Archive for Rational Mechanics and Analysis

, Volume 215, Issue 2, pp 497–529 | Cite as

The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions

  • Piotr Biler
  • Cyril ImbertEmail author
  • Grzegorz Karch


A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign-changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles—the well-known solutions of the classical porous medium equation.


Weak Solution Mild Solution Decay Estimate Differential Inequality Fourier Multiplier 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alikakos N.D.: An application of the invariance principle to reaction-diffusion equations. J. Differ. Equ. 33, 201–225 (1979)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Barles G., Chasseigne E., Imbert C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57, 213–246 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Biler, P., Imbert, C., Karch, G.: Finite speed of propagation for a non-local porous medium equation (preprint)Google Scholar
  4. 4.
    Biler P., Imbert C., Karch G.: Barenblatt profiles for a nonlocal porous medium equation. C. R., Math. Acad. Sci. Paris 349, 641–645 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Biler P., Karch G., Monneau R.: Nonlinear diffusion of dislocation density and self-similar solutions.. Commun. Math. Phys. 294, 145–168 (2010)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Caffarelli L., Vázquez J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Caffarelli L.A., Soria F., Vázquez J.L.: Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. (JEMS) 15, 1701–1746 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Caffarelli L.A., Vázquez J.L.: Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29, 1393–1404 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Carrillo J.A., Jüngel A., Markowich P.A., Toscani G., Unterreiter A.: Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133, 1–82 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Córdoba A., Córdoba D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys., 249, 511–528 (2004)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A fractional porous medium equation. Adv. Math. 226, 1378–1409 (2011)Google Scholar
  12. 12.
    de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A general fractional porous medium equation. Commun. Pure Appl. Math. 65, 1242–1284 (2012)Google Scholar
  13. 13.
    Droniou J., Imbert C.: Fractal first order partial differential equations. Arch. Ration. Mech. Anal. 182, 299–331 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Dyda B.: Fractional calculus for power functions and eigenvalues of the fractional laplacian. Fract. Calc. Appl. Anal. 15, 535–555 (2012)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Getoor R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Huang, Y.: Explicit Barenblatt profiles for fractional porous medium equations. arXiv:1312.0469 [math.AP]. (2013)
  17. 17.
    Imbert C., Mellet A.: Existence of solutions for a higher order non-local equation appearing in crack dynamics.. Nonlinearity 24, 3487–3514 (2011)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Karch G., Miao C., Xu X.: On convergence of solutions of fractal Burgers equation toward rarefaction waves. SIAM J. Math. Anal. 39, 1536–1549 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Ladyzhenskaya, O., Solonnikov, V., Ural’tseva, N.: Linear and Quasi-Linear Equations of Parabolic Type. Translated from the Russian by S. Smith, p. 648. Translations of Mathematical Monographs. 23. American Mathematical Society (AMS), Providence, RI. XI, 1968Google Scholar
  20. 20.
    Liskevich, V.A., Semenov, Y.A.: Some problems on Markov semigroups. In: Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Vol. 11 of Math. Top., pp. 163–217. Akademie Verlag, Berlin, 1996Google Scholar
  21. 21.
    Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966Google Scholar
  22. 22.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences. Springer, New York, 1983Google Scholar
  23. 23.
    Rakotoson J.M., Temam R.: An optimal compactness theorem and application to elliptic-parabolic systems. Appl. Math. Lett. 14, 303–306 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, vol. 3 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, 1996Google Scholar
  25. 25.
    Stan, D., del Teso, F., Vázquez, J.L.: Finite and infinite speed of propagation for porous medium equations with fractional pressure. arXiv:1311.7007 [math.AP]. (2013)
  26. 26.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970Google Scholar
  27. 27.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press, Princeton Mathematical Series, No. 32, Princeton, 1971Google Scholar
  28. 28.
    Taylor, M.E.: Tools for PDE, vol. 81 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2000 (pseudodifferential operators, paradifferential operators, and layer potentials)Google Scholar
  29. 29.
    Taylor, M.E.: Partial Differential Equations. III: Nonlinear Equations, 2nd edn, xxii, p. 715. Applied Mathematical Sciences 117. Springer, New York, 2011Google Scholar
  30. 30.
    Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006 (equations of porous medium type)Google Scholar
  31. 31.
    Vázquez, J.L.: The Porous Medium Equation. Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007 (mathematical theory)Google Scholar
  32. 32.
    Vázquez, J.L.: Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. Preprint arXiv:1205.6332v1. (2012)
  33. 33.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995 (reprint of the second (1944) edition)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland
  2. 2.CNRS, UMR 8050Université Paris-Est CréteilCréteil CedexFrance

Personalised recommendations