{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-nonlocal-interaction-equation-near","title":"The nonlocal-interaction equation near attracting manifolds","arxiv_id":"2106.01823","date":"2021-06-03","proceeding":null,"authors":["Francesco S. Patacchini","Dejan Slepčev"],"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 to the $\\Gamma$-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on $\\mathcal{M}$. Uniqueness, on the other hand, is established using a stability argument. 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.","url_abs":"https://arxiv.org/abs/2106.01823v1","url_pdf":"https://arxiv.org/pdf/2106.01823v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-nonlocal-interaction-equation-near","repo_url":"https://github.com/francesco-patacchini/interaction-equation-attracting-manifolds","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}