Funzionali entropia ed equilibrio di sistemi di molte particelle

G. Toscani

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 3, page 509-524
  • ISSN: 0392-4041

Abstract

top
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.

How to cite

top

Toscani, G.. "Funzionali entropia ed equilibrio di sistemi di molte particelle." Bollettino dell'Unione Matematica Italiana 1.3 (2008): 509-524. <http://eudml.org/doc/290482>.

@article{Toscani2008,
author = {Toscani, G.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {ita},
month = {10},
number = {3},
pages = {509-524},
publisher = {Unione Matematica Italiana},
title = {Funzionali entropia ed equilibrio di sistemi di molte particelle},
url = {http://eudml.org/doc/290482},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Toscani, G.
TI - Funzionali entropia ed equilibrio di sistemi di molte particelle
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/10//
PB - Unione Matematica Italiana
VL - 1
IS - 3
SP - 509
EP - 524
LA - ita
UR - http://eudml.org/doc/290482
ER -

References

top
  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. Zbl0982.35113MR1842428DOI10.1081/PDE-100002246
  2. ARNOLD, A. - MARKOWICH, P.A. - TOSCANI, G. - UNTERREITER, A., On Generalized Csiszar-Kullback Inequalities. Monatsh. Math., 131 (2000), 235-253. Zbl1015.94003MR1801751DOI10.1007/s006050070013
  3. BAKRY, D. - EMERY, M., Diffusions hypercontractives. In Sém. Proba. XIX, 1123Lecture Notes in Math.Springer, 1985, 177-206. MR889476DOI10.1007/BFb0075847
  4. BERNIS, F. - FRIEDMAN, A., Higher order nonlinear degenerate parabolic equations, J. Diff. Eqns.83 (1990), 179-206. Zbl0702.35143MR1031383DOI10.1016/0022-0396(90)90074-Y
  5. BOBYLEV, A.V., The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Mathematical physics reviews, 7 (1988), 111-233. Zbl0850.76619MR1128328
  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. Zbl0055.23609
  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. Zbl1077.35028MR2110436DOI10.1090/S0002-9947-04-03528-7
  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. Zbl0899.76317MR1171145DOI10.1007/BF01049721
  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. Zbl0984.35027MR1853037DOI10.1007/s006050170032
  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. Zbl1011.35025MR1901093DOI10.1088/0951-7715/15/3/303
  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. Zbl1073.35127MR2053570DOI10.4171/RMI/376
  13. CARRILLO, J.A. - TOSCANI, G., Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations. Mathem. Methods Appl. Sciences, 21 (1998), 1269-1286. Zbl0922.35131MR1639292DOI10.1002/(SICI)1099-1476(19980910)21:13<1269::AID-MMA995>3.3.CO;2-F
  14. CARRILLO, J.A. - TOSCANI, G., Asymptotic L 1 -decay of the porous medium equation to self-similarity. Indiana Univ. Math. J., 46 (2000), 113-142. Zbl0963.35098MR1777035DOI10.1512/iumj.2000.49.1756
  15. CARRILLO, J.A. - TOSCANI, G., Large-time asymptotics for strong solutions of the thin film equation. Commun. Math. Phys., 225 (2002), 113-142. Zbl0990.35054MR1888873DOI10.1007/s002200100591
  16. CARRILLO, J.A. - VÁZQUEZ, J.L., Fine asymptotics for fast diffusion equations. Comm. Partial Differential Equations, 28 (2003), 1023-1056. Zbl1036.35100MR1986060DOI10.1081/PDE-120021185
  17. CARRILLO, J.A. - VÁZQUEZ, J.L., Asymptotic complexity in filtration equations. J. Evol. Equ., 7 (2007), 471-495. MR2328935DOI10.1007/s00028-006-0298-z
  18. CERCIGNANI, C., H-theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech., 34 (1982), 231-241. Zbl0538.76068MR715658
  19. CERCIGNANI, C., Mathematical methods in kinetic theory. Second edition. Plenum Press, New York, 1990. Zbl0726.76083MR1069558DOI10.1007/978-1-4899-7291-0
  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. Zbl0762.94001MR1122806DOI10.1002/0471200611
  22. CSISZAR, I., Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung., 2 (1967), 299-318. Zbl0157.25802MR219345
  23. DESVILLETTES, L., Entropy production rate and convergence in kinetic equations. Comm. Math. Phys., 26 (1989), 687-702. Zbl0688.76057MR1006301
  24. DESVILLETTES, L. - VILLANI, C., On the Spatially Homogeneous Landau Equation for Hard Potentials. Part II: H-Theorem and Applications. Comm. Partial Differential Equations25, n. 1-2 (2000), 261-298. Zbl0951.35130MR1737548DOI10.1080/03605300008821513
  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. Zbl1029.82032MR1787105DOI10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q
  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. Zbl1162.82316MR2116276DOI10.1007/s00222-004-0389-9
  27. DIPERNA, R.J., Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc., 292 (1985), 383-420. Zbl0606.35052MR808729DOI10.2307/2000221
  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. Zbl0698.45010MR1014927DOI10.2307/1971423
  29. FISHER, R., Theory of statistical estimation. Math. Proc. Cambridge Philos. Soc., 22 (1925), 700-725. Zbl51.0385.01
  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. Zbl0939.35035MR1750607DOI10.1016/S0893-9659(99)00044-0
  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). MR2533926DOI10.1007/s00205-008-0186-5
  32. GROSS, L., Logarithmic Sobolev inequalities. Amer. J. of Math., 97 (1975), 1061-1083. Zbl0318.46049MR420249DOI10.2307/2373688
  33. JUENGEL, A. - MARKOWICH, P. - TOSCANI, G., Decay rates for solutions of degenerate parabolic systems. Electron. J. Diff. Eqs. Conf., 06 (2001), 189-202. Zbl0964.35085MR1804774
  34. JUENGEL, A. - MATTHES, D., An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity, 19 (2006), 633-659. Zbl1091.35031MR2209292DOI10.1088/0951-7715/19/3/006
  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. Zbl1029.35033MR2048659DOI10.1007/s00033-003-1026-y
  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. Zbl0015.38202
  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. Zbl0268.35062MR350216
  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. Zbl1024.35040MR1974458DOI10.1081/PDE-120019384
  40. KAMENOMOSTSKAYA, S., The asymptotic behavior of the solution of the filtration equation. Israel J. Math., 14 (1973), 76-87. Zbl0254.35054MR315292DOI10.1007/BF02761536
  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. Zbl1139.82326MR1812873
  42. NASH, J., Continuity of solutions of parabolic and elliptic equations. Amer. J. Math., 80 (1958), 931-954. Zbl0096.06902MR100158DOI10.2307/2372841
  43. OTTO, F., The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eq., 26 (2001), 101-174. Zbl0984.35089MR1842429DOI10.1081/PDE-100002243
  44. RISKEN, H., The Fokker-Planck equation. Methods of solution e applications. Second edition. Springer Series in Synergetics, 18Springer-Verlag, Berlin, 1989. Zbl0665.60084MR987631DOI10.1007/978-3-642-61544-3
  45. SHANNON, C.E., Collected papers. Edited by N. J. A. Sloane e Aaron D. Wyner. IEEE Press, New York, 1993. Zbl0846.01022MR1216351
  46. STAM, A., Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control, 2 (1959), 101-112. Zbl0085.34701MR109101
  47. TOSCANI, G., Sur l'inégalité logarithmique de Sobolev. C.R. Acad. Sc. Paris, 324 (1997), 689-694. MR1447044DOI10.1016/S0764-4442(97)86991-1
  48. TOSCANI, G., Remarks on entropy and equilibrium states. Appl. Math. Letters, 12 (1999), 19-25. Zbl0940.35168MR1750055DOI10.1016/S0893-9659(99)00096-8
  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. Zbl0944.35066MR1700142DOI10.1007/s002200050631
  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. Zbl0958.82044MR1675367DOI10.1023/A:1004589506756
  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. Zbl1034.82032MR1751701DOI10.1023/A:1018623930325
  52. VILLANI, C., Cercignani's conjecture is sometimes true and always almost true. Commun. Math. Phys., 234 (2003), 455-490. Zbl1041.82018MR1964379DOI10.1007/s00220-002-0777-1

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.