Unified approach to fluid approximation of linear kinetic equations with heavy tails

Abstract: The rigorous fluid approximation of linear kinetic equations was first obtained in the late 70s when the equilibrium distribution decays faster than polynomials. In this case the limit is a diffusion equation. In the case of heavy tail equilibrium distribution (with infinite variance), the first rigorous derivation was obtained in 2011 in my joint paper with Mellet and Mischler, in the case of scattering operators. The limit shows then anomalous diffusion; it is governed by a fractional diffusion equation. Lebeau and Puel proved last year the first similar result for Fokker-Planck operator, in dimension 1 and assuming that the equilibrium distribution has finite mass. Fournier and Tardif gave an alternative probabilistic proof, more general (covering any dimension and infinite-mass equilibrium distribution) but non-constructive. We present a unified quantitative PDE approach that obtains constructively the limit for Fokker-Planck operators in dimensions greater than 2, but also recovers and unifies the previous works. This is a joint work with Emeric Bouin (Université Paris-Dauphine).

analysis of PDEs

Audience: researchers in the topic

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