Fractional diffusion limit for collisional kinetic equations: A moments method

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 [14] 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 [14]). We believe that it also provides the relevant tool to address nonlinear problems.