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
equilibria satisfy a {\em generalised} weighted mass condition
and can have infinite mass. 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
characterization of the diffusion coefficient was not known,
for Lévy-Fokker-Planck operators with general equilibria, and
in cases where the equilibrium has infinite mass. It also
unifies and generalises the results of ten previous papers with
a quantitative method, and our estimates on the fluid
approximation error seem novel in these cases.
Origine : Fichiers produits par l'(les) auteur(s)