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Quantitative fluid approximation in transport theory: a unified approach

Abstract : 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03005595
Contributor : Emeric Bouin Connect in order to contact the contributor
Submitted on : Thursday, June 3, 2021 - 10:21:01 AM
Last modification on : Wednesday, November 17, 2021 - 12:27:08 PM

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  • HAL Id : hal-03005595, version 3
  • ARXIV : 2011.07836

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Emeric Bouin, Clément Mouhot. Quantitative fluid approximation in transport theory: a unified approach. 2021. ⟨hal-03005595v3⟩

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