Archive for Rational Mechanics and Analysis

, Volume 209, Issue 3, pp 1055–1088 | Cite as

Dimensionality of Local Minimizers of the Interaction Energy

  • D. Balagué
  • J. A. Carrillo
  • T. Laurent
  • G. Raoul
Article

Abstract

In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated with a repulsive–attractive potential. We show how the dimensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin.

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References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008Google Scholar
  2. 2.
    Balagué, D., Carrillo, J.A., Laurent, T., Raoul, G.: Nonlocal interactions by repulsive–attractive potentials: radial ins/stability. Physica D (2013, to appear)Google Scholar
  3. 3.
    Balagué, D., Carrillo, J.A., Yao, Y.: Confinement for repulsive–attractive kernels. PreprintGoogle Scholar
  4. 4.
    Bernoff, A.J., Topaz, C.M.: A primer of swarm equilibria. SIAM J. Appl. Dyn. Syst. 10(1), 212–250 (2011)Google Scholar
  5. 5.
    Bertozzi A., Carrillo J.A., Laurent T.: Blowup in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22, 683–710 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Bertozzi A., Garnett J., Laurent T.: Characterization of radially symmetric finite time blowup in multidimensional aggregation equations. SIAM J. Math. Anal. 44, 651–681 (2012)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bertozzi A., Laurent T.: Finite-time blow-up of solutions of an aggregation equation in \({\mathbb{R}^n}\). Commun. Math. Phys. 274, 717–735 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bertozzi A., Laurent T., Rosado J.: Lp theory for the multidimensional aggregation equation. Commun. Pure Appl. Math. 64(1), 45–83 (2011)MathSciNetMATHCrossRefGoogle 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(supp01), 1140005 (2012)Google Scholar
  10. 10.
    Carrillo J.A., Di Francesco M., Figalli A., Laurent T., Slepcev D.: Confinement in nonlocal interaction equations. Nonlinear Anal. 75(2), 550–558 (2012)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Carrillo J.A., Di Francesco M., Figalli A., Laurent T., Slepcev D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156, 229–271 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Carrillo J.A., McCann R.J., Villani C.: Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19(3), 971–1018 (2003)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    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, 217–263 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    D’Orsogna M.R., Chuang Y., Bertozzi A., Chayes L.: Self-propelled particles with soft-core interactions: patterns, stability and collapse. Phys. Rev. Lett. 96, 104302 (2006)ADSCrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications, 2nd edn. Wiley, Hoboken, 2003Google Scholar
  17. 17.
    Fellner K., Raoul G.: Stable stationary states of non-local interaction equations. Math. Models Methods Appl. Sci. 20(12), 2267–2291 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Fellner, K., Raoul, G.: Stability of stationary states of non-local equations with singular interaction potentials. Math. Comput. Model., 53(7-8):1436–1450, 2011Google Scholar
  19. 19.
    Fetecau R.C., Huang Y., Kolokolnikov T.: Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24(10), 2681–2716 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Givens C.R., Shortt R.M.: A class of Wasserstein metrics for probability distributions. Mich. Math. J. 31(2), 231–240 (1984)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hagan M.F., Chandler D.: Dynamic pathways for viral capsid assembly. Biophys. J. 91, 42–54 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    Kolokolnikov, T., Huang, Y., Pavlovski, M.: Singular patterns for an aggregation model with a confining potential. Physica D (in press)Google Scholar
  23. 23.
    Kolokonikov T., Sun H., Uminsky D., B ertozzi A.: Stability of ring patterns arising from 2D particle interactions. Phys. Rev. E 84(1), 015203 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    Laurent T.: Local and global existence for an aggregation equation. Commun. Partial Differ. Equ. 32, 1941–1964 (2007)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge, 1995Google Scholar
  26. 26.
    McCann R.J.: Stable rotating binary stars and fluid in a tube. Houst. J. Math. 32(2), 603–631 (2006)MathSciNetMATHGoogle Scholar
  27. 27.
    Mogilner A., Edelstein-Keshet L.: A non-local model for a swarm. J. Math. Biol. 38, 534–570 (1999)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Mogilner A., Edelstein-Keshet L., Bent L., Spiros A.: Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol. 47(4), 353–389 (2003)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Papadimitrakis, M.: Notes on classical potential theory. see http://www.math.uoc.gr/dept/lnotes/papadimitrakis+notes-on-classical-potential-theory.pdf link to web page, 2003
  30. 30.
    Pérez, J., Ros, A.: Properly embedded minimal surfaces with finite total curvature. The Global Theory of Minimal Surfaces in Flat Spaces (Martina Franca, 1999). Lecture Notes in Mathematics, vol. 1775, Springer, Berlin, 15–66, 2002Google Scholar
  31. 31.
    Raoul G.: Non-local interaction equations: stationary states and stability analysis. Differ. Integral Equ. 25(5–6), 417–440 (2012)MathSciNetMATHGoogle Scholar
  32. 32.
    Rechtsman M.C., Stillinger F.H., Torquato S.: Optimized interactions for targeted self-assembly: application to a honeycomb lattice. Phys. Rev. Lett. 95(22), 228–301 (2005)CrossRefGoogle Scholar
  33. 33.
    Sun H., Uminsky D., Bertozzi A.L.: Stability and clustering of self-similar solutions of aggregation equations. J. Math. Phys. 53, 115610 (2012)MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Topaz C., Bernoff A., Logan S.S., Toolson W.: A model for rolling swarms of locusts. Eur. Phys. J. Spec. Top. 157, 93–109 (2008)CrossRefGoogle Scholar
  35. 35.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI, 2003Google Scholar
  36. 36.
    von Brecht J., Uminsky D.: On soccer balls and linearized inverse statistical mechanics. J. Nonlinear Sci. 22(6), 935–959 (2012)MathSciNetADSMATHCrossRefGoogle Scholar
  37. 37.
    von Brecht J., Uminsky D., Kolokolnikov T., Bertozzi A.: Predicting pattern formation in particle interactions. Math. Mod. Meth. Appl. Sci. 22, 1140002 (2012)CrossRefGoogle Scholar
  38. 38.
    Wales D.J.: Energy landscapes of clusters bound by short-ranged potentials. Chem. Eur. J. Chem. Phys. 11, 2491–2494 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • D. Balagué
    • 1
  • J. A. Carrillo
    • 2
  • T. Laurent
    • 3
  • G. Raoul
    • 4
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Department of MathematicsImperial College LondonLondonUK
  3. 3.Department of MathematicsUniversity of California-RiversideRiversideUSA
  4. 4.Centre d’Ecologie Fonctionnelle et Evolutive, UMR 5175, CNRSMontpellier Cedex 5France

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