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Global solutions of kinetic models and related questions. (English) Zbl 0799.35184

Cercignani, Carlo (ed.) et al., Nonequilibrium problems in many-particle systems. Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini, Italy, June 15-27, 1992. Berlin: Springer-Verlag. Lect. Notes Math. 1551, 58-86 (1993).
The paper reviews some recent progress on various kinetic equations which include Boltzmann equation and Vlasov models like Vlasov-Poisson or Vlasov-Maxwell system. The type of results discussed on the Cauchy problem associated with kinetic models are essentially existence and uniqueness results.
After the presentation of the models, some a priori estimates are given which allow to prove the existence of global weak solutions. The main tools that are used in the proof of existence are velocity averaging lemma and the construction of generalized ODE flows.
For the entire collection see [Zbl 0777.00027].
Reviewer: S.Totaro (Firenze)

MSC:

35Q35 PDEs in connection with fluid mechanics
82C70 Transport processes in time-dependent statistical mechanics
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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