Publication 21-CNA-017
The nonlocal-interaction equation near attracting manifolds
Francesco S. Patacchini
IFP Energies nouvelles
1 et 4 avenue de Bois-Pr´eau, 92852 Rueil-Malmaison, France
and
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15203, USA
francesco.patacchini@ifpen.fr
Dejan Slepčev
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
slepcev@andrew.cmu.edu
Abstract: 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 [9]. 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.
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