Preprint

Inserted: 14 mar 2023
Last Updated: 14 mar 2023

Year: 2023

Abstract:

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 the general framework of Lipschitz potentials, and provide a $O(\sqrt{\varepsilon})$ rate, where $\varepsilon$ is the diffusion parameter, when the potential is $\lambda-$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.

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