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

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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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References

[1]	H. Ahn, S.-Y. Ha, H. Park and W. Shim, Emergent behaviors of Cucker–Smale flocks on the hyperboloid, J. Math. Phys., 62 (2021), Paper No. 082702, 22 pp. arXiv: 2007.02556. doi: 10.1063/5.0020923.
[2]	L. Alasio, M. Bruna and J. A. Carrillo, The role of a strong confining potential in a nonlinear Fokker-Planck equation, Nonlinear Analysis, 193 (2020), 111480, 28 pp. doi: 10.1016/j.na.2019.03.003.
[3]	L. Ambrosio and N. Gigli, A user's guide to optimal transport, in Modelling and Optimisation of Flows on Networks, vol. 2062 of Lecture Notes in Math., Springer, Heidelberg, (2013), 1–155. doi: 10.1007/978-3-642-32160-3_1.
[4]	L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.
[5]	A. L. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83. doi: 10.1002/cpa.20334.
[6]	P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc, New York, 1999. doi: 10.1002/9780470316962.
[7]	J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539. doi: 10.1142/S0218202511005131.
[8]	J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds, 1–46, CISM Courses and Lect., 553, Springer, Vienna, (2014). doi: 10.1007/978-3-7091-1785-9_1.
[9]	J. A. Carrillo, F. S. Patacchini, P. Sternberg and G. Wolansky, Convergence of a particle method for diffusive gradient flows in one dimension, SIAM J. Math. Anal., 48 (2016), 3708-3741. doi: 10.1137/16M1077210.
[10]	J. A. Carrillo, D. Slepčev and L. Wu, Nonlocal interaction equations on uniformly prox-regular sets, Discrete Contin. Dyn. Syst., 36 (2016), 1209-1247. doi: 10.3934/dcds.2016.36.1209.
[11]	K. Craig and I. Topaloglu, Convergence of regularized nonlocal interaction energies, SIAM J. Math. Anal., 48 (2016), 34-60. doi: 10.1137/15M1013882.
[12]	S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, Optimal Transport Theory and Applications, (2014), 100–144. doi: 10.1017/CBO9781107297296.007.
[13]	R. C. Fetecau, S.-Y. Ha and H. Park, An intrinsic aggregation model on the special orthogonal group $SO(3)$: Well-posedness and collective behaviours, J. Nonlinear Sci., 31 (2021), Paper No. 74, 61 pp. doi: 10.1007/s00332-021-09732-2.
[14]	R. C. Fetecau, H. Park and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation modelwith intrinsic interactions on sphere and other manifolds, preprint, arXiv: 2004.06951, (2020).
[15]	R. C. Fetecau and B. Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426. doi: 10.3934/jgm.2019020.
[16]	N. García Trillos, M. Gerlach, M. Hein and D. Slepčev, Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace-Beltrami operator, Found. Comput. Math., 20 (2020), 827-887. doi: 10.1007/s10208-019-09436-w.
[17]	S.-Y. Ha and D. Kim, A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116. doi: 10.1137/18M1205996.
[18]	S.-Y. Ha, D. Kim, J. Lee and S. E. Noh, Particle and kinetic models for swarming particles on a sphere and stability properties, J. Stat. Phys., 174 (2019), 622-655. doi: 10.1007/s10955-018-2169-8.
[19]	S. Lisini, Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces, ESAIM: Control Optim. Calc. Var., 15 (2009), 712-740. doi: 10.1051/cocv:2008044.
[20]	P. Niyogi, S. Smale and S. Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom., 39 (2008), 419-441. doi: 10.1007/s00454-008-9053-2.
[21]	F. S. Patacchini and D. Slepčev, GitHub repository for present paper with open source code, https://github.com/francesco-patacchini/interaction-equation-attracting-manifolds.
[22]	J. Rataj and L. Zajíček, On the structure of sets with positive reach, Math. Nachr., 290 (2017), 1806-1829. doi: 10.1002/mana.201600237.
[23]	E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.
[24]	S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.
[25]	L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries, Comm. Partial Differential Equations, 40 (2015), 1241-1281. doi: 10.1080/03605302.2015.1015033.