The nonlocal-interaction equation near attracting manifolds

TL;DR

This paper investigates how nonlocal-interaction equations constrained to manifolds can be approximated by classical equations in Euclidean space with external potentials, providing theoretical convergence results and numerical insights into geometric effects.

Contribution

It introduces new approximation schemes for nonlocal-interaction equations on manifolds and proves their convergence, extending well-posedness and stability analysis.

Findings

Approximation by external potential strongly attracting to the manifold.

Convergence of iterative projection-based scheme.

Numerical experiments illustrating geometric effects on dynamics.

Abstract

We study the approximation of the nonlocal-interaction equation restricted to a compact manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on $\mathcal{M}$ can be approximated by the classical nonlocal-interaction equation on $\mathbb{R}^d$ by adding an external potential which strongly attracts to $\mathcal{M}$. The proof relies on the Sandier--Serfaty approach to the $\Gamma$-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on $\mathcal{M}$. Uniqueness, on the other hand, is established using a stability argument. We also provide an another approximation to the interaction equation on $\mathcal{M}$, based on iterating approximately solving an interaction equation on $\mathbb{R}^d$ and projecting to $\mathcal{M}$. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.