Potential theory and optimal convergence rates in fast nonlinear diffusion

Journal des Mathematiques Pures et Appliquees

Volumn 86, Issue 1, 2006, Pages 42-67

  Kim, Yong Jung ^a   McCann, Robert J ^b  

^a Korea Advanced Institute of Science and Technology (KAIST)   (South Korea)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Fast nonlinear diffusion; Large time convergence rate; Moments; Newtonian potential

Indexed keywords

EID: 33745714343     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2006.01.002     Document Type: Article

References (61)

1. Angenent S.B. Large time asymptotics for the porous medium equation. Nonlinear Diffusion Equations and Their Equilibrium States I. Math. Sci. Res. Inst. Publ. vol. 12 (1988), Springer-Verlag, New York 21-34
2. N. C. R. Acad. Sci. Paris Ser. A-B 288 (1979) A103-A105
3. ∞-regularity and concavity of flows in one-dimensional porous media. Arch. Rational Mech. Anal. 99 (1987) 329-348
4. Bakry D., and Emery M. Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math. vol. 1123 (1985), Springer-Verlag, New York 177-206
5. Barenblatt G.I. On some unsteady motions of a liquid or gas in a porous medium. Akad. Nauk SSSR Prikl. Mat. Mekh. 16 (1952) 67-78
6. J.A. Carrillo, M. DiFrancesco, G. Toscani, Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc., submitted for publication
7. Carrillo J.A., Jüngel A., Markowich P.A., Toscani G., and Unterreiter A. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133 (2001) 1-82
8. Carrillo J.A., Lederman C., Markowich P.A., and Toscani G. Poincaré inequalities for linearizations of very fast diffusion equations. Nonlinearity 15 (2002) 565-580
9. J.A. Carrillo, R.J. McCann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal., submitted for publication
10. 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49 (2000) 113-141
11. Carrillo J.A., and Vázquez J.L. Fine asymptotics for fast diffusion equations. Comm. Partial Differential Equations 28 (2003) 1023-1056
12. Chasseigne E., and Vázquez J.L. Theory of extended solutions for fast diffusion equations in optimal classes of data. Radiation from singularities. Arch. Rational Mech. Anal. 164 (2002) 133-187
13. Chayes J.T., Osher S.J., and Ralston J.V. On singular diffusion equations with applications to self-organized criticality. Comm. Pure Appl. Math. 46 (1993) 1363-1377
14. Dahlberg B.E.J., and Kenig C. Nonnegative solutions of the porous medium equation. Comm. Partial Differential Equations 9 (1984) 409-437
15. Dahlberg B.E.J., and Kenig C. Nonnegative solutions of generalized porous medium equations. Rev. Mat. Iberoamericana 2 (1986) 267-305
16. Dahlberg B.E.J., and Kenig C. Nonnegative solutions of fast diffusion equations. Rev. Mat. Iberoamericana 4 (1988) 11-29
17. Dahlberg B.E.J., and Kenig C. Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders. J. Amer. Math. Soc. 1 (1988) 401-412
18. Dahlberg B.E.J., and Kenig C. Weak solutions of the porous medium equation in a cylinder. Trans. Amer. Math. Soc. 336 (1993) 701-709
19. Dahlberg B.E.J., and Kenig C. Weak solutions of the porous medium equation. Trans. Amer. Math. Soc. 336 (1993) 711-725
20. Daskalopoulos P. The Cauchy problem for variable coefficient porous medium equations. Potential Anal. 7 (1997) 485-516
21. t = Δ log u in higher dimensions. Math. Ann. 131 (1999) 189-206
22. Daskalopoulos P., and del Pino M.A. Nonradial solvability structure of super-diffusive nonlinear parabolic equations. Trans. Amer. Math. Soc. 354 (2002) 1583-1599
23. ∞-regularity of interface in degenerated diffusion: a geometrical approach. Duke Math. J. 108 (2001) 295-327
24. Denzler J., and McCann R.J. Phase transitions and symmetry breaking in singular diffusion. Proc. Natl. Acad. Sci. USA 100 (2003) 6922-6925
25. Denzler J., and McCann R.J. Fast diffusion to self-similarity: complete spectrum, long time asymptotics, and numerology. Arch. Rational Mech. Anal. 175 (2005) 301-342
26. Dolbeault J., and del Pino M. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002) 847-875
27. ∞ intermediate asymptotics for scalar conservation laws. Asymptot. Anal. 41 (2005) 189-213
28. Esteban J.R., Rodríguez A., and Vázquez J.L. A nonlinear heat equation with singular diffusivity. Comm. Partial Differential Equations 13 (1988) 985-1039
29. Friedman A. Partial Differential Equations of Parabolic Type (1964), Prentice Hall, Englewood Cliffs, NJ
30. Friedman A., and Kamin S. The asymptotic behaviour of a gas in an n-dimensional porous medium. Trans. Amer. Math. Soc. 262 (1980) 551-563
31. Galaktionov V.A., and Peletier L.A. Asymptotic behavior near finite time extinction for the fast diffusion equation. Arch. Rational Mech. Anal. 139 (1997) 83-98
32. Galaktionov V.A., Peletier L.A., and Vazquez J.L. Asymptotics of the fast-diffusion equation with critical exponent. SIAM J. Math. Anal. 31 (2000) 1157-1174
33. m when 0 < m < 1. Trans. Amer. Math. Soc. 291 (1985) 145-158
34. Kamin (Kamenomostskaya) S. The asymptotic behavior of the solution of the filtration equation. Israel J. Math. 14 (1973) 76-87
35. Kamin S., and Vazquez J.L. Fundamental solutions and asymptotic behavior for the p-Laplacian equation. Rev. Mat. Iberoamericana 4 (1988) 339-354
36. Kamin S., and Vazquez J.L. Asymptotic behavior of the solutions of the porous medium equation with changing signs. SIAM J. Math. Anal. 22 (1991) 34-45
37. Kim Y.-J. Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to N-waves. J. Differential Equations 192 (2003) 202-224
38. Kim Y.-J., and McCann R.J. Sharp decay rates for the fastest conservative diffusions. C. R. Acad. Sci. Paris 341 (2005) 157-162
39. Kim Y.-J., and Ni W.-M. On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation. Indiana Univ. Math. J. 51 (2002) 727-752
40. King J. Self-similar behaviour for the equation of fast nonlinear diffusion. Phil. Trans. Roy. Soc. London A 343 (1993) 337-375
41. H. Koch, Non-Euclidean singular integrals and the porous medium equation, Habilitation thesis, Universität Heidelberg, 1999
42. Ladyženskaja O.A., Solonnikov V.A., and Ural'ceva N.N. Linear and Quasilinear Equations of Parabolic Type. (in Russian). Translated from the Russian by S. Smith. Transl. Math. Monogr. vol. 23 (1967), Amer. Math. Soc., Providence, RI
43. Lederman C., and Markowich P.A. On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass. Comm. Partial Differential Equations 28 (2001) 301-332
44. Lee K.-A., and Vázquez J.L. Geometrical properties of solutions of the porous medium equation for large times. Indiana Univ. Math. J. 52 (2003) 991-1016
45. Lieb E.H., and Loss M. Analysis. Grad. Stud. in Math. vol. 14 (1997), Amer. Math. Soc., Providence, RI
46. R.J. McCann, D. Slepčev, Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not., in press
47. Newman W. A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I. J. Math. Phys. 25 (1984) 3120-3123
48. Otto F. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001) 101-174
49. Otto F., and Villani C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361-400
50. Pattle R.E. Diffusion from an instantaneous point source with concentration dependent coefficient. Quart. J. Mech. Appl. Math. 12 (1959) 407-409
51. t - Δ φ (u) = 0 with initial datum a measure. Nonlinear Anal. 6 (1982) 175-187
52. Pierre M. Nonlinear fast diffusion with measures as data. Pitman Res. Notes Math. Ser. vol. 149 (1987) 179-188
53. (N - 2) / (N + 2). Indiana Univ. Math. J. 50 (2001) 612-628
54. Ralston J. A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II. J. Math. Phys. 25 (1984) 3124-3127
55. t = Δ φ{symbol} (u) et applications. C. R. Acad. Sci. Paris Sér. I Math. 295 (1982) 71-74
56. Vázquez J.L. Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium. Trans. Amer. Math. Soc. 277 (1983) 507-527
57. Vázquez J.L. An introduction to the mathematical theory of the Porous Medium Equation. Shape Optimization and Free Boundaries. Montreal, PQ, 1990. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. vol. 380 (1992), Kluwer Acad. Publ., Dordrecht 347-389
58. Vázquez J.L. Asymptotic behaviour for the porous medium equation posed in the whole space. J. Evol. Equations 3 (2003) 67-118
59. Vázquez J.L., and Zuazua E. Complexity of large time behaviour of evolution equations with bounded data. Chinese Ann. Math. Ser. B 23 (2002) 293-310
60. Witelski T.P., and Bernoff A.J. Self-similar asymptotics for linear and nonlinear diffusion equations. Stud. Appl. Math. 100 (1998) 153-193
61. Zel'dovich Ya.B., and Barenblatt G.I. The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations. Sov. Phys. Dokl. 3 (1958) 44-47