Vanishing viscosity limit for aggregation-diffusion equations

• Published in: Journal de l'École polytechnique. Mathématiques Vol. 11; pp. 1123 - 1179
• Main Authors: Lagoutière, Frédéric, Santambrogio, Filippo, Tran Tien, Sébastien
• Format: Journal Article
• Language: English
• Published: École polytechnique 08-10-2024
• Subjects: Mathematics
• ISSN: 2429-7100; 2270-518X
• DOI: 10.5802/jep.275

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 Op ? εq 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.