Title: Fractional diffusion limit for collisional kinetic equations: A moments method

Authors: A. Mellet

Issue: Volume 59 (2010), Issue 4, 1333-1360

Abstract: This paper is devoted to hydrodynamic limits of linear kinetic equations. We consider situations in which the thermodynamical equilibrium is described by a heavy-tail distribution function rather than a maxwellian distribution. A similar problem was addressed in [A. Mellet, S. Mischler, C. Mouhot, \emph{Fractional diffusion limit for collisional kinetic equations}, preprint, 2008] using Fourier transform and it was shown that the long time/small mean free path behavior of the solution of the kinetic equation is described by a fractional diffusion equation. In this paper, we propose a different method to obtain similar results. This method is somewhat reminiscent of the so-called ``moments method'' which plays an important role in kinetic theory. This new method allows us to consider space dependent collision operators (which could not be treated in [the work cited above]). We believe that it also provides the relevant tool to address nonlinear problems.