Archive for Rational Mechanics and Analysis

, Volume 227, Issue 1, pp 1–67 | Cite as

Congested Aggregation via Newtonian Interaction

  • Katy Craig
  • Inwon Kim
  • Yao Yao


We consider a congested aggregation model that describes the evolution of a density through the competing effects of nonlocal Newtonian attraction and a hard height constraint. This provides a counterpoint to existing literature on repulsive–attractive nonlocal interaction models, where the repulsive effects instead arise from an interaction kernel or the addition of diffusion. We formulate our model as the Wasserstein gradient flow of an interaction energy, with a penalization to enforce the constraint on the height of the density. From this perspective, the problem can be seen as a singular limit of the Keller–Segel equation with degenerate diffusion. Two key properties distinguish our problem from previous work on height constrained equations: nonconvexity of the interaction kernel (which places the model outside the scope of classical gradient flow theory) and nonlocal dependence of the velocity field on the density (which causes the problem to lack a comparison principle). To overcome these obstacles, we combine recent results on gradient flows of nonconvex energies with viscosity solution theory. We characterize the dynamics of patch solutions in terms of a Hele-Shaw type free boundary problem and, using this characterization, show that in two dimensions patch solutions converge to a characteristic function of a disk in the long-time limit, with an explicit rate on the decay of the energy. We believe that a key contribution of the present work is our blended approach, combining energy methods with viscosity solution theory.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexander, D.; Kim, I.; Yao, Y.: Quasi-static evolution and congested crowd transport. Nonlinearity 27(4), 823–858 (2014). doi: 10.1088/0951-7715/27/4/823 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L.; Gigli, N.; Savaré, G.: Gradient flows in metric spaces and in the space of probability measures, 2nd edn. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)zbMATHGoogle Scholar
  3. 3.
    Ambrosio, L.; Serfaty, S.: A gradient flow approach to an evolution problem arising in superconductivity. Commun. Pure Appl. Math. 61(11), 1495–1539 (2008). doi: 10.1002/cpa.20223 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balagué, D.; Carrillo, J.; Laurent, T.; Raoul, G.: Nonlocal interactions by repulsive-attractive potentials: radial ins/stability. Phys. D. 260, 5–25 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balagué, D.; Carrillo, J.A.; Yao, Y.: Confinement for repulsive-attractive kernels. Discrete Contin. Dyn. Syst. Ser. B 19(5), 1227–1248 (2014). doi: 10.3934/dcdsb.2014.19.1227 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benedetto, D.; Caglioti, E.; Carrillo, J.A.; Pulvirenti, M.: A non-Maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys. 91(5–6), 979–990 (1998). doi: 10.1023/A:1023032000560 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bertozzi, A.L.; Carrillo, J.A.; Laurent, T.: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22(3), 683–710 (2009). doi: 10.1088/0951-7715/22/3/009 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bertozzi, A.L.; Kolokolnikov, T.; Sun, H.; Uminsky, D.; von Brecht, J.: Ring patterns and their bifurcations in a nonlocal model of biological swarms. Commun. Math. Sci. 13(4), 955–985 (2015). doi: 10.4310/CMS.2015.v13.n4.a6 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bertozzi, A.L., Laurent, T., Léger, F.: Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions. Math. Models Methods Appl. Sci. 22(suppl. 1), 1140,005, 39, 2012. doi: 10.1142/S0218202511400057
  10. 10.
    Blanchet, A.: A gradient flow approach to the Keller–Segel systems. to appear in RIMS Kokyuroku's lecture notes, preprint at
  11. 11.
    Blanchet, A.; Carlen, E.A.; Carrillo, J.A.: Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model. J. Funct. Anal. 262(5), 2142–2230 (2012). doi: 10.1016/j.jfa.2011.12.012 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Burchard, A.; Chambers, G.R.: Geometric stability of the Coulomb energy. Calc. Var. Partial Differ. Equ. 54(3), 3241–3250 (2015). doi: 10.1007/s00526-015-0900-8 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Burger, M.; Fetecau, R.; Huang, Y.: Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion. SIAM J. Appl. Dyn. Syst. 13(1), 397–424 (2014). doi: 10.1137/130923786 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Caffarelli, L., Salsa, S.: A geometric approach to free boundary problems, Graduate Studies in Mathematics, vol. 68. American Mathematical Society, Providence, RI, 2005. doi: 10.1090/gsm/068
  15. 15.
    Caffarelli, L., Vazquez, J.L.: Viscosity solutions for the porous medium equation. In: Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, pp. 13–26. Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/pspum/065/1662747
  16. 16.
    Carrillo, J.A., Hittmeir, S., Volzone, B., Yao, Y.: Nonlinear aggregation–diffusion equations: radial symmetry and long time asymptotics, in preparationGoogle Scholar
  17. 17.
    Carrillo, J.A.; Lisini, S.; Mainini, E.: Uniqueness for Keller-Segel-type chemotaxis models. Discrete Contin. Dyn. Syst. 34(4), 1319–1338 (2014). doi: 10.3934/dcds.2014.34.1319 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Carrillo, J.A.; McCann, R.J.; Villani, C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179(2), 217–263 (2006). doi: 10.1007/s00205-005-0386-1 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chuang, Y.L., Huang, Y., D'Orsogna, M., Bertozzi, A.: Multi-vehicle flocking: scalability of cooperative control algorithms using pairwise potentials. IEEE International Conference on Robotics and Automation, pp. 2292–2299, 2007Google Scholar
  20. 20.
    Craig, K.: Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions, preprint at
  21. 21.
    Doye, J.P.K.; Wales, D.J.; Berry, R.S.: The effect of the range of the potential on the structures of clusters. J. Chem. Phys. 103, 4234–4249 (1995)ADSCrossRefGoogle Scholar
  22. 22.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fellner, K.; Raoul, G.: Stable stationary states of non-local interaction equations. Math. Models Methods Appl. Sci. 20(12), 2267–2291 (2010). doi: 10.1142/S0218202510004921 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fetecau, R.C.; Huang, Y.: Equilibria of biological aggregations with nonlocal repulsive-attractive interactions. Phys. D 260, 49–64 (2013). doi: 10.1016/j.physd.2012.11.004 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fetecau, R.C.; Huang, Y.; Kolokolnikov, T.: Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24(10), 2681–2716 (2011). doi: 10.1088/0951-7715/24/10/002 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fusco, N., Maggi, F., Pratelli, A.: Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. 8(5), 51–71, 2009Google Scholar
  27. 27.
    Jordan, R.; Kinderlehrer, D.; Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998). doi: 10.1137/S0036141096303359 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Judovič, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066, 1963Google Scholar
  29. 29.
    Keller, E., Segel, L.: Initiation of slide mold aggregation viewed as an instability. J. Theor. Biol. 26, 1970Google Scholar
  30. 30.
    Kim, I., Pozar, N.: Porous medium equation to Hele-Shaw flow with general initial density, preprint at
  31. 31.
    Kim, I.; Yao, Y.: The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle. SIAM J. Math. Anal. 44(2), 568–602 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kim, I.C.: Uniqueness and existence results on the Hele-Shaw and the Stefan problems. Arch. Ration. Mech. Anal. 168(4), 299–328 (2003). doi: 10.1007/s00205-003-0251-z MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kim, I.C.; Lei, H.K.: Degenerate diffusion with a drift potential: a viscosity solutions approach. Discrete Contin. Dyn. Syst. 27(2), 767–786 (2010). doi: 10.3934/dcds.2010.27.767 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lieb, E.H.; Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (1997)Google Scholar
  35. 35.
    Lieb, E.H., Yau, H.T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112(1), 147–174, 1987.
  36. 36.
    Lin, F.; Zhang, P.: On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete Contin. Dyn. Syst. 6(1), 121–142 (2000). doi: 10.3934/dcds.2000.6.121 MathSciNetzbMATHGoogle Scholar
  37. 37.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(2), 109–145, 1984.
  38. 38.
    Loeper, G.: Uniqueness of the solution to the Vlasov-Poisson system with bounded density. J. Math. Pures Appl. (9) 86(1), 68–79, 2006. doi: 10.1016/j.matpur.2006.01.005
  39. 39.
    Masmoudi, N.; Zhang, P.: Global solutions to vortex density equations arising from sup-conductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4), 441–458 (2005). doi: 10.1016/j.anihpc.2004.07.002 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Maury, B.; Roudneff-Chupin, A.; Santambrogio, F.: A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821 (2010). doi: 10.1142/S0218202510004799 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Maury, B.; Roudneff-Chupin, A.; Santambrogio, F.; Venel, J.: Handling congestion in crowd motion modeling. Netw. Heterog. Media 6(3), 485–519 (2011). doi: 10.3934/nhm.2011.6.485 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mellet, A., Perthame, B., Quiros, F.: A Hele-Shaw problem for tumor growth, preprint at
  43. 43.
    Perea, L.; Gómez, G.; Elosegui, P.: Extension of the Cucker-Smale control law to space flight formations. AIAA J. Guid. Control Dyn. 32, 527–537 (2009)ADSCrossRefGoogle Scholar
  44. 44.
    Poupaud, F.: Diagonal defect measures, adhesion dynamics and Euler equation. Methods Appl. Anal. 9(4), 533–561 (2002). doi: 10.4310/MAA.2002.v9.n4.a4 MathSciNetzbMATHGoogle Scholar
  45. 45.
    Rechtsman, M.; Stillinger, F.; Torquato, S.: Optimized interactions for targeted self-assembly: application to a honeycomb lattice. Phys. Rev. Lett. 95(22), 228301 (2005)ADSCrossRefGoogle Scholar
  46. 46.
    Santambrogio, F.: Optimal transport for applied mathematicians. Progress in Nonlinear Differential Equations and their Applications, vol. 87. Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2. Calculus of variations, PDEs, and modeling
  47. 47.
    Sugiyama, Y.: Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems. Differ. Integral Equ. 19(8), 841–876 (2006)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Sun, H., Uminsky, D., Bertozzi, A.L.: Stability and clustering of self-similar solutions of aggregation equations. J. Math. Phys. 53(11), 115,610, 18, 2012. doi: 10.1063/1.4745180
  49. 49.
    Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(4), 697–718, 1976Google Scholar
  50. 50.
    Topaz, C.M.; Bertozzi, A.L.; Lewis, M.A.: A nonlocal continuum model forbiological aggregation. Bull. Math. Biol. 68(7), 1601–1623 (2006). doi: 10.1007/s11538-006-9088-6 MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Vázquez, J.L.: The Porous Medium Equation. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theoryGoogle Scholar
  52. 52.
    Villani, C.: Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003)zbMATHGoogle Scholar
  53. 53.
    Wales, D.: Energy landscapes of clusters bound by short-ranged potentials. Chem. Eur. J. Chem. Phys. 11, 2491–2494 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA
  2. 2.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA
  3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations