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 & Continuous Dynamical Systems - A, 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.  doi: 10.1088/0951-7715/27/4/823.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Second edition, (2008).   Google Scholar

[3]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria,, SIAM J. Appl. Dyn. Syst., 10 (2011), 212.  doi: 10.1137/100804504.  Google Scholar

[4]

M. Bounkhel, Regularity Concepts in Nonsmooth Analysis. Theory and Applications,, Springer Optimization and Its Applications, (2012).  doi: 10.1007/978-1-4614-1019-5.  Google Scholar

[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.  doi: 10.1215/00127094-2010-211.  Google Scholar

[6]

J. A. Carrillo, S. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials,, Nonlinear Anal., 100 (2014), 122.  doi: 10.1016/j.na.2014.01.010.  Google Scholar

[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.   Google Scholar

[8]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program., 104 (2005), 347.  doi: 10.1007/s10107-005-0619-y.  Google Scholar

[9]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation,, J. Differential Equations, 226 (2006), 135.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar

[10]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces,, Springer Monographs in Mathematics, (2007).   Google Scholar

[11]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[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.  doi: 10.1142/S0218202510004799.  Google Scholar

[13]

B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling,, Netw. Heterog. Media, 6 (2011), 485.  doi: 10.3934/nhm.2011.6.485.  Google Scholar

[14]

J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires,, C. R. Acad. Sci. Paris, 255 (1962), 238.   Google Scholar

[15]

R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions,, Trans. Amer. Math. Soc., 352 (2000), 5231.  doi: 10.1090/S0002-9947-00-02550-2.  Google Scholar

[16]

R. T. Rockafellar, Convex Analysis,, Princeton Mathematical Series, (1970).   Google Scholar

[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.  doi: 10.1140/epjst/e2008-00633-y.  Google Scholar

[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).  doi: 10.1371/journal.pcbi.1002642.  Google Scholar

[19]

J. Venel, A numerical scheme for a class of sweeping processes,, Numer. Math., 118 (2011), 367.  doi: 10.1007/s00211-010-0329-0.  Google Scholar

[20]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003).   Google Scholar

[21]

C. Villani, Optimal Transport. Old and New,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[22]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries,, Comm. Partial Differential Equations, 40 (2015), 1241.  doi: 10.1080/03605302.2015.1015033.  Google Scholar

[23]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries: Compactly supported initial data case,, in preparation., ().   Google Scholar

show all references

References:
[1]

D. Alexander, I. Kim and Y. Yao, Quasi-static evolution and congested crowd transport,, Nolinearity, 27 (2014), 823.  doi: 10.1088/0951-7715/27/4/823.  Google Scholar

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures,, Second edition, (2008).   Google Scholar

[3]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria,, SIAM J. Appl. Dyn. Syst., 10 (2011), 212.  doi: 10.1137/100804504.  Google Scholar

[4]

M. Bounkhel, Regularity Concepts in Nonsmooth Analysis. Theory and Applications,, Springer Optimization and Its Applications, (2012).  doi: 10.1007/978-1-4614-1019-5.  Google Scholar

[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.  doi: 10.1215/00127094-2010-211.  Google Scholar

[6]

J. A. Carrillo, S. Lisini and E. Mainini, Gradient flows for non-smooth interaction potentials,, Nonlinear Anal., 100 (2014), 122.  doi: 10.1016/j.na.2014.01.010.  Google Scholar

[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.   Google Scholar

[8]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process,, Math. Program., 104 (2005), 347.  doi: 10.1007/s10107-005-0619-y.  Google Scholar

[9]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation,, J. Differential Equations, 226 (2006), 135.  doi: 10.1016/j.jde.2005.12.005.  Google Scholar

[10]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces,, Springer Monographs in Mathematics, (2007).   Google Scholar

[11]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[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.  doi: 10.1142/S0218202510004799.  Google Scholar

[13]

B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling,, Netw. Heterog. Media, 6 (2011), 485.  doi: 10.3934/nhm.2011.6.485.  Google Scholar

[14]

J.-J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires,, C. R. Acad. Sci. Paris, 255 (1962), 238.   Google Scholar

[15]

R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions,, Trans. Amer. Math. Soc., 352 (2000), 5231.  doi: 10.1090/S0002-9947-00-02550-2.  Google Scholar

[16]

R. T. Rockafellar, Convex Analysis,, Princeton Mathematical Series, (1970).   Google Scholar

[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.  doi: 10.1140/epjst/e2008-00633-y.  Google Scholar

[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).  doi: 10.1371/journal.pcbi.1002642.  Google Scholar

[19]

J. Venel, A numerical scheme for a class of sweeping processes,, Numer. Math., 118 (2011), 367.  doi: 10.1007/s00211-010-0329-0.  Google Scholar

[20]

C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics, (2003).   Google Scholar

[21]

C. Villani, Optimal Transport. Old and New,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[22]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries,, Comm. Partial Differential Equations, 40 (2015), 1241.  doi: 10.1080/03605302.2015.1015033.  Google Scholar

[23]

L. Wu and D. Slepčev, Nonlocal interaction equations in environments with heterogeneities and boundaries: Compactly supported initial data case,, in preparation., ().   Google Scholar

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