muskat problem in 2D
HTML articles powered by AMS MathViewer

by Bogdan–Vasile Matioc PDF

Trans. Amer. Math. Soc. 370 (2018), 7511-7556 Request permission

Abstract:

We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the mathematical model as an evolution problem for the sharp interface between the fluids, we show that the Muskat problem with surface tension is a quasilinear parabolic problem, whereas, in the absence of surface tension effects, the Rayleigh–Taylor condition identifies a domain of parabolicity for the fully nonlinear problem. Based upon these aspects, we then establish the local well-posedness for arbitrary large initial data in $H^s$, $s>2$, if surface tension is taken into account, respectively for arbitrary large initial data in $H^2$ that additionally satisfy the Rayleigh–Taylor condition if surface tension effects are neglected. We also show that the problem exhibits the parabolic smoothing effect and we provide criteria for the global existence of solutions.

References

Herbert Amann, Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal. 92 (1986), no. 2, 153–192. MR 816618, DOI 10.1007/BF00251255
Herbert Amann, Dynamic theory of quasilinear parabolic equations. I. Abstract evolution equations, Nonlinear Anal. 12 (1988), no. 9, 895–919. MR 960634, DOI 10.1016/0362-546X(88)90073-9
Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126. MR 1242579, DOI 10.1007/978-3-663-11336-2_{1}
Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385, DOI 10.1007/978-3-0348-9221-6
David M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math. 15 (2004), no. 5, 597–607. MR 2128613, DOI 10.1017/S0956792504005662
David M. Ambrose, The zero surface tension limit of two-dimensional interfacial Darcy flow, J. Math. Fluid Mech. 16 (2014), no. 1, 105–143. MR 3171344, DOI 10.1007/s00021-013-0146-1
Sigurd B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 1-2, 91–107. MR 1059647, DOI 10.1017/S0308210500024598
B. V. Bazaliy and N. Vasylyeva, The two-phase Hele-Shaw problem with a nonregular initial interface and without surface tension, Zh. Mat. Fiz. Anal. Geom. 10 (2014), no. 1, 3–43, 152, 155 (English, with English, Russian and Ukrainian summaries). MR 3236960, DOI 10.15407/mag10.01.003
Jacob Bear, Dynamics of fluids in porous media, Dover Publications, New York, 1988.
Luigi C. Berselli, Diego Córdoba, and Rafael Granero-Belinchón, Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound. 16 (2014), no. 2, 175–213. MR 3231970, DOI 10.4171/IFB/317
Ángel Castro, Diego Córdoba, Charles Fefferman, and Francisco Gancedo, Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal. 208 (2013), no. 3, 805–909. MR 3048596, DOI 10.1007/s00205-013-0616-x
Ángel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, and María López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2) 175 (2012), no. 2, 909–948. MR 2993754, DOI 10.4007/annals.2012.175.2.9
Angel Castro, Diego Córdoba, Charles L. Fefferman, Francisco Gancedo, and María López-Fernández, Turning waves and breakdown for incompressible flows, Proc. Natl. Acad. Sci. USA 108 (2011), no. 12, 4754–4759. MR 2792311, DOI 10.1073/pnas.1101518108
C. H. Arthur Cheng, Rafael Granero-Belinchón, and Steve Shkoller, Well-posedness of the Muskat problem with $H^2$ initial data, Adv. Math. 286 (2016), 32–104. MR 3415681, DOI 10.1016/j.aim.2015.08.026
Peter Constantin, Diego Córdoba, Francisco Gancedo, Luis Rodríguez-Piazza, and Robert M. Strain, On the Muskat problem: global in time results in 2D and 3D, Amer. J. Math. 138 (2016), no. 6, 1455–1494. MR 3595492, DOI 10.1353/ajm.2016.0044
Peter Constantin, Diego Córdoba, Francisco Gancedo, and Robert M. Strain, On the global existence for the Muskat problem, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 201–227. MR 2998834, DOI 10.4171/JEMS/360
Peter Constantin, Vlad Vicol, Roman Shvydkoy, and Francisco Gancedo, Global regularity for 2D Muskat equations with finite slope, Ann. Inst. H. Poincaré, Anal. Nonlinéaire 34 (2017), no. 4 1041–1074.
Antonio Córdoba, Diego Córdoba, and Francisco Gancedo, Interface evolution: the Hele-Shaw and Muskat problems, Ann. of Math. (2) 173 (2011), no. 1, 477–542. MR 2753607, DOI 10.4007/annals.2011.173.1.10
Antonio Córdoba, Diego Córdoba, and Francisco Gancedo, Porous media: the Muskat problem in three dimensions, Anal. PDE 6 (2013), no. 2, 447–497. MR 3071395, DOI 10.2140/apde.2013.6.447
Diego Córdoba and Francisco Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Comm. Math. Phys. 273 (2007), no. 2, 445–471. MR 2318314, DOI 10.1007/s00220-007-0246-y
Diego Córdoba and Francisco Gancedo, Absence of squirt singularities for the multi-phase Muskat problem, Comm. Math. Phys. 299 (2010), no. 2, 561–575. MR 2679821, DOI 10.1007/s00220-010-1084-x
Diego Córdoba Gazolaz, Rafael Granero-Belinchón, and Rafael Orive-Illera, The confined Muskat problem: differences with the deep water regime, Commun. Math. Sci. 12 (2014), no. 3, 423–455. MR 3144991, DOI 10.4310/CMS.2014.v12.n3.a2
Joachim Escher, The Dirichlet-Neumann operator on continuous functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 2, 235–266. MR 1288366
Joachim Escher, Anca-Voichita Matioc, and Bogdan-Vasile Matioc, A generalized Rayleigh-Taylor condition for the Muskat problem, Nonlinearity 25 (2012), no. 1, 73–92. MR 2864377, DOI 10.1088/0951-7715/25/1/73
Joachim Escher and Bogdan-Vasile Matioc, On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results, Z. Anal. Anwend. 30 (2011), no. 2, 193–218. MR 2793001, DOI 10.4171/ZAA/1431
Joachim Escher, Bogdan-Vasile Matioc, and Christoph Walker, The domain of parabolicity for the Muskat problem, Indiana Univ. Math. J. (2018), no. 2, 679-737.
Joachim Escher and Gieri Simonett, Maximal regularity for a free boundary problem, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 463–510. MR 1356871, DOI 10.1007/BF01210620
Joachim Escher and Gieri Simonett, Analyticity of the interface in a free boundary problem, Math. Ann. 305 (1996), no. 3, 439–459. MR 1397432, DOI 10.1007/BF01444233
Joachim Escher and Gieri Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal. 28 (1997), no. 5, 1028–1047. MR 1466667, DOI 10.1137/S0036141095291919
E. B. Fabes, M. Jodeit Jr., and N. M. Rivière, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math. 141 (1978), no. 3-4, 165–186. MR 501367, DOI 10.1007/BF02545747
Avner Friedman and Youshan Tao, Nonlinear stability of the Muskat problem with capillary pressure at the free boundary, Nonlinear Anal. 53 (2003), no. 1, 45–80. MR 1992404, DOI 10.1016/S0362-546X(02)00286-9
Francisco Gancedo and Robert M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014), no. 2, 635–639. MR 3181769, DOI 10.1073/pnas.1320554111
Javier Gómez-Serrano and Rafael Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof, Nonlinearity 27 (2014), no. 6, 1471–1498. MR 3215843, DOI 10.1088/0951-7715/27/6/1471
Rafael Granero-Belinchón, Global existence for the confined Muskat problem, SIAM J. Math. Anal. 46 (2014), no. 2, 1651–1680. MR 3196945, DOI 10.1137/130912529
Rafael Granero-Belinchón and Steve Shkoller, Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability, (2016), preprint. arXiv:1611.06147.
G. M. Homsy, Viscous fingering in porous media, Ann. Rev. Fluid Mech. 19 (1987), 271–311.
Jiaxing Hong, Youshan Tao, and Fahuai Yi, Muskat problem with surface tension, J. Partial Differential Equations 10 (1997), no. 3, 213–231. MR 1471114
Jian Ke Lu, Boundary value problems for analytic functions, Series in Pure Mathematics, vol. 16, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1279172
Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547, DOI 10.1007/978-3-0348-9234-6
Bogdan-Vasile Matioc, The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results, to appear in Anal. PDE (2018), arXiv:1610.05546.
Yves Meyer and Ronald Coifman, Wavelets, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, Cambridge, 1997. Calderón-Zygmund and multilinear operators; Translated from the 1990 and 1991 French originals by David Salinger. MR 1456993
Takafumi Murai, Boundedness of singular integral operators of Calderón type. VI, Nagoya Math. J. 102 (1986), 127–133. MR 846134, DOI 10.1017/S0027763000000477
M. Muskat, Two fluid systems in porous media. The encroachment of water into an oil sand, Physics 5 (1934), 250–264.
Jan Prüss, Yuanzhen Shao, and Gieri Simonett, On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension, Interfaces Free Bound. 17 (2015), no. 4, 555–600. MR 3450740, DOI 10.4171/IFB/354
Jan Prüss and Gieri Simonett, Moving interfaces and quasilinear parabolic evolution equations, Monographs in Mathematics, vol. 105, Birkhäuser/Springer, [Cham], 2016. MR 3524106, DOI 10.1007/978-3-319-27698-4
Jan Prüss and Gieri Simonett, On the Muskat problem, Evol. Equ. Control Theory 5 (2016), no. 4, 631–645. MR 3603251, DOI 10.3934/eect.2016022
P. G. Saffman and Geoffrey 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). MR 97227, DOI 10.1098/rspa.1958.0085
Michael Siegel, Russel E. Caflisch, and Sam Howison, Global existence, singular solutions, and ill-posedness for the Muskat problem, Comm. Pure Appl. Math. 57 (2004), no. 10, 1374–1411. MR 2070208, DOI 10.1002/cpa.20040
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Spencer Tofts, On the existence of solutions to the Muskat problem with surface tension, J. Math. Fluid Mech. 19 (2017), no. 4, 581–611. MR3714494
Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI 10.1016/0022-1236(84)90066-1
Fahuai Yi, Local classical solution of Muskat free boundary problem, J. Partial Differential Equations 9 (1996), no. 1, 84–96. MR 1384001