Nonlocal-interaction equations on uniformly prox-regular sets

TL;DR

This paper establishes the well-posedness and stability of nonlocal-interaction equations on uniformly prox-regular domains, analyzing how domain geometry and potential convexity influence solution behavior and aggregation.

Contribution

It proves existence, uniqueness, and stability of solutions for nonlocal-interaction equations on nonconvex domains using gradient flow theory, extending previous results to more general geometries.

Findings

Solutions are unique and stable under mild domain regularity conditions.

Domain geometry and potential convexity significantly affect long-term aggregation behavior.

Quantitative estimates relate domain shape and energy convexity to solution stability.

Abstract

We study the well-posedness of a class of nonlocal-interaction equations on general domains $\Omega\subset \mathbb{R}^d$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $\lambda$-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity. Our approach is based on the theory of gradient flows in spaces of probability measures.