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On the existence and convergence of solutions of Boltzmann equations. (English) Zbl 0813.45003

The author gives a brief survey of some recent results on the Boltzmann equation concerning the global existence of a weak solution [cf. R. J. DiPerna and P. L. Lions, Ann. Math., II. Ser. 130, No. 2, 321- 366 (1989; Zbl 0698.45010)] and the convergence of a sequence of solutions [cf. the author, J. Math. Kyoto Univ. 34, No. 2, 391-427 and 429-461 (1994)].
Reviewer: T.Aktosun (Fargo)

MSC:

45K05 Integro-partial differential equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics

Citations:

Zbl 0698.45010
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Full Text: DOI

References:

[1] Boltzmann, L., Weitere studien über das wärmegleichgenicht unfer gasmoleküler, Sitzungsberichte der Akademie der Wissenschaften, Wien, 66, 275-370 (1978) · JFM 04.0566.01
[2] T. Carleman,Problémes mathématiques dans la théorie cinétique des gaz. Notes rédigées par L. Carleson et O. Frostma, Publications mathématiques de l’Institut Mittag Leffler, Almquist and Wikselles, Uppsala, 1957. · Zbl 0077.23401
[3] Cercignani, C., The Boltzmann equation and its applications (1988), Berlin: Springer, Berlin · Zbl 0646.76001
[4] DiPerna, R. J.; Lions, P. L., On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130, 321-366 (1989) · Zbl 0698.45010 · doi:10.2307/1971423
[5] DiPerna, R. J.; Lions, P. L., Global solutions of Boltzmann’s equation and the entropy inequality, Arch. Rat. Mech. Anal., 114, 47-55 (1191) · Zbl 0724.45011 · doi:10.1007/BF00375684
[6] P.L. Lions,On Boltzmann and Landau equations, to appear, in Phil. Trans., Roy. Soc. · Zbl 0809.35137
[7] P.L. Lions,Compactness in Boltzmann’s equation via Fourier integral operators and applications, Part I. Preprint. · Zbl 0831.35139
[8] P.L. Lions,Compactness in Boltzmann’s equation via Fourier integral operators and applications. Part II. Preprint. · Zbl 0831.35139
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