The Nonlocal-Interaction Equation Near Attracting Manifolds

Francesco S Patacchini (1, 2) , Dejan Slepčev (2)

1 IFPEN - IFP Energies nouvelles
2 CMU - Carnegie Mellon University [Pittsburgh]

We study the approximation of the nonlocal-interaction equation restricted to a compact manifold M embedded in Rd, 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 Rd by adding an external potential which strongly attracts to M. The proof relies on the Sandier–Serfaty approach [23,24] to the Γ-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 Rd 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.

Mots clés: Interaction Equation; Positive Reach; Manifold with Boundary; Γ-Convergence; Gradient Flow

Domaines: Sciences de l'ingénieur [physics]; Sciences de l'environnement

Francesco S Patacchini, Dejan Slepčev. The Nonlocal-Interaction Equation Near Attracting Manifolds. Discrete and Continuous Dynamical Systems - Series A, 2022, 42 (2), pp.903-929. ⟨10.3934/dcds.2021142⟩. ⟨hal-03563678⟩