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Un roman d'une grande beauté, drôle, fin, extrêmement lumineux sur des sujets difficiles : la perte de
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Ensemble il ne diront plus Sayonara mais Mata Ne !
The book is focused on the recent developments on the mathematical theory of partial differential equations of kinetic type. During the last few years,...
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Résumé
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.
Sommaire
INTRODUCTION TO THE MATHEMATICAL THEORY OF KINETIC EQUATIONS
Theory of characteristics
The kinetic transport equation
Weak solutions for Vlasov-Poisson and Vlasov-Maxwell
Classical solutions for Vlasov-Poisson and Vlasov-Maxwell
Averaging lemmas
Dispersion lemmas
Relations between averaging lemmas and dispersion
References
FROM KINETIC TO MACROSCOPIC MODELS
Lecture 1 : The Boltzmann or BGK equations and their Euler limit
The Boltzmann equation
Fundamental conservation laws
Boltzmann's H Theorem and Maxwell's distributions
The Knudsen number
Hilbert's expansion I : the compressible Euler limit
The moment closure method 1 : the compressible Euler limit
The Gibbs principle and BGK type equations
Lecture 2 : Navier-Stokes asymptotics
Grad's cutoff assumption
The linearized collision integral
Symmetries of the linearized collision integral
Hilbert's expansion H : the compressible Navier-Stokes asymptotics The von Karman relation between the Reynolds, Mach and Knudsen numbers
The moment closure method II : the incompressible Euler and Navier-Stokes limits
Ghost effects
The mathematical derivation of hydrodynarnic limits
Velocity Averaging
Hyperbolic vs parabolic type limits
Mathematical proof for hydrodynamic limits of hyperbolic type : the Perthame-Tadmor kinetic model leading to the Hopf equation Mathematical proof for hydrodynamic limits of parabolic type : an example from radiative transfer
Mathematical proof of the incompressible Euler limit based on the relative entropy
Other examples in the literature
FROM PARTICLES TO TRANSPORT EQUATIONS
The Hard Sphere system and formal derivation of the Boltzmann equation
The Lorentz model
Derivation of the Boltzmann equation for hard spheres