Darcy’s Flow with Prescribed Contact Angle: Well-Posedness and Lubrication Approximation

Archive for Rational Mechanics and Analysis

Volumn 218, Issue 2, 2015, Pages 589-646

  Knüpfer, Hans ^a   Masmoudi, Nader ^b  

^a UNIVERSITY OF HEIDELBERG   (Germany)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84938553281     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-015-0868-8     Document Type: Article

References (46)

1. Ambrose, D.M.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Eur. J. Appl. Math. 15(5), 597–607 (2004). y:10.1017/S0956792504005662
2. Ambrose D.M., Masmoudi N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 1287–1315 (2005)
3. Angenent S.: Analyticity of the interface of the porous media equation after the waiting time. Proc. Am. Math. Soc. 102(2), 329–336 (1988)
4. Bazaliy B.V., Friedman A.: The Hele-Shaw problem with surface tension in a half-plane. J. Differ. Equ. 216(2), 439–469 (2005)
5. Bazaliy B.V., Friedman A.: The Hele-Shaw problem with surface tension in a half-plane: a model problem. J. Differ. Equ. 216(2), 387–438 (2005)
6. Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York, 1972
7. 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)
8. Bertozzi A.L., Pugh M.: The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions. Commun. Pure Appl. Math. 49(2), 85–123 (1996)
9. Bertsch M., Giacomelli L., Karali G.: Thin-film equations with “partial wetting” energy: existence of weak solutions. Phys. D 209(1–4), 17–27 (2005)
10. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Lopez-Fernandez, M.: Rayleigh–Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. (2)175(2), 909–948 (2012). doi:10.4007/annals.2011.173.1.10
11. Constantin P., Pugh M.: Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6(3), 393–415 (1993)
12. Córdoba, A., Córdoba, D., Gancedo, F.: Interface evolution: the Hele-Shaw and Muskat problems. Ann. Math. (2)173(1), 477–542 (2011). doi:10.4007/annals.2011.173.1.10
13. Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007). doi:10.1090/S0894-0347-07-00556-5
14. Daskalopoulos P., Hamilton R.: Regularity of the free boundary for the porous medium equation. J. Am. Math. Soc. 11(4), 899–965 (1998)
15. Duchon, J., Robert, R.: Évolution d’une interface par capillarité et diffusion de volume. I. Existence locale en temps. Ann. Inst. H. Poincaré Anal. Non Linéaire1(5), 361–378 (1984)
16. Eidelman, S.D.: Parabolic Systems. Translated from the Russian by Scripta Technica, London. North-Holland Publishing Co., Amsterdam, 1969
17. Escher J., Simonett G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Differ. Equ. 2(4), 619–642 (1997)
18. Escher, J., Simonett, G.: Classical solutions for the quasi-stationary Stefan problem with surface tension. Differential Equations, Asymptotic Analysis, and Mathematical Physics (Potsdam, 1996), Math. Res., Vol. 100. Akademie, Berlin, 98–104, 1997
19. Germain, P., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. (2)175(2), 691–754 (2012)
20. Giacomelli, L., Gnann, M.V., Knüpfer, H., Otto, F.: Well-posedness for the Navier-slip thin-film equation in the case of complete wetting. J. Differ. Equ. 257(1), 15–81 (2014). doi:10.1016/j.jde.2014.03.010
21. Giacomelli, L., Knüpfer, H.: A free boundary problem of fourth order: classical solutions in weighted Hölder spaces. Comm. Partial Differ. Equ. 35(11), 2059–2091 (2010). doi:10.1080/03605302.2010.494262
22. Giacomelli L., Knüpfer H., Otto F.: Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equ. 245(6), 1454–1506 (2008)
23. Giacomelli L., Otto F.: Variational formulation for the lubrication approximation of the Hele-Shaw flow. Calc. Var. Partial Differ. Equ. 13(3), 377–403 (2001)
24. Giacomelli L., Otto F.: Rigorous lubrication approximation. Interfaces Free Bound. 5(4), 483–529 (2003)
25. Kawarada H., Koshigoe H.: Unsteady flow in porous media with a free surface. Jpn. J. Ind. Appl. Math. 8(1), 41–84 (1991)
26. Knüpfer H.: Navier slip thin-film equation for partial wetting. Commun. Pure Appl. Math. 64(9), 1263–1296 (2011)
27. Knüpfer, H., Masmoudi, N.: Well-posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge. Comm. Math. Phys. 320(2), 395–424 (2013). doi:10.1007/s00220-013-1708-z
28. Koch, H.: Non-euclidean singular integrals and the porous medium equation. Ph.D. thesis, Dortmund (1999)
29. Kozlov, V.A., Mazya, V.G., Rossmann, J.: Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs, Vol. 52. Am. Math. Soc., Providence, 1997
30. Krylov, N.V.: Lectures on elliptic and parabolic equations in Sobolev spaces. Graduate Studies in Mathematics, Vol. 96. American Mathematical Society, Providence, 2008
31. Lannes D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18(3), 605–654 (2005)
32. Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. (2)162(1), 109–194 (2005)
33. Masmoudi, N., Rousset, F.: Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations (2012). arXiv:1202.0657
34. McGeough J., Rasmussen H.: On the derivation of the quasi-steady model in electrochemical machining. J. Inst. Math. Appl. 13(1), 13–21 (1974)
35. Otto F.: Lubrication approximation with prescribed nonzero contact angle. Commun. Partial Differ. Equ. 23(11–12), 2077–2164 (1998)
36. Prokert G.: Existence results for Hele-Shaw flow driven by surface tension. Eur. J. Appl. Math. 9(2), 195–221 (1998)
37. Ren, W., Hu, D., Weinan, E.: Continuum models for the contact line problem. Phys. Fluids22(10), 102103 (2010)
38. Reynolds O.: On the theory of lubrication and its application to mr. beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil. Proc. R. Soc. Lond. 40, 191–203 (1886)
39. Rubenstein, L.I.: The Stefan problem. Translated from the Russian by A. D. Solomon. Translations of Mathematical Monographs, Vol. 27. American Mathematical Society, Providence, 1971
40. Shatah, J., Zeng, C.: Geometry and a priori estimates for free boundary problems of the Euler equation. Commun. Pure Appl. Math. 61(5), 698–744 (2008). doi:10.1002/cpa.20213
41. 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). doi:10.1002/cpa.20040
42. p(0, T; B). Ann. Math. Pura Appl. 146(4), 65–96 (1987). doi:10.1007/BF01762360
43. Tabeling, P.: Growth and Form: Nonlinear Aspe. Springer, New York, 1991
44. Wu, S.: Global wellposedness of the 3-D full water wave problem. Invent. Math. 184(1), 125–220 (2011). doi:10.1007/s00222-010-0288-1
45. Young, T.: An essay on the cohesion of fluids. Philos. Trans. R. Soc. Lond. 95, 65–87 (1805). http://www.jstor.org/stable/107159
46. Zhang, P., Zhang, Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61(7), 877–940 (2008). doi:10.1002/cpa.20226