# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021142
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## The nonlocal-interaction equation near attracting manifolds

 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

Received  June 2021 Revised  July 2021 Early access September 2021

Fund Project: 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

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.

Citation: Francesco S. Patacchini, Dejan Slepčev. The nonlocal-interaction equation near attracting manifolds. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021142
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc, New York, 1999. doi: 10.1002/9780470316962.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] K. Craig and I. Topaloglu, Convergence of regularized nonlocal interaction energies, SIAM J. Math. Anal., 48 (2016), 34-60.  doi: 10.1137/15M1013882.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [15] R. C. Fetecau and B. Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.  doi: 10.3934/jgm.2019020.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc, New York, 1999. doi: 10.1002/9780470316962.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] K. Craig and I. Topaloglu, Convergence of regularized nonlocal interaction energies, SIAM J. Math. Anal., 48 (2016), 34-60.  doi: 10.1137/15M1013882.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [15] R. C. Fetecau and B. Zhang, Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.  doi: 10.3934/jgm.2019020.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Construction of $\mu_{\varepsilon}^n$
Dynamics of (5) approximated by (27) with domain ${\mathcal{M}} = [-1,1] \cup \{1.5\}$ for an attractive potential
Dynamics of (1) approximated by (30) with domain ${\mathcal{M}} = \overline B(0,1)$ for repulsive potentials with varying length scales
Dynamics of (1) approximated by (30) with a bean-shaped domain for a repulsive potential
Dynamics of (1) approximated by (30) with domain the boundary of a bean shape for a repulsive potential
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