Kinetic equations and asymptotic theory
Gauthier-Villars, 2000 - Science - 162 pages
The book is focused on the recent developments on the mathematical theory of partial differential equations of kinetic type. During the last few years, this domain has received a lot of attention and has given rise to several developments. The use of general advanced mathematical tools (regularity theory, compactness averaging lemmas, dispersion lemmas) is described together with more introductory topics. They are used to analyze the Boltzmann equation and its various hydrodynamical limits : convergence towards the Euler equations of incompressible fluids, models or scallings which allow to recover parabolic or hyperbolic limits. The last part in this book concerns the derivation of kinetic equations in the limit of large systems of interacting particles. Here, the purpose is to justify rigorously the so-called Boltzmann-Grad limit which allows to recover kinetic equations from the BBGKY hierarchy.
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Introduction to the mathematical theory of kinetic equations
From kinetic to macroscopic models
From particles to transport equations
1 other sections not shown
Appl assume assumption averaging lemmas BBGKY hierarchy BGK model Boltzmann equation Boltzmann equation 2.1 Bouchut bounded Cauchy problem collision integral collision operator Comm compact support computation conservation laws consider convergence cutoff defined denote derived Desvillettes dissipative solution divr dynamics energy entropy estimate Euler equation Euler system fluid global existence Golse F hard sphere heat conduction hierarchy hydrodynamic limits incompressible Euler equation incompressible Euler limit incompressible Navier-Stokes inequality initial data kinetic equations kinetic model kinetic theory lecture Lions P.-L macroscopic Math mathematical Maxwell's equations Maxwellian momentum Navier-Stokes equation nonlinear nonnegative notation number density obtain particle Perthame phase space Phys proof Proposition prove Pulvirenti renormalized renormalized solutions result right hand side satisfies scaled Boltzmann equation sense of distributions T3 x S2 taking limits term transport equation variable Velocity Averaging verifies Vlasov-Maxwell Vlasov-Poisson weak solutions