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Construction of $ \mu_{\varepsilon}^n $
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February
2022, 42(2): 903-929.
doi: 10.3934/dcds.2021142
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 |
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 [
Keywords:
Interaction equation,
positive reach,
manifold with boundary,
Γ-convergence,
gradient flow.
Mathematics Subject Classification:
35A01, 35A02, 35A15, 35D30, 45K05, 65M12.
Citation:
Francesco S. Patacchini, Dejan Slepčev. The nonlocal-interaction equation near attracting manifolds. Discrete and Continuous Dynamical Systems,
2022, 42
(2)
: 903-929.
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. |
| [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. |
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. |
| [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. |
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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