The nonlocal-interaction equation near attracting manifolds

Francesco S. Patacchini; Dejan Slepčev

doi:10.3934/dcds.2021142

Article Contents

2022, Volume 42, Issue 2: 903-929. Doi: 10.3934/dcds.2021142

This issue Previous Article Global stability solution of the 2D MHD equations with mixed partial dissipation Next Article Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

The nonlocal-interaction equation near attracting manifolds

Francesco S. Patacchini^1,2, and
Dejan Slepčev^2, ,

1.
IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France
2.
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

^* Corresponding author: Dejan Slepčev
^* Corresponding author: Dejan Slepčev

Received: June 2021

Revised: July 2021

Early access: September 2021

Published: February 2022

DS is grateful to NSF for support via grant DMS 1814991. DS and FSP are grateful to the Center for Nonlinear Analysis of CMU for its support

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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 [10]. 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.

Keywords:

Mathematics Subject Classification: 35A01, 35A02, 35A15, 35D30, 45K05, 65M12.

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Figure 1. Construction of $ \mu_{\varepsilon}^n $

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Figure 2. Dynamics of (5) approximated by (27) with domain $ {\mathcal{M}} = [-1,1] \cup \{1.5\} $ for an attractive potential

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Figure 3. Dynamics of (1) approximated by (30) with domain $ {\mathcal{M}} = \overline B(0,1) $ for repulsive potentials with varying length scales

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Figure 4. Dynamics of (1) approximated by (30) with a bean-shaped domain for a repulsive potential

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Figure 5. Dynamics of (1) approximated by (30) with domain the boundary of a bean shape for a repulsive potential

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