Quantitative fluid approximation in transport theory: a unified approach
Résumé
We propose a unified method for the large space-time scaling
limit of \emph{linear} collisional kinetic equations in the
whole space. The limit is of \emph{fractional} diffusion type
for heavy tail equilibria with slow enough decay, and of
diffusive type otherwise. The proof is constructive and the
fractional/standard diffusion matrix is obtained. The method combines energy estimates and quantitative spectral
methods to construct a `fluid mode'. The method is applied to
scattering models (without assuming detailed balance
conditions), Fokker-Planck operators and Lévy-Fokker-Planck
operators. It proves a series of new results, including the
fractional diffusive limit for Fokker-Planck operators in any
dimension, for which the formulas for the diffusion coefficient
were not known, for Lévy-Fokker-Planck operators with general
equilibria, and for scattering operators including some cases
of infinite mass equilibria. It also unifies and generalises
the results of previous papers with a quantitative method, and
our estimates on the fluid approximation error also seem novel.
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Origine : Fichiers produits par l'(les) auteur(s)