Nonlinear Diffusion with Fractional Laplacian Operators

  • Juan Luis VázquezEmail author
Part of the Abel Symposia book series (ABEL, volume 7)


We describe two models of flow in porous media including nonlocal (long-range) diffusion effects. The first model is based on Darcy’s law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that these special solutions describe the asymptotic behavior of a wide class of solutions.

The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1-contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity.


Porous Medium Obstacle Problem Fractional Diffusion Porous Medium Equation Schwartz Class 
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.



Author partially supported by Spanish Project MTM2008-06326-C02. The author is grateful to the referee and N. Guillén for carefully reading the manuscript.


  1. 1.
    Abe, S., Thurner, S.: Anomalous diffusion in view of Einstein’s 1905 theory of Brownian motion. Physica A 356(2–4), 403–407 (2005) CrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Mainini, E., Serfaty, S.: Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28(2), 217–246 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Serfaty, S.: A gradient flow approach to an evolution problem arising in superconductivity. Commun. Pure Appl. Math. 61(11), 1495–1539 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 116. Cambridge University Press, Cambridge (2009) zbMATHCrossRefGoogle Scholar
  5. 5.
    Aronson, D.G.: The porous medium equation. In: Nonlinear Diffusion Problems Montecatini Terme, 1985. Lecture Notes in Math., vol. 1224, pp. 1–46. Springer, Berlin (1986) CrossRefGoogle Scholar
  6. 6.
    Athanasopoulos, I., Caffarelli, L.A.: Optimal regularity of lower dimensional obstacle problems. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004); Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35(34), 49–66, 226; translation in J. Math. Sci. (N.Y.) 132(3), 274–284 (2006) Google Scholar
  7. 7.
    Athanasopoulos, I., Caffarelli, L.A.: Continuity of the temperature in boundary heat control problem. Adv. Math. 224(1), 293–315 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Athanasopoulos, I., Caffarelli, L.A., Salsa, S.: The structure of the free boundary for lower dimensional obstacle problems. Am. J. Math. 130(2), 485–498 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bachelier, L.: Théorie de la spéculation. Ann. Sci. Éc. Norm. Super. 3(17), 21–86 (1900) MathSciNetGoogle Scholar
  10. 10.
    Barenblatt, G.I.: On self-similar motions of a compressible fluid in a porous medium. Akad. Nauk SSSR, Prikl. Mat. Meh. 16, 679–698 (1952) (in Russian) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bass, R.F., Kassmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc. 357(2), 837–850 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375–388 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bendahmane, M., Karlsen, K.H.: Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations. SIAM J. Math. Anal. 36(2), 405–422 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bertoin, J.: Lévy processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996). ISBN: 0-521-56243-0 zbMATHGoogle Scholar
  15. 15.
    Bertozzi, A.L., Laurent, T.: The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels. Chin. Ann. Math., Ser. B 30(5), 463–482 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bertozzi, A.L., Carrillo, J.L., Laurent, T.: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22(3), 683–710 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bertozzi, A.L., Laurent, T., Rosado, J.: L p theory for the multidimensional aggregation equation. Commun. Pure Appl. Math. 64(1), 45–83 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Biler, P., Imbert, C., Karch, G.: Fractal porous media equation. arXiv:1001.0910
  19. 19.
    Biler, P., Karch, G., Monneau, R.: Nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Phys. 294(1), 145–168 (2010). MR2575479 MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95(2), 263–273 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Bonforte, M., Dolbeault, J., Grillo, G., Vázquez, J.L.: Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities. Proc. Natl. Acad. Sci. USA 107(38), 16459–16464 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Caffarelli, L.A.: Further regularity for the Signorini problem. Commun. Partial Differ. Equ. 4, 1067–1075 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Caffarelli, L.A., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171, 1903–1930 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Caffarelli, L.A., Vázquez, J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011). doi: 10.1007/s00205-011-0420-4. arXiv:1001.0410v2. MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Caffarelli, L.A., Vázquez, J.L.: Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst., Ser. A 29(4), 1393–1404 (2011). A special issue “Trends and Developments in DE/Dynamics, Part III” zbMATHGoogle Scholar
  27. 27.
    Caffarelli, L.A., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian. Invent. Math. 171(2), 425–461 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Caffarelli, L., Chan, C.-H., Vasseur, A.: Regularity theory for nonlinear integral operators. J. Am. Math. Soc. 24, 849–869 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Caffarelli, L.A., Soria, F., Vázquez, J.L.: Regularity of solutions of the fractional porous medium flow (in preparation) Google Scholar
  30. 30.
    Carrillo, J.A., Toscani, G.: Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113–141 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Cifani, S., Jakobsen, E.R.: Entropy solution theory for fractional degenerate convection-diffusion equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28(3), 413–441 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Chapman, J.S., Rubinstein, J., Schatzman, M.: A mean-field model for superconducting vortices. Eur. J. Appl. Math. 7(2), 97–111 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    De Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 3, 25–43 (1957) MathSciNetGoogle Scholar
  34. 34.
    Denzler, J., McCann, R.: Phase transitions and symmetry breaking in singular diffusion. Proc. Natl. Acad. Sci. USA 100, 6922–6925 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    De Pablo, A., Quirós, F., Rodriguez, A., Vázquez, J.L.: A fractional porous medium equation. Adv. Math. 226(2), 1378–1409 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    De Pablo, A., Quirós, F., Rodriguez, A., Vázquez, J.L.: A general fractional porous medium equation. arXiv:1104.0306v1 [math.AP]. Commun. Pure Appl. Math. (2011, to appear)
  37. 37.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Preprint (2011) Google Scholar
  38. 38.
    E, W.: Dynamics of vortex-liquids in Ginzburg–Landau theories with applications to superconductivity. Phys. Rev. B 50(3), 1126–1135 (1994) MathSciNetCrossRefGoogle Scholar
  39. 39.
    Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme gefordete Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. (Leipz.) 17, 549–560 (1905). English translation: Investigations on the Theory of Brownian Movement. Dover, New York (1956) zbMATHCrossRefGoogle Scholar
  40. 40.
    Friedman, A., Kamin, S.: The asymptotic behavior of gas in an N-dimensional porous medium. Trans. Am. Math. Soc. 262, 551–563 (1980). MR0586735 (81j:35054) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition zbMATHGoogle Scholar
  43. 43.
    Head, A.K.: Dislocation group dynamics II. Similarity solutions of the continuum approximation. Philos. Mag. 26, 65–72 (1972) CrossRefGoogle Scholar
  44. 44.
    Kassmann, M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. 34, 1–21 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167(3), 445–453 (2007). MR2276260 MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Jara, M.: Hydrodynamic limit of particle systems with long jumps. Preprint.
  47. 47.
    Jara, M.: Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps. Commun. Pure Appl. Math. 62(2), 198–214 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Jara, M., Komorowski, T., Olla, S.: Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19(6) (2009) 2270–2300 MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Elliptic Type. Nauka, Moscow (1964). Academic Press, New York (1968) (in Russian). MR 0244627 (39:5941) zbMATHGoogle Scholar
  50. 50.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Monographs, vol. 23. Am. Math. Soc., Providence (1968) Google Scholar
  51. 51.
    Landkof, N.S.: Foundations of Modern Potential Theory. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer, New York (1972). Translated from the Russian by A.P. Doohovskoy zbMATHCrossRefGoogle Scholar
  52. 52.
    Lee, K.A., Vázquez, J.L.: Geometrical properties of solutions of the porous medium equation for large times. Indiana Univ. Math. J. 52(4), 991–1016 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Lin, F.H., Zhang, P.: On the hydrodynamic limit of Ginzburg–Landau vortices. Discrete Contin. Dyn. Syst. 6, 121–142 (2000) MathSciNetzbMATHGoogle Scholar
  54. 54.
    Lions, P.L., Mas-Gallic, S.: Une méthode particulaire déterministe pour des équations diffusives non linéaires. C. R. Acad. Sci. Paris, Sér. I 332, 369–376 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Preprint.
  56. 56.
    Oleinik, O.A., Kalashnikov, A.S., Chzou, Y.-I.: The Cauchy problem and boundary problems for equations of the type of unsteady filtration. Izv. Akad. Nauk SSSR, Ser. Mat. 22, 667–704 (1958) MathSciNetzbMATHGoogle Scholar
  57. 57.
    Peletier, L.A.: The porous media equation. In: Amann, H. (ed.) Application of Nonlinear Analysis in the Physical Sciences, pp. 229–242. Pitman, London (1981) Google Scholar
  58. 58.
    Serfaty, S., Vázquez, J.L.: Work in preparation Google Scholar
  59. 59.
    Signorini, A.: Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. Appl. 18, 95–139 (1959) MathSciNetzbMATHGoogle Scholar
  60. 60.
    Silvestre, L.E.: Hölder estimates for solutions of integro differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Silvestre, L.E.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 6–112 (2007) MathSciNetCrossRefGoogle Scholar
  62. 62.
    Smoluchowski, M.: Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. 21, 756–780 (1906) (in German). Previously, Bull. Int. Acad. Sci. Cracovie, 46 A (1906) (in Polish) zbMATHCrossRefGoogle Scholar
  63. 63.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970). MR0290095 (44 #7280) zbMATHGoogle Scholar
  64. 64.
    Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. 49, 33–44 (2009) MathSciNetzbMATHGoogle Scholar
  65. 65.
    Vázquez, J.L.: Asymptotic behaviour for the Porous Medium Equation posed in the whole space. J. Evol. Equ. 3, 67–118 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Vázquez, J.L.: Asymptotic behaviour for the PME in a bounded domain. The Dirichlet problem. Monatshefte Math. 142(1–2), 81–111 (2004) zbMATHCrossRefGoogle Scholar
  67. 67.
    Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and Its Applications, vol. 33. Oxford University Press, Oxford (2006) zbMATHCrossRefGoogle Scholar
  68. 68.
    Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007) zbMATHGoogle Scholar
  69. 69.
    Vázquez, J.L.: Perspectives in nonlinear diffusion. Between analysis, physics and geometry. In: Sanz-Solé, M., et al. (eds.) Proceedings of the International Congress of Mathematicians, ICM Madrid 2006, vol. 1, pp. 609–634. Eur. Math. Soc. Pub. House, Zurich (2007) CrossRefGoogle Scholar
  70. 70.
    Villani, C.: Topics in Optimal Transportation. Am. Math. Soc., Providence (2003) zbMATHGoogle Scholar
  71. 71.
    Vlahos, L., Isliker, H., Kominis, Y., Hizonidis, K.: Normal and anomalous Diffusion: a tutorial. In: Bountis, T. (ed.) Order and Chaos, vol. 10. Patras University Press, Patras (2008) Google Scholar
  72. 72.
    Weitzner, H., Zaslavsky, G.M.: Some applications of fractional equations. Chaotic transport and complexity in classical and quantum dynamics. Commun. Nonlinear Sci. Numer. Simul. 8(3–4), 273–281 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Zel’dovich, Ya.B., Kompaneets, A.S.: Towards a theory of heat conduction with thermal conductivity depending on the temperature. In: Collection of Papers Dedicated to 70th Anniversary of A.F. Ioffe, pp. 61–72. Izd. Akad. Nauk SSSR, Moscow (1950) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Dpto. de MatemáticasUniversidad Autónoma de MadridMadridSpain

Personalised recommendations