Mathematical tools for kinetic equations. (English) Zbl 1151.82351

Perthame, Benoît

Bull. Am. Math. Soc., New Ser. 41, No. 2, 205-244 (2004).

The author presents general methods for linear kinetic equations, which covers time decay and dispersion effects as Strichartz inequalities moment lemmas – all of these improve the obvious integrability derived from conservation laws. It is well-known that in the context of the linear equation it is possible to gain regularity to averaging lemmas. The author gives several statements beginning with the simplest regularizing effect \(H^{1/2}\). These tools have been used to treat nonlinear models in the last few years. The author presents in this context the Vlasov equation of plasma physics, scattering models and the Boltzmann equation. Moreover, the author deals with also asymptotic problems and the derivation of macroscopic models especially through the diffusion, hyperbolic and high field limits.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

82B40	Kinetic theory of gases in equilibrium statistical mechanics
76P05	Rarefied gas flows, Boltzmann equation in fluid mechanics
35F10	Initial value problems for linear first-order PDEs

Keywords:

kinetic equation; regularity; hydrodynamic limit; Strichartz-type inequality

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