On Rosenau-Type approximations to fractional diffusion equations

Owing to the Rosenau argument, originally proposed to obtain a regularized version of the Chapman-Enskog expansion of hydrodynamics, we introduce a non-local linear kinetic equation which approximates a fractional diffusion equation. We then show that the solution to this approximation, apart of a rapidly vanishing in time perturbation, approaches the fundamental solution of the fractional diffusion (a Lévy stable law) at large times. The research that led to the present paper was partially supported by a grant of the group GNFM of INdAM.