Kinetic Equations and Asymptotic Theory

François Bouchut

,

François Golse

,

Mario Pulvirenti

,

Benoît Perthame

,

Laurent Desvillettes

Note moyenne 
François Bouchut et François Golse - Kinetic Equations and Asymptotic Theory.
The book is focused on the recent developments on the mathematical theory of partial differential equations of kinetic type. During the last few years,... Lire la suite
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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
    • Other results.

Caractéristiques

  • Date de parution
    09/06/2000
  • Editeur
  • Collection
    Series in Applied Mathematics
  • ISBN
    2-84299-110-9
  • EAN
    9782842991104
  • Présentation
    Broché
  • Nb. de pages
    162 pages
  • Poids
    0.36 Kg
  • Dimensions
    17,0 cm × 24,0 cm × 1,0 cm

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