Relaxation limit of the aggregation equation with pointy potential

TL;DR

This paper studies the relaxation limit of the aggregation equation with a pointy potential in one dimension, analyzing convergence, providing estimates, and exploring numerical schemes for measure-valued solutions.

Contribution

It introduces a relaxation approximation for measure-valued solutions of the aggregation equation with a pointy potential and provides convergence analysis and numerical discretization methods.

Findings

Proves convergence of the relaxation approximation in one dimension with Newtonian potential.

Provides rigorous estimates of the convergence speed.

Develops uniformly accurate numerical schemes for the relaxation limit.

Abstract

This work is devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one dimensional space. The aggregation equation is by now widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigate an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigate the numerical discretization of this relaxation limit by uniformly accurate schemes.