Fractional diffusion limit for collisional kinetic equations: A moments method A. Mellet 76P0535B4026A33kinetic equationslinear Boltzmann equationasymptotic analysisanomalous diffusion limitfractional diffusionrelaxation equationanomalous diffusive time scale 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. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4128 10.1512/iumj.2010.59.4128 en Indiana Univ. Math. J. 59 (2010) 1333 - 1360 state-of-the-art mathematics http://iumj.org/access/