December  2016, 5(4): 631-645. doi: 10.3934/eect.2016022

On the Muskat problem

1. 

Institut für Mathematik Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5 D-60120, Halle, Germany

2. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240

Received  June 2016 Revised  September 2016 Published  October 2016

Of concern is the motion of two fluids separated by a free interface in a porous medium, where the velocities are given by Darcy's law. We consider the case with and without phase transition. It is shown that the resulting models can be understood as purely geometric evolution laws, where the motion of the separating interface depends in a non-local way on the mean curvature. It turns out that the models are volume preserving and surface area reducing, the latter property giving rise to a Lyapunov function. We show well-posedness of the models, characterize all equilibria, and study the dynamic stability of the equilibria. Lastly, we show that solutions which do not develop singularities exist globally and converge exponentially fast to an equilibrium.
Citation: Jan Prüss, Gieri Simonett. On the Muskat problem. Evolution Equations and Control Theory, 2016, 5 (4) : 631-645. doi: 10.3934/eect.2016022
References:
[1]

D. M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math., 15 (2004), 597-607. doi: 10.1017/S0956792504005662.

[2]

B. V. Bazaliy and N. Vasylyeva, The Muskat problem with surface tension and a nonregular initial interface, Nonlinear Anal., 74 (2011), 6074-6096. doi: 10.1016/j.na.2011.05.087.

[3]

L. C. Berselli, D. Córdoba and R. Granero-Belinchón, Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound., 16 (2014), 175-213. doi: 10.4171/IFB/317.

[4]

Á. Castro, D. Córdoba, C. Fefferman and F. Gancedo, Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal., 208 (2013), 805-909. doi: 10.1007/s00205-013-0616-x.

[5]

Á. Castro, D. Córdoba, C. Fefferman, F. Gancedo and M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175 (2012), 909-948. doi: 10.4007/annals.2012.175.2.9.

[6]

A. Castro, D. Córdoba, C. L. Fefferman, F. Gancedo and M. López-Fernández, Turning waves and breakdown for incompressible flows, Proc. Natl. Acad. Sci. USA, 108 (2011), 4754-4759. doi: 10.1073/pnas.1101518108.

[7]

C. H. A. Cheng, R. Granero-Belinchón and S. Shkoller, Well-posedness of the Muskat problem with $H^2$ initial data, Adv. Math., 286 (2016), 32-104. doi: 10.1016/j.aim.2015.08.026.

[8]

P. Constantin, D. Córdoba, F. Gancedo and R. M. Strain, On the global existence for the Muskat problem, J. Eur. Math. Soc. (JEMS), 15 (2013), 201-227. doi: 10.4171/JEMS/360.

[9]

P. Constantin, G. Francisco, S. Roman and V. Vicol, Global regularity for the 2D Muskat equations with finite slope, arXiv:1507.01386, 2015. doi: 10.1016/j.anihpc.2016.09.001.

[10]

A. Cordoba, D. Cordoba and F. Gancedo, The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces, Proc. Natl. Acad. Sci. USA, 106 (2009), 10955-10959. doi: 10.1073/pnas.0809874106.

[11]

A. Córdoba, D. Córdoba and F. Gancedo, Interface evolution: The Hele-Shaw and Muskat problems, Ann. of Math. (2), 173 (2011), 477-542. doi: 10.4007/annals.2011.173.1.10.

[12]

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. doi: 10.1007/s00220-007-0246-y.

[13]

D. Córdoba, J. Gómez-Serrano and A. Zlatoš, A note on stability shifting for the Muskat problem, Philos. Trans. A, 373 (2015), 20140278, 10pp. doi: 10.1098/rsta.2014.0278.

[14]

D. Córdoba, J. Gómez-Serrano and A. Zlatoš, A note on stability shifting for the Muskat problem II: Stable to unstable and back to stable, arXiv:1512.02564, 2015.

[15]

D. Córdoba Gazolaz, R. Granero-Belinchón and R. Orive-Illera, The confined Muskat problem: differences with the deep water regime, Commun. Math. Sci., 12 (2014), 423-455. doi: 10.4310/CMS.2014.v12.n3.a2.

[16]

M. Ehrnström, J. Escher and B.-V. Matioc, Steady-state fingering patterns for a periodic Muskat problem, Methods Appl. Anal., 20 (2013), 33-46. doi: 10.4310/MAA.2013.v20.n1.a2.

[17]

J. Escher, A.-V. Matioc and B.-V. Matioc, A generalized Rayleigh-Taylor condition for the Muskat problem, Nonlinearity, 25 (2012), 73-92. doi: 10.1088/0951-7715/25/1/73.

[18]

J. Escher and B.-V. Matioc, On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results, Z. Anal. Anwend., 30 (2011), 193-218. doi: 10.4171/ZAA/1431.

[19]

J. Escher, B.-V. Matioc and C. Walker, The domain of parabolicity for the Muskat problem, arXiv:1507.02601, 2015.

[20]

J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373.

[21]

A. Friedman and Y. Tao, Nonlinear stability of the Muskat problem with capillary pressure at the free boundary, Nonlinear Anal., 53 (2003), 45-80. doi: 10.1016/S0362-546X(02)00286-9.

[22]

J. Hong, Y. Tao and F. Yi, Muskat problem with surface tension, J. Partial Differential Equations, 10 (1997), 213-231.

[23]

M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Equ., 10 (2010), 443-463. doi: 10.1007/s00028-010-0056-0.

[24]

M. Muskat, Two fluid systems in porous media. The encroachment of water into an oil sand, Physics, 5 (1934), 250-264. doi: 10.1063/1.1745259.

[25]

M. Muskat and R. D. Wyckoff, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, London, 1937.

[26]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbb R^n$, Discrete Cont. Dyn. Sys. A, 33 (2013), 5407-5428. doi: 10.3934/dcds.2013.33.5407.

[27]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105, Birkhäuser, 2016. Available from: http://www.springer.com/us/book/9783319276977.

[28]

J. Prüss and G. Simonett, The Verigin problem with and without phase transition, Submitted, 2016.

[29]

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. doi: 10.1002/cpa.20040.

[30]

F. Yi, Local classical solution of Muskat free boundary problem, J. Partial Differential Equations, 9 (1996), 84-96.

[31]

F. Yi, Global classical solution of Muskat free boundary problem, J. Math. Anal. Appl., 288 (2003), 442-461. doi: 10.1016/j.jmaa.2003.09.003.

show all references

References:
[1]

D. M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J. Appl. Math., 15 (2004), 597-607. doi: 10.1017/S0956792504005662.

[2]

B. V. Bazaliy and N. Vasylyeva, The Muskat problem with surface tension and a nonregular initial interface, Nonlinear Anal., 74 (2011), 6074-6096. doi: 10.1016/j.na.2011.05.087.

[3]

L. C. Berselli, D. Córdoba and R. Granero-Belinchón, Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound., 16 (2014), 175-213. doi: 10.4171/IFB/317.

[4]

Á. Castro, D. Córdoba, C. Fefferman and F. Gancedo, Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal., 208 (2013), 805-909. doi: 10.1007/s00205-013-0616-x.

[5]

Á. Castro, D. Córdoba, C. Fefferman, F. Gancedo and M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175 (2012), 909-948. doi: 10.4007/annals.2012.175.2.9.

[6]

A. Castro, D. Córdoba, C. L. Fefferman, F. Gancedo and M. López-Fernández, Turning waves and breakdown for incompressible flows, Proc. Natl. Acad. Sci. USA, 108 (2011), 4754-4759. doi: 10.1073/pnas.1101518108.

[7]

C. H. A. Cheng, R. Granero-Belinchón and S. Shkoller, Well-posedness of the Muskat problem with $H^2$ initial data, Adv. Math., 286 (2016), 32-104. doi: 10.1016/j.aim.2015.08.026.

[8]

P. Constantin, D. Córdoba, F. Gancedo and R. M. Strain, On the global existence for the Muskat problem, J. Eur. Math. Soc. (JEMS), 15 (2013), 201-227. doi: 10.4171/JEMS/360.

[9]

P. Constantin, G. Francisco, S. Roman and V. Vicol, Global regularity for the 2D Muskat equations with finite slope, arXiv:1507.01386, 2015. doi: 10.1016/j.anihpc.2016.09.001.

[10]

A. Cordoba, D. Cordoba and F. Gancedo, The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces, Proc. Natl. Acad. Sci. USA, 106 (2009), 10955-10959. doi: 10.1073/pnas.0809874106.

[11]

A. Córdoba, D. Córdoba and F. Gancedo, Interface evolution: The Hele-Shaw and Muskat problems, Ann. of Math. (2), 173 (2011), 477-542. doi: 10.4007/annals.2011.173.1.10.

[12]

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. doi: 10.1007/s00220-007-0246-y.

[13]

D. Córdoba, J. Gómez-Serrano and A. Zlatoš, A note on stability shifting for the Muskat problem, Philos. Trans. A, 373 (2015), 20140278, 10pp. doi: 10.1098/rsta.2014.0278.

[14]

D. Córdoba, J. Gómez-Serrano and A. Zlatoš, A note on stability shifting for the Muskat problem II: Stable to unstable and back to stable, arXiv:1512.02564, 2015.

[15]

D. Córdoba Gazolaz, R. Granero-Belinchón and R. Orive-Illera, The confined Muskat problem: differences with the deep water regime, Commun. Math. Sci., 12 (2014), 423-455. doi: 10.4310/CMS.2014.v12.n3.a2.

[16]

M. Ehrnström, J. Escher and B.-V. Matioc, Steady-state fingering patterns for a periodic Muskat problem, Methods Appl. Anal., 20 (2013), 33-46. doi: 10.4310/MAA.2013.v20.n1.a2.

[17]

J. Escher, A.-V. Matioc and B.-V. Matioc, A generalized Rayleigh-Taylor condition for the Muskat problem, Nonlinearity, 25 (2012), 73-92. doi: 10.1088/0951-7715/25/1/73.

[18]

J. Escher and B.-V. Matioc, On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results, Z. Anal. Anwend., 30 (2011), 193-218. doi: 10.4171/ZAA/1431.

[19]

J. Escher, B.-V. Matioc and C. Walker, The domain of parabolicity for the Muskat problem, arXiv:1507.02601, 2015.

[20]

J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373.

[21]

A. Friedman and Y. Tao, Nonlinear stability of the Muskat problem with capillary pressure at the free boundary, Nonlinear Anal., 53 (2003), 45-80. doi: 10.1016/S0362-546X(02)00286-9.

[22]

J. Hong, Y. Tao and F. Yi, Muskat problem with surface tension, J. Partial Differential Equations, 10 (1997), 213-231.

[23]

M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Equ., 10 (2010), 443-463. doi: 10.1007/s00028-010-0056-0.

[24]

M. Muskat, Two fluid systems in porous media. The encroachment of water into an oil sand, Physics, 5 (1934), 250-264. doi: 10.1063/1.1745259.

[25]

M. Muskat and R. D. Wyckoff, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, London, 1937.

[26]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbb R^n$, Discrete Cont. Dyn. Sys. A, 33 (2013), 5407-5428. doi: 10.3934/dcds.2013.33.5407.

[27]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105, Birkhäuser, 2016. Available from: http://www.springer.com/us/book/9783319276977.

[28]

J. Prüss and G. Simonett, The Verigin problem with and without phase transition, Submitted, 2016.

[29]

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. doi: 10.1002/cpa.20040.

[30]

F. Yi, Local classical solution of Muskat free boundary problem, J. Partial Differential Equations, 9 (1996), 84-96.

[31]

F. Yi, Global classical solution of Muskat free boundary problem, J. Math. Anal. Appl., 288 (2003), 442-461. doi: 10.1016/j.jmaa.2003.09.003.

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