The nonlocal-interaction equation near attracting manifolds

Abstract

We study the approximation of the nonlocal-interaction equation restricted to a compact manifold \begin{document}$ {\mathcal{M}} $\end{document} embedded in \begin{document}$ {\mathbb{R}}^d $\end{document}, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on \begin{document}$ {\mathcal{M}} $\end{document} can be approximated by the classical nonlocal-interaction equation on \begin{document}$ {\mathbb{R}}^d $\end{document} by adding an external potential which strongly attracts to \begin{document}$ {\mathcal{M}} $\end{document}. The proof relies on the Sandier–Serfaty approach [23,24] to the \begin{document}$ \Gamma $\end{document}-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on \begin{document}$ {\mathcal{M}} $\end{document}, which was shown [10]. We also provide an another approximation to the interaction equation on \begin{document}$ {\mathcal{M}} $\end{document}, based on iterating approximately solving an interaction equation on \begin{document}$ {\mathbb{R}}^d $\end{document} and projecting to \begin{document}$ {\mathcal{M}} $\end{document}. 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.