The nonlocal-interaction equation near attracting manifolds

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.