Long-time dynamics for a simple aggregation equation on the sphere

TL;DR

This paper thoroughly analyzes the long-term behavior of a basic model for vector alignment on the sphere, demonstrating convergence to an aligned state with explicit rates in certain cases.

Contribution

It provides a complete analysis of the asymptotic dynamics of a simple alignment model at both particle and continuum levels, including convergence proofs and rates.

Findings

Unconditional convergence to an aligned state.

Explicit convergence rates for symmetric initial data.

Complete characterization of long-time behavior.

Abstract

We give a complete study of the asymptotic behavior of a simple model of alignment of unit vectors, both at the level of particles , which corresponds to a system of coupled differential equations, and at the continuum level, under the form of an aggregation equation on the sphere. We prove unconditional convergence towards an aligned asymptotic state. In the cases of the differential system and of symmetric initial data for the partial differential equation, we provide precise rates of convergence.