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This paper is devoted to the grazing collision limit of the inelastic Kac model introduced in [A. Pulvirenti and G. Toscani, J. Statist. Phys., 14 (2004), pp. 1453-1480], when the equilibrium distribution function is a heavy-tailed Lévy-type distribution with infinite variance. We prove that solutions in an appropriate domain of attraction of the equilibrium distribution converge to solutions of a Fokker--Planck equation with a fractional diffusion operator. [ABSTRACT FROM AUTHOR]

Abstract: We prove the propagation of regularity, uniformly in time, for the scaled solutions of the one-dimensional dissipative Maxwell models introduced in [D. Ben-Avraham, E. Ben-Naim, K. Lindenberg, A. Rosas, Self-similarity in random collision processes, Phys. Rev. E 68 (2003) R050103]. This result together with the weak convergence towards the stationary state proven in [L. Pareschi, G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, J. Stat. Phys. 124 (2–4) (2006) 747–779] implies the strong convergence in Sobolev norms and in the norm towards it depending on the regularity of the initial data. As a consequence, the original nonscaled solutions are also proved to be convergent in towards the corresponding self-similar homogeneous cooling state. The proof is based on the (uniform in time) control of the tails of the Fourier transform of the solution, and it holds for a large range of values of the mixing parameters. In particular, in the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity. [Copyright &y& Elsevier]