Fractional diffusion limit for collisional kinetic equations

Antoine Mellet (1) , Stéphane Mischler (2) , Clément Mouhot (2)

1 Department of Mathematics [College Park]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.

Mots clés: fractional diffusion; fractional heat equation; anomalous heat transport; linear Boltzmann equation; relaxation equation; linear BGK equation; diffusion limit; anomalous diffusion limit; anomalous diffusive time scale; mathematical derivation

Domaines: Equations aux dérivées partielles [math.AP]

Antoine Mellet, Stéphane Mischler, Clément Mouhot. Fractional diffusion limit for collisional kinetic equations. Archive for Rational Mechanics and Analysis, 2011, 199 (2), pp.493-525. ⟨10.1007/s00205-010-0354-2⟩. ⟨hal-00321478v2⟩