Vanishing viscosity limit for aggregation-diffusion equations

This article is devoted to the convergence analysis of the diffusive approximation of the measure-valued solutions to the so-called aggregation equation, which is now widely used to model collective motion of a population directed by an interaction potential. We prove, over the whole space in any dimension, a uniform-in-time convergence in Wasserstein distance in all finite-time intervals, in the general framework of Lipschitz continuous potentials, and provide an O(ε) rate, where ε is the diffusion parameter, when the potential is λ-convex. We give an extension to some repulsive potentials and prove sharp convergence rates of the steady states towards the Dirac mass, under some uniform attractiveness assumptions.