Global regularity for 2D Muskat equations with finite slope

Constantin, Peter; Gancedo, Francisco; Shvydkoy, Roman; Vicol, Vlad

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 1041-1074.

We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class $W^{2,p}$, $1<p\le \infty$. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

[1] Ambrose, D.M. Well-posedness of two-phase Hele-Shaw flow without surface tension, Eur. J. Appl. Math., Volume 15 (2004) no. 5, pp. 597–607 | DOI | MR | Zbl

[2] Ambrose, D.M. Well-posedness of two-phase Darcy flow in 3D, Q. Appl. Math., Volume 65 (2007) no. 1, pp. 189–203 | DOI | MR | Zbl

[3] Ambrose, D.M. The zero surface tension limit of two-dimensional interfacial Darcy flow, J. Math. Fluid Mech., Volume 16 (2014) no. 1, pp. 105–143 | DOI | MR | Zbl

[4] Berselli, L.C.; Córdoba, D.; Granero-Belinchón, R. Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound., Volume 16 (2014) no. 2, pp. 175–213 | DOI | MR | Zbl

[5] Bear, J. Dynamics of Fluids in Porous Media, Courier Dover Publications, 1972 | Zbl

[6] Beck, T.; Sosoe, P.; Wong, P. Duchon–Robert solutions for the Rayleigh–Taylor and Muskat problems, J. Differ. Equ., Volume 256 (2014) no. 1, pp. 206–222 | DOI | MR | Zbl

[7] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F.; López-Fernández, M. Rayleigh–Taylor breakdown for the Muskat problem with applications to water waves, Ann. Math. (2), Volume 175 (2012) no. 2, pp. 909–948 | DOI | MR | Zbl

[8] Castro, A.; Cordoba, D.; Fefferman, C.; Gancedo, F. Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal., Volume 208 (2013) no. 3, pp. 805–909 | DOI | MR | Zbl

[9] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F. Splash singularities for the one-phase Muskat problem in stable regimes, Arch. Ration. Mech. Anal., Volume 222 (2016) no. 1, pp. 213–243 | DOI | MR | Zbl

[10] Constantin, P.; Córdoba, D.; Gancedo, F.; Rodriguez-Piazza, L.; Strain, R.M. On the Muskat problem: global in time results in 2D and 3D, Am. J. Math. (2014) https://www.press.jhu.edu/journals/american_journal_of_mathematics/future_publications.html (in press, see) | arXiv | MR

[11] Constantin, P.; Córdoba, D.; Gancedo, F.; Strain, R.M. On the global existence for the Muskat problem, J. Eur. Math. Soc., Volume 15 (2013), pp. 201–227 | DOI | MR | Zbl

[12] Constantin, P.; Pugh, M. Global solutions for small data to the Hele-Shaw problem, Nonlinearity, Volume 6 (1993) no. 3, pp. 393–415 | DOI | MR | Zbl

[13] Constantin, P.; Tarfulea, A.; Vicol, V. Long time dynamics of forced critical SQG, Commun. Math. Phys., Volume 335 (2015) no. 1, pp. 93–141 | DOI | MR | Zbl

[14] Constantin, P.; Vicol, V. Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., Volume 22 (2012) no. 5, pp. 1289–1321 | DOI | MR | Zbl

[15] Córdoba, A.; Córdoba, D.; Gancedo, F. Porous media: the Muskat problem in three dimensions, Anal. PDE, Volume 6 (2013) no. 2, pp. 447–497 | DOI | MR | Zbl

[16] Córdoba, D.; Gancedo, F. Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Commun. Math. Phys., Volume 273 (2007) no. 2, pp. 445–471 | DOI | MR | Zbl

[17] Córdoba, D.; Gancedo, F. A maximum principle for the Muskat problem for fluids with different densities, Commun. Math. Phys., Volume 286 (2009) no. 2, pp. 681–696 | DOI | MR | Zbl

[18] Córdoba, D.; Gancedo, F. Absence of squirt singularities for the multi-phase Muskat problem, Commun. Math. Phys., Volume 299 (2010) no. 2, pp. 561–575 | DOI | MR | Zbl

[19] Córdoba, D.; Gómez-Serrano, J.; Zlatos, A. A note in stability shifting for the Muskat problem, Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci., Volume 373 (2015) no. 2050 | MR | Zbl

[20] Córdoba, D.; Granero-Belinchón, R.; Orive-Illera, R. The confined Muskat problem: differences with the deep water regime, Commun. Math. Sci., Volume 12 (2014) no. 3, pp. 423–455 | MR | Zbl

[21] Córdoba, D.; Pernas-Castaño, T. Non-splat singularity for the one-phase Muskat problem, Trans. Am. Math. Soc. (2014) (in press) | arXiv | DOI | MR | Zbl

[22] Chen, X. The Hele-Shaw problem and area-preserving curve-shortening motions, Arch. Ration. Mech. Anal., Volume 123 (1993) no. 2, pp. 117–151 | DOI | MR | Zbl

[23] Cheng, C.H.; Granero-Belinchón, R.; Shkoller, S. Well-posedness of the Muskat problem with $H^2$ initial data, Adv. Math., Volume 286 (2016), pp. 32–104 | DOI | MR | Zbl

[24] Darcy, H. Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, 1856

[25] Escher, J.; Matioc, B.-V. On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results, Z. Anal. Anwend., Volume 30 (2011) no. 2, pp. 193–218 | MR | Zbl

[26] Escher, J.; Simonett, G. Classical solutions for Hele-Shaw models with surface tension, Adv. Differ. Equ., Volume 2 (1997) no. 4, pp. 619–642 | MR | Zbl

[27] Gancedo, F.; Strain, R.M. Absence of splash singularities for SQG sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci., Volume 111 (2014) no. 2, pp. 635–639 | DOI | MR | Zbl

[28] Granero-Belinchón, R. Global existence for the confined Muskat problem, SIAM J. Math. Anal., Volume 46 (2014) no. 2, pp. 1651–1680 | DOI | MR | Zbl

[29] Gómez-Serrano, J.; Granero-Belinchón, R. On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof, Nonlinearity, Volume 27 (2014) no. 6, pp. 1471–1498 | DOI | MR | Zbl

[30] Guo, Y.; Hallstrom, C.; Spirn, D. Dynamics near unstable, interfacial fluids, Commun. Math. Phys., Volume 270 (2007) no. 3, pp. 635–689 | MR | Zbl

[31] Knüpfer, H.; Masmoudi, N. Darcy's flow with prescribed contact angle: well-posedness and lubrication approximation, Arch. Ration. Mech. Anal. (2015), pp. 1–58 | DOI | MR | Zbl

[32] Muscalu, C.; Schlag, W. Classical and Multilinear Harmonic Analysis. Vol. II, Cambridge Studies in Advanced Mathematics, vol. 138, Cambridge University Press, Cambridge, 2013 | MR | Zbl

[33] Muskat, M. Two fluid systems in porous media. The encroachment of water into an oil sand, J. Appl. Phys., Volume 5 (1934) no. 9, pp. 250–264 | JFM

[34] Otto, F. Evolution of microstructure in unstable porous media flow: a relaxational approach, Commun. Pure Appl. Math., Volume 52 (1999) no. 7, pp. 873–915 | DOI | MR | Zbl

[35] Siegel, M.; Caflisch, R.E.; Howison, S. Global existence, singular solutions, and ill-posedness for the Muskat problem, Commun. Pure Appl. Math., Volume 57 (2004) no. 10, pp. 1374–1411 | DOI | MR | Zbl

[36] 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. Lond. Ser. A, Volume 245 (1958), pp. 312–329 | MR | Zbl

[37] Székelyhidi, L. Jr. Relaxation of the incompressible porous media equation, Ann. Sci. Éc. Norm. Supér. (4), Volume 45 (2012) no. 3, pp. 491–509 | Numdam | MR | Zbl