Vanishing viscosity limit for aggregation-diffusion equations

Lagoutière, Frédéric; Santambrogio, Filippo; Tran Tien, Sébastien

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1123-1179

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 $O\left(\sqrt{\epsilon }\right)$ rate, where $\epsilon$ 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.

Cet article est consacré à l’analyse de convergence de l’approximation diffusive des solutions à valeur mesure de l’équation d’agrégation, largement utilisée pour modéliser le mouvement collectif d’une population dirigée par un potentiel d’interaction. Nous prouvons, dans l’espace entier en n’importe quelle dimension d’espace, dans le cadre général des potentiels lipschitziens, la convergence uniforme en temps sur tout intervalle borné, en distance de Wasserstein, et avec un ordre de convergence $1/2$ lorsque le potentiel est $\lambda$-convexe. Nous étendons ces résultats à certains potentiels répulsifs et prouvons des taux de convergence optimaux des états stationnaires vers la masse de Dirac, sous certaines hypothèses d’attractivité uniforme.

Classification : 35K20, 35B40, 35A35, 35Q92, 49Q22
Keywords: Aggregation-diffusion equations, asymptotic analysis, numerical analysis, Wasserstein distance
Mots-clés : Équations d’agrégation-diffusion, analyse asymptotique, analyse numérique, distance de Wasserstein

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