THE NONLOCAL-INTERACTION EQUATION NEAR ATTRACTING MANIFOLDS

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