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 [23,24] to the $ \Gamma $-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on $ {\mathcal{M}} $, which was shown [10]. 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.