# American Institute of Mathematical Sciences

March  2016, 36(3): 1209-1247. doi: 10.3934/dcds.2016.36.1209

## Nonlocal-interaction equations on uniformly prox-regular sets

 1 Department of Mathematics, Imperial College, London, London SW7 2AZ 2 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States, United States

Received  April 2014 Revised  June 2015 Published  August 2015

We study the well-posedness of a class of nonlocal-interaction equations on general domains $\Omega\subset \mathbb{R}^{d}$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $\lambda$-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity. Our approach is based on the theory of gradient flows in spaces of probability measures.
Citation: José A. Carrillo, Dejan Slepčev, Lijiang Wu. Nonlocal-interaction equations on uniformly prox-regular sets. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1209-1247. doi: 10.3934/dcds.2016.36.1209
##### References:
 [1] D. Alexander, I. Kim and Y. Yao, Quasi-static evolution and congested crowd transport, Nolinearity, 27 (2014), 823-858. doi: 10.1088/0951-7715/27/4/823. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [3] A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250. doi: 10.1137/100804504. [4] M. Bounkhel, Regularity Concepts in Nonsmooth Analysis. Theory and Applications, Springer Optimization and Its Applications, 59, Springer, New York, 2012. doi: 10.1007/978-1-4614-1019-5. [5] J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211. [6] J. A. Carrillo, S. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials, Nonlinear Anal., 100 (2014), 122-147. doi: 10.1016/j.na.2014.01.010. [7] F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144. [8] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program., 104 (2005), 347-373. doi: 10.1007/s10107-005-0619-y. [9] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differential Equations, 226 (2006), 135-179. doi: 10.1016/j.jde.2005.12.005. [10] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, Springer, New York, 2007. [11] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [12] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [13] B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2011), 485-519. doi: 10.3934/nhm.2011.6.485. [14] J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris, 255 (1962), 238-240. [15] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: 10.1090/S0002-9947-00-02550-2. [16] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N. J., 1970. [17] C. Topaz, A. Bernoff, S. S. Logan and W. Toolson, A model for rolling swarms of locusts, Eur. Phys. J. Special Topics, 157 (2008), 93-109. doi: 10.1140/epjst/e2008-00633-y. [18] C. M. Topaz, M. R. D'Orsogna, L. Edelstein-Keshet and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput. Biol., 8 (2012), e1002642, 11pp. doi: 10.1371/journal.pcbi.1002642. [19] J. Venel, A numerical scheme for a class of sweeping processes, Numer. Math., 118 (2011), 367-400. doi: 10.1007/s00211-010-0329-0. [20] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. [21] C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [22] 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. [23] L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries: Compactly supported initial data case,, in preparation., ().

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##### References:
 [1] D. Alexander, I. Kim and Y. Yao, Quasi-static evolution and congested crowd transport, Nolinearity, 27 (2014), 823-858. doi: 10.1088/0951-7715/27/4/823. [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [3] A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250. doi: 10.1137/100804504. [4] M. Bounkhel, Regularity Concepts in Nonsmooth Analysis. Theory and Applications, Springer Optimization and Its Applications, 59, Springer, New York, 2012. doi: 10.1007/978-1-4614-1019-5. [5] J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211. [6] J. A. Carrillo, S. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials, Nonlinear Anal., 100 (2014), 122-147. doi: 10.1016/j.na.2014.01.010. [7] F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144. [8] J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program., 104 (2005), 347-373. doi: 10.1007/s10107-005-0619-y. [9] J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differential Equations, 226 (2006), 135-179. doi: 10.1016/j.jde.2005.12.005. [10] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, Springer, New York, 2007. [11] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [12] B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799. [13] B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2011), 485-519. doi: 10.3934/nhm.2011.6.485. [14] J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris, 255 (1962), 238-240. [15] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: 10.1090/S0002-9947-00-02550-2. [16] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N. J., 1970. [17] C. Topaz, A. Bernoff, S. S. Logan and W. Toolson, A model for rolling swarms of locusts, Eur. Phys. J. Special Topics, 157 (2008), 93-109. doi: 10.1140/epjst/e2008-00633-y. [18] C. M. Topaz, M. R. D'Orsogna, L. Edelstein-Keshet and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput. Biol., 8 (2012), e1002642, 11pp. doi: 10.1371/journal.pcbi.1002642. [19] J. Venel, A numerical scheme for a class of sweeping processes, Numer. Math., 118 (2011), 367-400. doi: 10.1007/s00211-010-0329-0. [20] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. [21] C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [22] 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. [23] L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries: Compactly supported initial data case,, in preparation., ().
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