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The nonlocal-interaction equation near attracting manifolds

  • * Corresponding author: Dejan Slepčev

    * Corresponding author: Dejan Slepčev

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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  • 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.

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


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

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

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

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

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

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