On the Thin Film Muskat and the Thin Film Stokes Equations

  • Gabriele Bruell
  • Rafael Granero-BelinchónEmail author


The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase thin film flows. In particular, we consider the two-phase thin film Muskat problem and the two-phase thin film approximation of the Stokes flow under the influence of both, capillary and gravitational forces. The existence of global weak solutions for medium size initial data in large function spaces is proved. Moreover, exponential decay results towards the equilibrium state are established, where the decay rate can be estimated by explicit constants depending on the physical parameters of the system. Eventually, it is shown that if the initial datum satisfies additional (low order) Sobolev regularity, we can propagate Sobolev regularity for the corresponding solution. The proofs are based on a priori energy estimates in Wiener and Sobolev spaces.


Muskat problem moving interfaces two-phase thin film approximation free-boundary problems stokes flow 

Mathematics Subject Classification

35K25 35D30 35R35 35Q35 76B03 



RGB was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). GB recognizes the support of Grant No. 250070 of the Research Council of Norway. Part of the research leading to results presented here was conducted during a short stay of GB at Institut Camille Jordan under Projects DYFICOLTI and ANR-13-BS01-0003-01 support.

Compliance with ethical standards

Conflict of interest

The authors declares that there is no conflict of interest.


  1. 1.
    Ambrose, D.M.: Well-posedness of two-phase Hele–Shaw flow without surface tension. Eur. J. Appl. Math. 15(5), 597–607 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beretta, E., Bertsch, M., Dal Passo, R.: Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Ration. Mech. Anal. 129(2), 175–200 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bernis, F., Friedman, A.: Higher order nonlinear degenerate parabolic equations. J. Differ. Equ. 83(1), 179–206 (1990)MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Bertozzi, A.L., Pugh, M.C.: The lubrication approximation for thin viscous films : regularity and long-time behavior of weak solutions. Commun. Pure Appl. Math. 49(2), 85–123 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Castro, A., Cordoba, D., Fefferman, R., Gancedo, F., Lopez-Fernandez, M.: Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175, 909–948 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cerminara, M., Fasano, A.: Modelling the dynamics of a geothermal reservoir fed by gravity driven flow through overstanding saturated rocks. J. Volcanol. Geotherm. Res. 233, 37–54 (2012)CrossRefADSGoogle Scholar
  7. 7.
    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. 138(6), 1455–1494 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Constantin, P., Cordoba, D., Gancedo, F., Strain, R.M.: On the global existence for the Muskat problem. J. Eur. Math. Soc. 15, 201–227 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Constantin, P., Gancedo, F., Shvydkoy, R., Vicol, V.: Global regularity for 2D Muskat equations with finite slope. To appear in Annales de l’Institut Henri Poincare (C) Non Linear Analysis (2016)Google Scholar
  10. 10.
    Cordoba, D., Gancedo, F.: Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Commun. Math. Phys. 273(2), 445–471 (2007)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Escher, J., Laurençot, Ph, Matioc, B.-V.: Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(4), 583–598 (2011)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Escher, J., Matioc, A.-V., Matioc, B.-V.: A generalized Rayleigh-Taylor condition for the Muskat problem. Nonlinearity 25(1), 73–92 (2012)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Escher, J., Matioc, A.-V., Matioc, B.-V.: Modelling and analysis of the Muskat problem for thin fluid layers. J. Math. Fluid Mech. 14(2), 267–277 (2012)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Escher, J., Matioc, A.-V., Matioc, B.-V.: Thin-film approximations of the two-phase Stokes problem. Nonlinear Anal. Theory Methods Appl. 76, 1–13 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Escher, J., Matioc, B.-V.: On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results. Zeitschrift für Analysis und ihre Anwendungen 30(2), 193–218 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Escher, J., Matioc, B.-V.: Existence and stability of solutions for a strongly coupled system modelling thin fluid films. NoDEA Nonlinear Differ. Equ. Appl. 20, 1–17 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Escher, J., Matioc, B.-V.: Non-negative global weak solutions for a degenerated parabolic system approximating the two-phase Stokes problem. J. Differ. Equ. 256(8), 2659–2676 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Escher, J., Matioc, B.-V., Walker, C.: The domain of parabolicity for the Muskat problem. Indiana Univ. Math. J. 67, 679–737 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gancedo, F., Garcia-Juarez, E., Patel, N., Strain, R.M.: On the Muskat problem with viscosity jump: global in time results (2017). preprint arXiv:1710.11604
  20. 20.
    Gancedo, F., Strain, R.M.: Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci. 111(2), 635–639 (2014)MathSciNetCrossRefADSGoogle Scholar
  21. 21.
    Gaver, D.P., Grotberg, J.B.: The dynamics of a localized surfactant on a thin film. J. Fluid Mech. 214, 127–148 (1990)CrossRefADSGoogle Scholar
  22. 22.
    Kawarada, H., Koshigoe, H.: Unsteady flow in porous media with a free surface. Jpn. J. Ind. Appl. Math. 8(1), 41–84 (1991)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Laurençot, Ph, Matioc, B.-V.: A gradient flow approach to a thin film approximation of the Muskat problem. Calc. Var. Partial Differ. Equ. 47(1–2), 319–341 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Laurençot, Ph, Matioc, B.-V.: A thin film approximation of the Muskat problem with gravity and capillary forces. J. Math. Soc. Jpn. 66(4), 1043–1071 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Laurençot, Ph, Matioc, B.-V.: Finite speed of propagation and waiting time for a thin-film Muskat problem. Proc. R. Soc. Edinb. Sect. A Math. 147(4), 813–830 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Laurençot, Ph, Matioc, B.-V.: Self-similarity in a thin film Muskat problem. SIAM J. Math. Anal. 49(4), 2790–2842 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Matioc, B.-V.: Non-negative global weak solutions for a degenerate parabolic system modelling thin films driven by capillarity. Proc. R. Soc. Edinb. Sect. A Math. 142(5), 1071–1085 (2012)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Muskat, M.: The flow of fluids through porous media. J. Appl. Phys. 8(4), 274–282 (1937)CrossRefADSGoogle Scholar
  29. 29.
    Pruess, J., Simonett, G.: On the Muskat flow. Evol. Equ. Control Theory 5, 631–645 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Siegel, M., Caflisch, R.E., Howison, S.: Global existence, singular solutions, and ill-posedness for the Muskat problem. Commun. Pure Appl. Math. 57(10), 1374–1411 (2004)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Simon, J.: Compact sets in the space \(L^{p}(O, T; B)\). Annali di Matematica Pura ed Applicata 146(1), 65–96 (1986)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain

Personalised recommendations