Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases

Abstract

This work deals with the propagation of chaos without cut-off for some aggregation-diffusion models known as fractional Keller Segel equations. The diffusion considered here is given by the fractional Laplacian operator −(−Δ)a2 with a∈(1,2) and the singularity of the aggregation kernel behaves like |x|1−α with α∈]1,a]. In the Diffusion Dominated case α∈(1,a), we give a propagation of chaos result, thanks to the Γ lower semi-continuity of the fractional Fisher information, already known for the classical Fisher information, using a result of [20]. In the fair competition case a=α, we only prove a convergence/consistency result in a sub-critical sensitivity regime, similarly as the result obtained for the classical Keller-Segel equation in [16].