Nonlinear diffusion with fractional laplacian operators

Nonlinear Partial Differential Equations: The Abel Symposium 2010

Volumn , Issue , 2012, Pages 271-298

  Vázquez, Juan Luis ^a  

^a UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

ELLIPTIC OBSTACLE PROBLEM; EXISTENCE OF SOLUTIONS; FLOW IN POROUS MEDIA; FRACTIONAL DERIVATIONS; FRACTIONAL LAPLACIAN; FRACTIONAL LAPLACIAN OPERATORS; SELF-SIMILAR SOLUTION; SPEED OF PROPAGATION;

MATHEMATICAL OPERATORS; PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS;

LAPLACE TRANSFORMS;

EID: 84880861203     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1007/978-3-642-25361-4_15     Document Type: Conference Paper

References (77)

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)
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)
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)
4. Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Studies in Advanced Mathematics, Vol. 116. Cambridge University Press, Cambridge (2009)
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)
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);
7. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35(34), 49-66, 226;
8. translation in J. Math. Sci. (N.Y.) 132(3), 274-284 (2006)
9. Athanasopoulos, I., Caffarelli, L.A.: Continuity of the temperature in boundary heat control problem. Adv. Math. 224(1), 293-315 (2010)
10. 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)
11. Bachelier, L.: Théorie de la spéculation. Ann. Sci. Éc. Norm. Super. 3(17), 21-86 (1900)
12. 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)
13. Bass, R.F., Kassmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc. 357(2), 837-850 (2005)
14. Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375-388 (2002)
15. Bendahmane, M., Karlsen, K.H.: Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations. SIAM J. Math. Anal. 36(2), 405-422 (2004)
16. Bertoin, J.: Lévy processes. Cambridge Tracts in Mathematics, Vol. 121. Cambridge University Press, Cambridge (1996). ISBN: 0-521-56243-0
17. 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)
18. 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)
19. p theory for the multidimensional aggregation equation. Commun. Pure Appl. Math. 64(1), 45-83 (2011)
20. Biler, P., Imbert, C., Karch, G.: Fractal porous media equation. arXiv:1001.0910
21. Biler, P., Karch, G., Monneau, R.: Nonlinear diffusion of dislocation density and self-similar solutions. Commun. Math. Phys. 294(1), 145-168 (2010). MR2575479
22. Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95(2), 263-273 (1960)
23. 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)
24. Caffarelli, L.A.: Further regularity for the Signorini problem. Commun. Partial Differ. Equ. 4, 1067-1075 (1979)
25. Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7-9), 1245-1260 (2007)
26. Caffarelli, L.A., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasigeostrophic equation. Ann. Math. 171, 1903-1930 (2010)
27. 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.
28. 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"
29. 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)
30. Caffarelli, L., Chan, C.-H., Vasseur, A.: Regularity theory for nonlinear integral operators. J. Am. Math. Soc. 24, 849-869 (2011)
31. Caffarelli, L.A., Soria, F., Vázquez, J.L.: Regularity of solutions of the fractional porous medium flow (in preparation)
32. 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113-141 (2000)
33. 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)
34. Chapman, J.S., Rubinstein, J., Schatzman, M.: A mean-field model for superconducting vortices. Eur. J. Appl. Math. 7(2), 97-111 (1996)
35. 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)
36. Denzler, J., McCann, R.: Phase transitions and symmetry breaking in singular diffusion. Proc. Natl. Acad. Sci. USA 100, 6922-6925 (2003)
37. De Pablo, A., Quirós, F., Rodriguez, A., Vázquez, J.L.: A fractional porous medium equation. Adv. Math. 226(2), 1378-1409 (2011)
38. 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)
39. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker's guide to the fractional Sobolev spaces. Preprint (2011)
40. E, W.: Dynamics of vortex-liquids in Ginzburg-Landau theories with applications to superconductivity. Phys. Rev. B 50(3), 1126-1135 (1994)
41. 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).
42. English translation: Investigations on the Theory of Brownian Movement. Dover, New York (1956)
43. 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)
44. Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75-90 (1961)
45. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition
46. Head, A.K.: Dislocation group dynamics II. Similarity solutions of the continuum approximation. Philos. Mag. 26, 65-72 (1972)
47. Kassmann, M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. 34, 1-21 (2009)
48. 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
49. Jara, M.: Hydrodynamic limit of particle systems with long jumps. Preprint. http://arxiv.org/abs/0805.1326
50. 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)
51. Jara, M., Komorowski, T., Olla, S.: Limit theorems for additive functionals of a Markov chain. Ann. Appl. Probab. 19(6) (2009) 2270-2300
52. 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)
53. 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)
54. 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
55. 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)
56. Lin, F.H., Zhang, P.: On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete Contin. Dyn. Syst. 6, 121-142 (2000)
57. 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)
58. Mellet, A., Mischler, S., Mouhot, C.: Fractional diffusion limit for collisional kinetic equations. Preprint. http://arxiv.org/abs/0809.2455
59. 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)
60. 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)
61. Serfaty, S., Vázquez, J.L.: Work in preparation
62. Signorini, A.: Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. Appl. 18, 95-139 (1959)
63. Silvestre, L.E.: Hölder estimates for solutions of integro differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155-1174 (2006)
64. 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)
65. Smoluchowski, M.: Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. 21, 756-780 (1906) (in German).
66. Previously, Bull. Int. Acad. Sci. Cracovie, 46 A (1906) (in Polish)
67. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, Vol. 30. Princeton University Press, Princeton (1970). MR0290095 (44 #7280)
68. Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. 49, 33-44 (2009)
69. Vázquez, J.L.: Asymptotic behaviour for the Porous Medium Equation posed in the whole space. J. Evol. Equ. 3, 67-118 (2003)
70. Vázquez, J.L.: Asymptotic behaviour for the PME in a bounded domain. The Dirichlet problem. Monatshefte Math. 142(1-2), 81-111 (2004)
71. 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)
72. Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2007)
73. 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)
74. Villani, C.: Topics in Optimal Transportation. Am. Math. Soc., Providence (2003)
75. 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)
76. 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)
77. 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)