Kinetic equations and asymptotic theory. Ed. by Benoît Perthame and Laurent Desvillettes. (English) Zbl 0979.82048

Series in Applied Mathematics (Paris). 4. Paris: Gauthier-Villars/ Elsevier. 162 p. (2000).
This book presents the essentials of some courses given during ‘session états de la recherche’ initiated by the Société Mathématique de France. The book is focused on the recent developments on the mathematical theory of kinetic equations. The modern developments are combined with an elementary introduction to the subject. The first part of the book supplies general mathematical tools. An overview of the classical Vlasov-Poisson and Vlasov-Maxwell systems as well as the theory of transport equations, a priori estimates, averaging lemmas, dispersion estimates are given.
The second part is concerned with the recent progress in global convergence results towards the Euler equations in incompressible fluids, models or scalings which allow to recover parabolic or hyperbolic limits. The last issue presented in the book describes the derivation of kinetic equations in the limit of large systems of interacting particles. Rigorous justification of the Boltzmann-Grad limit which allows to recover the Boltzmann equation from the BBGKY hierarchy is the main purpose. The methods used combine probability and PDE theories.


82C40 Kinetic theory of gases in time-dependent statistical mechanics
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35Q30 Navier-Stokes equations
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics