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Global solutions of kinetic models and related questions

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Nonequilibrium Problems in Many-Particle Systems

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References

  1. V.I. Agoshkov, Spaces of functions with differential-difference characteristics and smoothness of solutions of the transport equation. Soviet Math. Dokl, 29 (1984), p. 662–666.

    MathSciNet  MATH  Google Scholar 

  2. L. Arkeryd, On the Boltzmann equation. Arch. Rat. Mech. Anal., 45 (1972), p. 1–34.

    MathSciNet  MATH  Google Scholar 

  3. C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. Convergence proofs for the Boltzmann equation. Preprint, announced in C.R. Acad. Sci. Paris, 309 (1989), p. 727–732.

    MathSciNet  MATH  Google Scholar 

  4. J. Bergh and J. Löfstrom, Interpolation spaces. An introduction. Springer, Berlin, 1976.

    Book  MATH  Google Scholar 

  5. L. Boltzmann, Weitere studien über das wärme gleichgenicht unfer gasmoleküler. Sitzungsberichte der Akademie der Wissenschaften, Wien, 66 (1872), p. 275–370. Translation: Further studies on the thermal equilibrium of gaz molecules, 88–174; In Kinetic Theory 2, ed. S.G. Brush, Pergamon, Oxford, 1966.

    Google Scholar 

  6. T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz. Notes rédigées par L. Carleson et O. Frostman, Publications mathématiques de l'Institut Mittag-Leffler, Almquist and Wikselles, Uppsala, 1957.

    MATH  Google Scholar 

  7. C. Cercignani, The Boltzman equation and its applications. Springer, Berlin, 1988.

    Book  MATH  Google Scholar 

  8. S. Chapman and T.G. Cowling, The mathematical theory of non-uniform gases. Cambridge Univ. Press, 1939; 2nd edit., 1952.

    Google Scholar 

  9. P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Preprint.

    Google Scholar 

  10. L. Desvillettes, On an asymptotics of the Boltzman equation when the collisions become grazing. Preprint.

    Google Scholar 

  11. R.J. DiPerna and P.L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math., 130 (1989), p.321–366.

    Article  MathSciNet  MATH  Google Scholar 

  12. R.J. DiPerna and P.L. Lions, Global solutions of Boltzmann's equation and the Entropy inequality. Arch. Rat. Mech. Anal., 114 (1991), p. 47–55.

    Article  MathSciNet  MATH  Google Scholar 

  13. R.J. DiPerna and P.L. Lions, On the Fokker-Planck-Boltzmann equation. Comm. Math. Phys., 120 (1988), p. 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  14. R.J. DiPerna and P.L. Lions, Solutions globales d'quations du type Vlasov-Poisson. C.R. Acad. Sci. Paris, 307 (1988), p. 655–658.

    MathSciNet  MATH  Google Scholar 

  15. R.J. DiPerna and P.L. Lions, Global weak solutions of Vlasov-Maxwell equations. Comm. Pure Appl. Math., XLII (1989), p. 729–757

    Article  MathSciNet  MATH  Google Scholar 

  16. R.J. DiPerna and P.L. Lions, Global weak solutions of kinetic equations. Sem. Matematico Torino, 46 (1988), p. 259–288.

    MathSciNet  MATH  Google Scholar 

  17. R.J. DiPerna and P.L. Lions, Ordinary differential equations, Sobolev spaces and transport theory. Invent. Math., 98 (1989), p. 511–547.

    Article  MathSciNet  MATH  Google Scholar 

  18. R.J. DiPerna, P.L. Lions and Y. Meyer, L p regularity of velocity averages. Ann. I.H.P. Anal. Non Lin., 8 (1991), p. 271–287.

    MathSciNet  MATH  Google Scholar 

  19. P. Gérard, P.L. Lions and T. Paul, work in preparation.

    Google Scholar 

  20. R. Glassey, personnal communication.

    Google Scholar 

  21. F. Golse, P.L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76 (1988), p. 110–125.

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et applications au calcul de la limite de la valeur propre principale d'un opérateur de transport. C.R. Acad. Sci. Paris, 301 (1985), p. 341–344.

    MathSciNet  MATH  Google Scholar 

  23. H. Grad, Principles of the kinetic theory of gases, in Flügge's Handbuch der Physik XII, Springer, Berlin, (1958), p. 205–294.

    Google Scholar 

  24. R. Hamdache, Existence in the large and asymptotic behaviour for the Boltzmann equation. Japan J. Appl. Math., 2 (1985), p. 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Illner and M. Shinbrot, The Boltzmann equation. Global existence for a rare gas in an infinite vacuum. Comm. Math. Phys., 95 (1984), p. 217–226.

    Article  MathSciNet  MATH  Google Scholar 

  26. E.M. Lifschitz and L.P. Pitaevski, Physical kinetics. Pergamon, Oxford, 1951.

    Google Scholar 

  27. P.L. Lions, On kinetic equations. In Proceedings of the International Congress of Mathematicians, Kyoto, 1990. Springer, Tokyo, 1991.

    MATH  Google Scholar 

  28. P.L. Lions, On Boltzman and Landau equations. Preprint.

    Google Scholar 

  29. P.L. Lions and T. Paul, Sur les mesures de Wigner. Preprint.

    Google Scholar 

  30. P.L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math., 105 (1991), p. 415–430.

    Article  MathSciNet  MATH  Google Scholar 

  31. P.L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations. Preprint.

    Google Scholar 

  32. J.C. Maxwell, On the dynamical theory of gases. Phil. Trans. Roy. Soc. London, 157 (1866), p. 49–88.

    Article  Google Scholar 

  33. T. Nishida and K. Imaï, Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ. R.I.M.S. Kyoto Univ., 12 (1976), p. 229–239.

    Article  MathSciNet  MATH  Google Scholar 

  34. K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Diff. Eq., 95 (1992), p. 281–303.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions. Preprint.

    Google Scholar 

  36. C. Truesdell and R.G. Muncaster, Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas. Academic Press, New-York, 1963.

    Google Scholar 

  37. S. Ukai, Solutions of the Boltzmann equation. In: Patterns and Waves. Qualitative analysis of differential equations. North-Holland, Amsterdam, (1986), p. 37–96.

    Chapter  Google Scholar 

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  1. Ceremade Université Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris Cedex 16

    P. L. Lions

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  1. P. L. Lions
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Carlo Cercignani Mario Pulvirenti

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© 1993 Springer-Verlag

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Lions, P.L. (1993). Global solutions of kinetic models and related questions. In: Cercignani, C., Pulvirenti, M. (eds) Nonequilibrium Problems in Many-Particle Systems. Lecture Notes in Mathematics, vol 1551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090929

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  • DOI: https://doi.org/10.1007/BFb0090929

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  • Print ISBN: 978-3-540-56945-9

  • Online ISBN: 978-3-540-47832-4

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