Asymptotic behaviour of a porous medium equation with fractional diffusion

Discrete and Continuous Dynamical Systems

Volumn 29, Issue 4, 2011, Pages 1393-1404

  Caffarelli, Luis A ^a   Vázquez, Juan Luis ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

As ymptotic behavior; Fractional Laplacian; Obstacle problem; Porous medium equation

Indexed keywords

EID: 79954553666     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2011.29.1393     Document Type: Article

References (23)

1. I. Athanasopoulos, L. A. Caffarelli and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math., 130 (2008), 485-498.
2. G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 16 (1952), 679-698.
3. P. Biler, C. Imbert and G. Karch, Fractal porous medium equation, preprint.
4. P. Biler, G. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Comm. Math. Phys., 294 (2010), 145-168.
5. L. A. Caffarelli, The obstacle problem revisited, The Journal of Fourier Analysis and Applications, 4 (1998), 383-402.
6. L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075.
7. L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
8. L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqns., 32 (2007), 1245-1260.
9. L. A. Caffarelli, F. Soria and J. L. V́azquez, Regularity of solutions of the fractional porous medium flow, in preparation.
10. L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.
11. L. A. Caffarelli and J. L. V́azquez, Nonlinear porous medium flow with fractional potential pressure, arXiv:1001.0410.
12. J. A. Carrillo and G. Toscani, Asymptotic L-decay of solutions ofthe porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113-142.
13. J. A. Carrillo, A. Jngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82. (Pubitemid 33585637)
14. M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81 (2002), 847-875.
15. A. Friedman, "Variational Principles and Free Boundary Problems," Wiley, New York, 1982.
16. A. Friedman and S. Kamin, The asymptotic behavior of gas in an N-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.
17. D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics, Springer, Berlin, 2001. (reprint ofthe 1998 edition)
18. A. K. Head, Dislocation group dynamics II. Similarity solutions of the continuum approximation, Phil. Mag., 26 (1972), 65-72.
19. N. S. Landkof, "Foundations Of Modern Potential Theory," Die Grundlehren der mathematischen Wissenschaften, Band 180 (translated from the Russian by A. P. Doohovskoy), Springer, New York, 1972.
20. L. E. Silvestre, Ḧolder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.
21. E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
22. J. L. V́azquez, "The Porous Medium Equation. Mathematical Theory," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
23. J. L. V́azquez, "Smoothing And Decay Estimates For Nonlinear Diffusion Equations. Equations Of Porous Medium Type," Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006.