Archive for Rational Mechanics and Analysis

, Volume 217, Issue 1, pp 155–230 | Cite as

Homogenization of the Hele-Shaw Problem in Periodic Spatiotemporal Media

  • Norbert Požár


We consider the homogenization of the Hele-Shaw problem in periodic media that are inhomogeneous both in space and time. After extending the theory of viscosity solutions into this context, we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous Hele-Shaw-type problem with a general, possibly nonlinear dependence of the free boundary velocity on the gradient. Moreover, the free boundaries converge locally uniformly in Hausdorff distance.


Free Boundary Viscosity Solution Boundary Data Hausdorff Distance Comparison Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Antontsev S.N., Gonçalves C.R., Meirmanov A.M.: Local existence of classical solutions to the well-posed Hele-Shaw problem. Port. Math. (N.S.) 59, 435–452 (2002)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Athanasopoulos I., Caffarelli L., Salsa S.: Regularity of the free boundary in parabolic phase-transition problems. Acta Math. 176, 245–282 (1996) doi: 10.1007/BF02551583 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baiocchi C.: Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl. 92(4), 107–127 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Barles G., Souganidis P.E.: A new approach to front propagation problems: theory and applications. Arch. Rational Mech. Anal. 141, 237–296 (1998) doi: 10.1007/s002050050077 CrossRefADSzbMATHMathSciNetGoogle Scholar
  5. 5.
    Caffarelli L.A.: A Harnack inequality approach to the regularity of free boundaries I, Lipschitz free boundaries are C 1,α. Rev. Mat. Iberoamericana. 3, 139–162 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Caffarelli L.A., Monneau R.: Counter-example in three dimension and homogenization of geometric motions in two dimension. Arch. Ration. Mech. Anal. 212, 503–574 (2014) doi: 10.1007/s00205-013-0712-y CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Caffarelli, L., Salsa, S.: A geometric approach to free boundary problems. Graduate Studies in Mathematics. vol. 68, American Mathematical Society, Providence, 2005Google Scholar
  8. 8.
    Caffarelli L.A., Souganidis P.E., Wang L.: Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. 58, 319–361 (2005) doi: 10.1002/cpa.20069 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Caffarelli, L., Vazquez, J.L.: Viscosity solutions for the porous medium equation. In: Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, pp. 13–26. American Mathematical Society, Providence 1999Google Scholar
  10. 10.
    Caselles V., Chambolle A.: Anisotropic curvature-driven flow of convex sets. Nonlinear Anal. 65, 1547–1577 (2006) doi: 10.1016/ CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1–67 (1992) doi: 10.1090/S0273-0979-1992-00266-5 CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Elliott C.M., Janovský V.: A variational inequality approach to Hele-Shaw flow with a moving boundary. Proc. Roy. Soc. Edinburgh Sect. A 88, 93–107 (1981) doi: 10.1017/S0308210500017315 CrossRefzbMATHGoogle Scholar
  13. 13.
    Evans L.C.: Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 120, 245–265 (1992) doi: 10.1017/S0308210500032121 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hele-Shaw H.S.: The flow of water. Nature 58, 34–36 (1898)CrossRefADSGoogle Scholar
  15. 15.
    Ibrahim H., Monneau R.: On the rate of convergence in periodic homogenization of scalar first-order ordinary differential equations. SIAM J. Math. Anal. 42, 2155–2176 (2010) doi: 10.1137/080738830 CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ishii H.: Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55, 369–384 (1987) doi: 10.1215/S0012-7094-87-05521-9 CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Jikov, V.V., Kozlov, S.M., Olenik, O.A.: Homogenization of differential operators and integral functionals. In: Yosifian, G.A. Translated from the Russian. Springer, Berlin, 1994Google Scholar
  18. 18.
    Kim I.C.: Uniqueness and existence results on the Hele-Shaw and the Stefan problems. Arch. Ration. Mech. Anal. 168, 299–328 (2003) doi: 10.1007/s00205-003-0251-z CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kim I.C.: Regularity of the free boundary for the one phase Hele-Shaw problem. J. Differ. Equ. 223, 161–184 (2006) doi: 10.1016/j.jde.2005.07.003 CrossRefADSzbMATHGoogle Scholar
  20. 20.
    Kim I.C.: Homogenization of the free boundary velocity. Arch. Ration. Mech. Anal. 185, 69–103 (2007) doi: 10.1007/s00205-006-0035-3 CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Kim I.C.: Homogenization of a model problem on contact angle dynamics. Comm. Partial Differential Equations 33, 1235–1271 (2008) doi: 10.1080/03605300701518273 CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Kim I.C.: Error estimates on homogenization of free boundary velocities in periodic media. Ann. Inst. H. Poincaré Anal Non Linéaire. 26, 999–1019 (2009) doi: 10.1016/j.anihpc.2008.10.004 CrossRefADSzbMATHGoogle Scholar
  23. 23.
    Kim I.C., Mellet A.: Homogenization of a Hele-Shaw problem in periodic and random media. Arch. Ration. Mech. Anal. 194, 507–530 (2009) doi: 10.1007/s00205-008-0161-1 CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Kim I.C., Mellet A.: Homogenization of one-phase Stefan-type problems in periodic and random media. Trans. Am. Math. Soc. 362, 4161–4190 (2010) doi: 10.1090/S0002-9947-10-04945-7 CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kim I.C., Požár N.: Viscosity solutions for the two-phase Stefan problem. Comm. Partial Differential Equations. 36, 42–66 (2011) doi: 10.1080/03605302.2010.526980 CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Kim I.C., Požár N.: Nonlinear elliptic-parabolic problems. Arch. Ration. Mech. Anal. 210, 975–1020 (2013) doi: 10.1007/s00205-013-0663-3 CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Lions P.L., Papanicolaou G., Varadhan S.R.S. Homogenization of Hamilton–Jacobi equations, unpublishedGoogle Scholar
  28. 28.
    Louro B., Rodrigues J.-F.: Remarks on the quasisteady one phase Stefan problem. Proc. Roy. Soc. Edinburgh Sect. A. 102, 263–275 (1986) doi: 10.1017/S0308210500026354 CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Piccinini L.C.: Homogeneization problems for ordinary differential equations. Rend. Circ. Mat. Palermo 27(2), 95–112 (1978) doi: 10.1007/BF02843869 CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Požár N.: Long-time behavior of a Hele-Shaw type problem in random media. Interfaces Free Bound. 13, 373–395 (2011)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Primicerio M.: Stefan-like problems with space-dependent latent heat. Meccanica 5, 187–190 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Primicerio, M., Rodrigues, J.-F.: The Hele-Shaw problem with nonlocal injection condition. In: Nonlinear mathematical problems in industry, II, Iwaki, 1992, GAKUTO Int. Ser. Math. Sci. Appl., vol. 2, pp. 375–390. Gakkōtosho, Tokyo, 1993Google Scholar
  33. 33.
    Richardson S.: Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618 (1972)CrossRefADSzbMATHGoogle Scholar
  34. 34.
    Rodrigues J.-F. Obstacle problems in mathematical physics. Notas de Matemática [Mathematical Notes], 114. North-Holland Mathematics Studies, vol. 134, North-Holland Publishing Co., Amsterdam, 1987Google Scholar
  35. 35.
    Roubíček T.: The Stefan problem in heterogeneous media. Ann. Inst. H. Poincaré Anal. Non Linéaire. 6, 481–501 (1989)zbMATHGoogle Scholar
  36. 36.
    Schwab R.W.: Stochastic homogenization of Hamilton–Jacobi equations in stationary ergodic spatio-temporal media. Indiana Univ. Math. J. 58, 537–581 (2009) doi: 10.1512/iumj.2009.58.3455 CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Souganidis P.E.: Stochastic homogenization of Hamilton–Jacobi equations and some applications. Asymptot. Anal. 20, 1–11 (1999)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Steinbach, J.: A variational inequality approach to free boundary problems with applications in mould filling. International Series of Numerical Mathematics, vol. 136, Birkhäuser, Basel, 2002Google Scholar
  39. 39.
    Tartar, L. The general theory of homogenization. Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer, Berlin, 2009Google Scholar
  40. 40.
    Wang L.: On the regularity theory of fully nonlinear parabolic equations. I. Comm. Pure Appl. Math. 45, 27–76 (1992) doi: 10.1002/cpa.3160450103 CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan
  2. 2.Faculty of Mathematics and Physics,Institute of Science and EngineeringKanazawa UniversityKanazawaJapan

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