Quantitative fluid approximation in transport theory: a unified approach
Résumé
We propose a unified method for the large space-time scaling limit of linear collisional kinetic equations in the whole space. The limit is of 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 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, and for Lévy-Fokker-Planck operators with general equilibria. It also unifies and generalises the results of ten previous papers with a quantitative method; the estimates on the fluid approximation error seem novel in these cases.
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