Kinetic approach to long time behavior of linearized fast diffusion equations

Journal of Statistical Physics

Volumn 128, Issue 4, 2007, Pages 883-925

  Cáceres, María J ^a   Toscani, Giuseppe ^b  

^a UNIVERSITY OF GRANADA   (Spain)

^b UNIVERSITY OF PAVIA   (Italy)

Author keywords

Fast diffusion equation; Maxwell models

Indexed keywords

EID: 36148942468     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-007-9329-6     Document Type: Article

References (33)

1. A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi, Kinetic models of inelastic gases. Mat. Mod. Meth. Appl. Sci. 12:965-983 (2002).
2. M. Bisi, J. A. Carrillo and G. Toscani, Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equilibria. J. Statist. Phys. 118:301-331 (2005).
3. M. Bisi, J. A. Carrillo and G. Toscani, Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model. J. Statist. Phys. 124(2-4):625-653 (2006).
4. A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules. Sov. Sci. Rev. 7:111-233 (1988).
5. E. A. Carlen, E. Gabetta and G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 199:521-546 (1999).
6. J. A. Carrillo, M. Di Francesco and G. Toscani, Strict Contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering. Proc. Amer. math. Soc. 135(2):353-363 (2007).
7. J. A. Carrillo, M. P. Gualdani and G. Toscani, Finite speed of propagation in porous media by mass transportation methods. C. R. Math. Acad. Sci. Paris 338(10):815-818 (2004).
8. J. A. Carrillo, C. Lederman, P. Markowich and G. Toscani, Poincaré Inequalities for Linearization of Very Fast Diffusion Equations, Nonlinearity 15:565-580 (2002).
9. J. A. Carrillo and G. Toscani, Exponential convergence toward equilibrium for homogeneous Fokker-planck type equations. Math. Meth Appl. Sci. 21:1269-1286 (1998).
10. 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49(1):113-142 (2000).
11. J. A. Carrillo and J. L. Vázquez, Fine asymptotics for fast diffusion equations. Comm. Partial Differential Equations 28:1023-1056 (2003).
12. C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases Springer Series in Applied Mathematical Sciences, Vol. 106, Springer-Verlag (1994).
13. D. Cordero-Erausquin and R. J. McCann, Accelerated diffusion to minimum entropy, personal communication (2005).
14. J. Denzler and R. J. McCann, Fast diffusion to self-similarity: complete spectrum, long time asymptotics and numerology. Arch. Rational Mech. Anal. 175:301-342 (2005).
15. L. Desvillettes About the regularizing properties of the non-cut-off Kac equation. Comm. Math. Phys. 168(2):417-440 (1995).
16. J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions. C. R. Acad. Sci. Paris Sér. I Math. 315(6):693-698 (1992).
17. E. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81:901-934 (1995).
18. T. Goudon, S. Junca and G. Toscani, Fourier-based distances and Berry-Esseen like inequalities for smooth densities. Monatsh. Math. 135(2):115-136 (2002).
19. Y. J. Kim and R. J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion. J. Math. Pures Appl. 86:42-67 (2006).
20. R. G. Laha and V. K. Rohatgi, Probability Theory, John Wiley and Sons, New York, (1979).
21. L. Landau, Die kinetische Gleichung für den Fall Coulombscher Wechselwirkung. Phys. Z. Sowjet. 10:154 (1936).
22. Trad. The transport equation in the case of Coulomb interactions, in D. ter Haar, (ed.) Collected papers of L.D. Landau, Pergamon Press, Oxford, p. 163-170 (1981)
23. R. J. McCann and D. Slepcev, Second-order asymptotics for the fast-diffusion equations,Int. Math. Res. Not. Art. ID 24947, 22 pp (2006).
24. L. Pareschi and G. Toscani Self-similarity and power-like tails in nonconservative kinetic models. J. Statist. Phys. 126(2-4):747-779 (2006).
25. A. Pulvirenti and G. Toscani, Asymptotic properties of the inelastic Kac model. J. Statist. Phys. 114:1453-1480 (2004).
26. G. Toscani, One-dimensional kinetic models with dissipative collisions. M2AN Math. Model. Numer. Anal. 34:1277-1292 (2000).
27. G. Toscani, A central limit theorem for solutions fo the porous medium equation. J. Evol. Equ. 5:185-203 (2005).
28. G. Toscani and C. Villani, Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas. J. Statist. Phys. 94:619-637 (1999).
29. J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space. J. Evol. Equ. 3:67-118.
30. J. L. Vázquez, Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium. Trans. Amer. Math. Soc. 277(2):507-527 (1983).
31. C. Villani, Contribution à l7'étude mathématique de séquations de Boltzmann et de Landau en théorie cinétique des gaz et des plasmas. PhD thesis, Univ. Paris-Dauphine (1998).
32. C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143:273-307 (1998).
33. T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations. Stud. Appl. Math. 100(2):153-193 (1998).