The nonlocal-interaction equation near attracting manifolds

<p style='text-indent:20px;'>We study the approximation of nonlocal-interaction equation restricted to a compact manifold <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> embedded in id="M2">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula>, and more generally sets with positive reach (i.e. prox-regular sets). We show that on id="M3">\begin{document}$ can be approximated by classical id="M4">\begin{document}$ adding an external potential which strongly attracts id="M5">\begin{document}$ $\end{document}</tex-math></inline-formula>. The proof relies Sandier–Serfaty approach [<xref ref-type="bibr" rid="b23">23</xref>,<xref rid="b24">24</xref>] id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence gradient flows. As by-product, we recover well-posedness for id="M7">\begin{document}$ was shown rid="b10">10</xref>]. also provide another interaction id="M8">\begin{document}$ based iterating approximately solving id="M9">\begin{document}$ projecting id="M10">\begin{document}$ convergence this scheme, together estimate rate convergence. Finally, conduct numerical experiments, both attractive-potential-based projection-based approaches, highlight effects geometry dynamics.</p>