Abstract

We propose and analyze a natural extension of the Moreau sweeping process: given a family of moving convex sets (C(t))t(C(t))_t, we look for the evolution of a probability density ρt\rho_t, constrained to be supported on C(t). We describe in detail three cases: in the first, particles do not interact with each other and stay at rest unless pushed by the moving boundary; in the second they interact via a maximal density constraint ρ ≤ 1, so that they are not only pushed by the boundary, but also by the other particles; in the third case particles are submitted to Brownian diffusion, reflected along the moving boundary. We prove existence, uniqueness and approximation results by using techniques from optimal transport, and we provide numerical illustrations.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Sweeping process
  • optimal transportation
  • Wasserstein distance
  • differential inclusion
  • subdifferential
  • rotor router model

Mathematics Subject Classification

35R37, 35R05, 49J45, 49J53

References

  1. D. Alexander, I. Kim, Y. Yao: Quasi-static evolution and congested crowd transport, Nonlinearity 27 (2014) 1–36.https://doi.org/10.1088/0951-7715/27/4/823Google Scholar
  2. L. Ambrosio: Minimizing movements, Rend. Accad. Naz. Sci. XL, Mem. Mat. Appl. (5) 19 (1995) 191–246.Google Scholar
  3. L. Ambrosio, G. Crippa: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, in: Transport Equations and Multi-D Hyperbolic Conservation Laws, F. Ancona et al. (ed.), Lecture Notes of the Unione Matematica Italiana 5, Springer, Berlin (2008) 3–57.https://doi.org/10.1007/978-3-540-76781-7_1Google Scholar
  4. L. Ambrosio, N. Gigli, G. Savaré: Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel (2005).Google Scholar
  5. L. Ambrosio, P. Tilli: Topics on Analysis in Metric Spaces, Oxford Lecture Series in Mathematics and its Applications 25, Oxford University Press, Oxford (2004).https://doi.org/10.1093/oso/9780198529385.001.0001Google Scholar
  6. F. Bernicot, J. Venel: Differential inclusions with proximal normal cones in Banach spaces, J. Convex Analysis 17 (2010) 451–484.https://doi.org/10.68381/jca17032Google Scholar
  7. F. Bernicot, J. Venel: Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme, J. Differ. Equations (4–5) (2011) 1195–1224.https://doi.org/10.1016/j.jde.2011.03.010Google Scholar
  8. M. Bounkhel, L. Thibault: Nonconvex sweeping process and prox-regularity in Hilbert space, J. Nonlinear Convex Anal. 6 (2005) 359–374.Google Scholar
  9. A. Boulkhemair, A. Chakib: On the uniform Poincaré inequality, Commun. Partial Differ. Equations 32 (2007) 1439–1447.https://doi.org/10.1080/03605300600910241Google Scholar
  10. Y. Brenier: Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci., Paris, Sér. I 305(19) (1987) 805–808.Google Scholar
  11. G. Colombo, V. V. Goncharov: The sweeping processes without convexity, Set-Valued Anal. 7 (1999) 357–374.https://doi.org/10.1023/A:1008774529556Google Scholar
  12. G. Colombo, M. D. P. Monteiro Marques: Sweeping by a continuous prox-regular set, J. Differ. Equations 187 (2003) 46–62.https://doi.org/10.1016/S0022-0396(02)00021-9Google Scholar
  13. G. De Philippis, A. R. Mészáros, F. Santambrogio, B. Velichkov: BV estimates in optimal transportation and applications, Arch. Ration. Mech. Anal. 219 (2016) 829–860.https://doi.org/10.1007/s00205-015-0909-3Google Scholar
  14. S. Di Marino, A. R. Mészáros: Uniqueness issues for evolutive equations with density constraints, preprint, arXiv:1507.02900 (2015).Google Scholar
  15. J. L. Doob: Classical Potential Theory and its Probabilistic Counterpart, Grundlehren der Mathematischen Wissenschaften 262, Springer, New York (1984).https://doi.org/10.1007/978-1-4612-5208-5_15Google Scholar
  16. J. F. Edmond, L. Thibault: Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program. 104 (2005) 347–373.https://doi.org/10.1007/s10107-005-0619-yGoogle Scholar
  17. J. F. Edmond, L. Thibault: BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differ. Equations 226 (2006) 135–179.https://doi.org/10.1016/j.jde.2005.12.005Google Scholar
  18. B. Gustafsson, M. Sakai: Properties of some balayage operators, with applications to quadrature domains and moving boundary problems, Nonlinear Anal. 22 (1994) 1221–1245.https://doi.org/10.1016/0362-546X(94)90107-4Google Scholar
  19. R. Jordan, D. Kinderlehrer, F. Otto: The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998) 1–17.https://doi.org/10.1137/S0036141096303359Google Scholar
  20. A. Lefebvre-Lepot, B. Maury: Numerical Modeling of Fluid-Grain Interactions; Close Interaction of Immersed Grains; Chapters 11 and 12 in: Discrete Numerical Modeling of Ganular Materials, Arang Radjai, Frederic Dubois (eds.), ISTE, London (2011) 303–346.Google Scholar
  21. L. Levine, Y. Peres: Spherical asymptotics for the rotor-router model in Zd, Indiana Univ. Math. J. 57(1) (2008) 431–449.https://doi.org/10.1512/iumj.2008.57.3022Google Scholar
  22. G. Loeper: Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. 86(1) (2006) 68–79.https://doi.org/10.1016/j.matpur.2006.01.005Google Scholar
  23. J. Matousek: Lectures on Discrete Geometry, Graduate Texts in Mathematics 212, Springer, New York (2002).https://doi.org/10.1007/978-1-4613-0039-7Google Scholar
  24. B. Maury, A. Roudneff-Chupin, F. Santambrogio: A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci. 20 (2010) 1787–1821.https://doi.org/10.1142/S0218202510004799Google Scholar
  25. B. Maury, A. Roudneff-Chupin, F. Santambrogio: Congestion-driven dendritic growth, Discrete Contin. Dyn. Syst. 34(4) (2014) 1575–1604.https://doi.org/10.3934/dcds.2014.34.1575Google Scholar
  26. B. Maury, A. Roudneff-Chupin, F. Santambrogio, J. Venel: Handling congestion in crowd motion modeling, Netw. Heterog. Media 6(3) (2011) 485–519.https://doi.org/10.3934/nhm.2011.6.485Google Scholar
  27. B. Maury, J. Venel: A discrete contact model for crowd motion, ESAIM, Math. Model. Numer. Anal. 45 (2011) 145–168.https://doi.org/10.1051/m2an/2010035Google Scholar
  28. R. J. Mc Cann: A convexity principle for interacting gases, Adv. Math. 128(1) (1997) 153–159.https://doi.org/10.1006/aima.1997.1634Google Scholar
  29. A. R. Mészáros: Mean Field Games with Density Constraints, MSc Thesis, École Polytechnique, Palaiseau (2012).Google Scholar
  30. A. R. Mészáros, F. Santambrogio: A diffusive model for macroscopic crowd motion with density constraints, preprint (2015); available at: http://cvgmt.sns.it/paper/2644/.Google Scholar
  31. J. J. Moreau: On unilateral constraints, friction and plasticity, in: New Variational Techniques in Mathematical Physics (Bressanone, 1973), G. Capriz, G. Stampacchia (eds.), Cremonese, Roma (1974) 173–322.Google Scholar
  32. J.-J. Moreau: Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equations 26 (1977) 346–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  33. J.-J. Moreau: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci., Paris 255 (1962) 238–240.Google Scholar
  34. F. Otto: The geometry of dissipative evolution equations: the porous medium equation, Commun. Partial Differ. Equations 26(1–2) (2001) 101–174.https://doi.org/10.1081/PDE-100002243Google Scholar
  35. P. Pécol, S. Dal Pont, S. Erlicher, P. Argoul: Smooth/non-smooth contact modeling of human crowds movement: numerical aspects and application to emergency evacuations, Physica A: Statist. Mech. Appl. 388 (2009) 3600–3608.Google Scholar
  36. R. A. Poliquin, R. T. Rockafellar: Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996) 1805–1838.https://doi.org/10.1090/S0002-9947-96-01544-9Google Scholar
  37. R. A. Poliquin, R. T. Rockafellar, L. Thibault: Local differentiability of distance functions, Trans. Amer. Math. Soc. 352 (2000) 5231–5249.https://doi.org/10.1090/S0002-9947-00-02550-2Google Scholar
  38. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  39. F. Santambrogio: Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications 87, Birkhäuser, Basel (2015).https://doi.org/10.1007/978-3-319-20828-2Google Scholar
  40. R. Schneider: Curvature measures of convex bodies, Ann. Mat. Pura Appl. 116(1) (1978) 101–134.https://doi.org/10.1007/BF02413869Google Scholar
  41. D. E. Stewart: Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem, Arch. Ration. Mech. Anal. 145 (1998) 215–260.https://doi.org/10.1007/s002050050129Google Scholar
  42. L. Thibault: Sweeping process with regular and nonregular sets, J. Differ. Equations 193 (2003) 1–26.https://doi.org/10.1016/S0022-0396(03)00129-3Google Scholar
  43. J. Venel: A numerical scheme for a class of sweeping processes, Numer. Math. 118(2) (2011) 367–400.https://doi.org/10.1007/s00211-010-0329-0Google Scholar
  44. C. Villani: Topics in Optimal Transportation, Graduate Studies in Mathematics 58, AMS, Providence (2003).https://doi.org/10.1090/gsm/058Google Scholar

Suggested citation

S. Di Marino, B. Maury, F. Santambrogio. “Measure Sweeping Processes.” Journal of Convex Analysis 23 (2016), No. 2, 567–601. https://doi.org/10.68381/jca23021

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2016. Rights now held by Banach Press.