Funzionali entropia ed equilibrio di sistemi di molte particelle
Toscani, G.
Bollettino dell'Unione Matematica Italiana, Tome 1 (2008), p. 509-524 / Harvested from Biblioteca Digitale Italiana di Matematica

In this lecture, we discuss a number of questions related to entropy functionals and their use in the study of the asymptotic behavior of both kinetic and diffusion equations. Moreover, we introduce and discuss examples of the link between entropy production estimates and the field of functional inequalities.

Publié le : 2008-10-01
@article{BUMI_2008_9_1_3_509_0,
     author = {G. Toscani},
     title = {Funzionali entropia ed equilibrio di sistemi di molte particelle},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {1},
     year = {2008},
     pages = {509-524},
     zbl = {1191.35203},
     mrnumber = {2455328},
     language = {it},
     url = {http://dml.mathdoc.fr/item/BUMI_2008_9_1_3_509_0}
}
Toscani, G. Funzionali entropia ed equilibrio di sistemi di molte particelle. Bollettino dell'Unione Matematica Italiana, Tome 1 (2008) pp. 509-524. http://gdmltest.u-ga.fr/item/BUMI_2008_9_1_3_509_0/

[1] Arnold, A. - Markowich, P. A. - Toscani, G. - Unterreiter, A., On Convex Sobolev Inequalities and the Rate of Convergence to Equilibrium for Fokker-Planck Type Equations. Comm. Partial Differential Equations, 26 (2001), 43-100. | MR 1842428 | Zbl 0982.35113

[2] Arnold, A. - Markowich, P.A. - Toscani, G. - Unterreiter, A., On Generalized Csiszar-Kullback Inequalities. Monatsh. Math., 131 (2000), 235-253. | MR 1801751 | Zbl 1015.94003

[3] Bakry, D. - Emery, M., Diffusions hypercontractives. In Sém. Proba. XIX, 1123 Lecture Notes in Math. Springer, 1985, 177-206. | MR 889476

[4] Bernis, F. - Friedman, A., Higher order nonlinear degenerate parabolic equations, J. Diff. Eqns. 83 (1990), 179-206. | MR 1031383 | Zbl 0702.35143

[5] Bobylev, A.V., The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Mathematical physics reviews, 7 (1988), 111-233. | MR 1128328 | Zbl 0850.76619

[6] Boltzmann, L., Weitere Studien uber das Warmegleichgewicht unter Gasmolekulen. Sitz. der Akademie der Wissenschaften66, (1872), 275-370 , Lectures on Gas Theory. University of California Press, Berkeley, 1964. Translated by S.G. Brush. Reprint of the 1896-1898 Edition. Reprinted by Dover Publications, 1995.

[7] Bhatnagar, P.L. - Gross, E.P. - Krook, M., A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525. | Zbl 0055.23609

[8] Cáceres, M.J. - Carrillo, J.A. - Toscani, G., Long-time behavior for a nonlinear fourth-order parabolic equation. Trans. Amer. Math. Soc., 357 (2005), 1161-1175. | MR 2110436 | Zbl 1077.35028

[9] Carlen, E.A. - Carvalho, M., Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Statist. Phys., 67 (1992), 575-608. | MR 1171145 | Zbl 0899.76317

[10] Carrillo, J.A. - Juengel, A. - Markowich, P. - Toscani, G. - Unterreiter, A., Entropy production methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math., 133 (2001) 1-82. | MR 1853037 | Zbl 0984.35027

[11] Carrillo, J.A. - Lederman, C. - Markowich, P.A. - Toscani, G., Poincarè Inequalities for Linearizations of Very Fast Diffusion Equations., Nonlinearity, 15 (2001), 1-16. | MR 1901093 | Zbl 1011.35025

[12] Carrillo, J.A. - Mccann, R.J. - Villani, C., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 1-48. | MR 2053570 | Zbl 1073.35127

[13] Carrillo, J.A. - Toscani, G., Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations. Mathem. Methods Appl. Sciences, 21 (1998), 1269-1286. | MR 1639292 | Zbl 0922.35131

[14] Carrillo, J.A. - Toscani, G., Asymptotic L1-decay of the porous medium equation to self-similarity. Indiana Univ. Math. J., 46 (2000), 113-142. | MR 1777035 | Zbl 0963.35098

[15] Carrillo, J.A. - Toscani, G., Large-time asymptotics for strong solutions of the thin film equation. Commun. Math. Phys., 225 (2002), 113-142. | MR 1888873 | Zbl 0990.35054

[16] Carrillo, J.A. - Vázquez, J.L., Fine asymptotics for fast diffusion equations. Comm. Partial Differential Equations, 28 (2003), 1023-1056. | MR 1986060 | Zbl 1036.35100

[17] Carrillo, J.A. - Vázquez, J.L., Asymptotic complexity in filtration equations. J. Evol. Equ., 7 (2007), 471-495. | MR 2328935

[18] Cercignani, C., H-theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech., 34 (1982), 231-241. | MR 715658 | Zbl 0538.76068

[19] Cercignani, C., Mathematical methods in kinetic theory. Second edition. Plenum Press, New York, 1990. | MR 1069558 | Zbl 0726.76083

[20] Cherasekhar, S., Principles of Stellar Dynamics. Courier Dover Publ.New York, 1942

[21] Cover, T.M. - Thomas, J.A., Elements of Information Theory. J. Wiley & Sons Inc., New York, 1991. | MR 1122806 | Zbl 0762.94001

[22] Csiszar, I., Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung., 2 (1967), 299-318. | MR 219345 | Zbl 0157.25802

[23] Desvillettes, L., Entropy production rate and convergence in kinetic equations. Comm. Math. Phys., 26 (1989), 687-702. | MR 1006301 | Zbl 0688.76057

[24] Desvillettes, L. - Villani, C., On the Spatially Homogeneous Landau Equation for Hard Potentials. Part II: H-Theorem and Applications. Comm. Partial Differential Equations 25, n. 1-2 (2000), 261-298. | MR 1737548 | Zbl 0951.35130

[25] Desvillettes, L. - Villani, C., On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math., 54 (2001), 1-42. | MR 1787105 | Zbl 1029.82032

[26] Desvillettes, L. - Villani, C., On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math., 159 (2005), 245-316. | MR 2116276 | Zbl 1162.82316

[27] Diperna, R.J., Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc., 292 (1985), 383-420. | MR 808729 | Zbl 0606.35052

[28] Diperna, R.J. - Lions, P.L., On the Cauchy problem for the Boltzmann equation: Global existence and weak stability. Ann. of Math. (2) 130 (1989), 312-366. | MR 1014927 | Zbl 0698.45010

[29] Fisher, R., Theory of statistical estimation. Math. Proc. Cambridge Philos. Soc., 22 (1925), 700-725. | Zbl 51.0385.01

[30] Gabetta, E. - Markowich, P.A. - Unterreiter, A., A note on the entropy production of the radiative transfer equation. Appl. Math. Lett., 12 (1999), 111-116. | MR 1750607 | Zbl 0939.35035

[31] Gianazza, U. - Savaré, G. - Toscani, G., The Wasserstein gradient flow of the Fisher information and the Quantum Drift-Diffusion equation. Arch. Rat. Mech. Anal. (in press). | MR 2533926

[32] Gross, L., Logarithmic Sobolev inequalities. Amer. J. of Math., 97 (1975), 1061-1083. | MR 420249 | Zbl 0318.46049

[33] Juengel, A. - Markowich, P. - Toscani, G., Decay rates for solutions of degenerate parabolic systems. Electron. J. Diff. Eqs. Conf., 06 (2001), 189-202. | MR 1804774 | Zbl 0964.35085

[34] Juengel, A. - Matthes, D., An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity, 19 (2006), 633-659. | MR 2209292 | Zbl 1091.35031

[35] Juengel, A. - Toscani, G., Exponential decay in time of solutions to a nonlinear fourth-order parabolic equation. Z. Angew. Math. Phys., 54 (2003), 377-386. | MR 2048659 | Zbl 1029.35033

[36] Kullback, S., A lower bound for discrimination information in terms of variation. IEEE Trans. Inf. The., 4 (1967), 126-127.

[37] Landau, L., Die kinetische Gleichung fur den Fall Coulombscher Wechselwirkung. Phys. Z. Sovjet., 10 (1936), 154-164. | Zbl 0015.38202

[38] Lax, P.D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, 11. Society for Industrial and Applied Mathematics, Philadelphia, Pa., (1973), v+48. | MR 350216 | Zbl 0268.35062

[39] Lederman, C. - Markowich, P.A., On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass. Comm. Partial Differential Equations, 28 (2003), 301-332. | MR 1974458 | Zbl 1024.35040

[40] Kamenomostskaya, S., The asymptotic behavior of the solution of the filtration equation. Israel J. Math., 14 (1973), 76-87. | MR 315292 | Zbl 0254.35054

[41] Markowich, P.A. - Villani, C., On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis. Mat. Contemp., 19 (2000), 1-29. | MR 1812873 | Zbl 1139.82326

[42] Nash, J., Continuity of solutions of parabolic and elliptic equations. Amer. J. Math., 80 (1958), 931-954. | MR 100158 | Zbl 0096.06902

[43] Otto, F., The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eq., 26 (2001), 101-174. | MR 1842429 | Zbl 0984.35089

[44] Risken, H., The Fokker-Planck equation. Methods of solution e applications. Second edition. Springer Series in Synergetics, 18Springer-Verlag, Berlin, 1989. | MR 987631 | Zbl 0665.60084

[45] Shannon, C.E., Collected papers. Edited by N. J. A. Sloane e Aaron D. Wyner. IEEE Press, New York, 1993. | MR 1216351 | Zbl 0846.01022

[46] Stam, A., Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control, 2 (1959), 101-112. | MR 109101 | Zbl 0085.34701

[47] Toscani, G., Sur l'inégalité logarithmique de Sobolev. C.R. Acad. Sc. Paris, 324 (1997), 689-694. | MR 1447044

[48] Toscani, G., Remarks on entropy and equilibrium states. Appl. Math. Letters, 12 (1999), 19-25. | MR 1750055 | Zbl 0940.35168

[49] Toscani, G. - Villani, C., Sharp entropy production bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203 (1999), 667-706. | MR 1700142 | Zbl 0944.35066

[50] Toscani, G. - Villani, C., Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Statist. Phys., 94 (1999), 619-637. | MR 1675367 | Zbl 0958.82044

[51] Toscani, G. - Villani, C., On the trend to equilibrium for some dissipative systems with slowing increasing a priori bounds. J. Statist. Phys., 98 (2000), 1279-1309. | MR 1751701 | Zbl 1034.82032

[52] Villani, C., Cercignani's conjecture is sometimes true and always almost true. Commun. Math. Phys., 234 (2003), 455-490. | MR 1964379 | Zbl 1041.82018