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. 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 with 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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