Abstract
The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.
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Albi G., Balagué D., Carrillo J.A., von Brecht J.: Stability analysis of flock and mill rings for 2nd order models in swarming. SIAM J. Appl. Math. 74, 794–818 (2014)
Balagué D., Carrillo J.A., Laurent T., Raoul G.: Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability. Physica D 260, 5–25 (2013)
Balagué D., Carrillo J.A., Laurent T., Raoul G.: Dimensionality of local minimizers of the interaction energy. Arch. Rat. Mech. Anal. 209(3), 1055–1088 (2013)
Balagué D., Carrillo J.A., Yao Y.: Confinement for repulsive-attractive kernels. DCDS-B 19, 1227–1248 (2014)
Bernoff A.J., Topaz C.M.: A primer of swarm equilibria. SIAM J. Appl. Dyn. Syst. 10(1), 212–250 (2011)
Bertozzi A., Carrillo J.A., Laurent T.: Blowup in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22, 683–710 (2009)
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)
Blank I.: Sharp results for the regularity and stability of the free boundary in the obstacle problem. Indiana Univ. Math. J. 50, 1077–1112 (2001)
Brézis H., Kinderlehrer D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1974)
Caffarelli L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 44, 383–402 (1998)
Caffarelli L.A.: A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets. Boll. Un. Mat. Ital. A 18.1, 109–113 (1981)
Caffarelli L.A., Dolbeault J., Markowich P.A., Schmeiser C.: On Maxwellian equilibria of insulated semiconductors. Interfaces Free Bound. 2, 331–339 (2000)
Caffarelli L.A., Friedman A.: A singular perturbation problem for semiconductors. Boll. Un. Mat. Ital. B 7.1, 409–421 (1987)
Caffarelli L.A., Salsa S., Silvestre L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Inventiones Mathematicae 171, 425–461 (2008)
Caffarelli L.A., Vázquez J.L.: Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal. 202, 537–565 (2011)
Caffarelli L.A., Vázquez J.L.: Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29, 1393–1404 (2011)
Cañizo J.A., Carrillo J.A., Patacchini F.S.: Existence of global minimisers for the interaction energy. Arch. Rat. Mech. Anal. 217, 1197–1217 (2015)
Carrillo J.A., Chertock A., Huang Y.: A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. Commun. Comput. Phys. 17, 233–258 (2015)
Carrillo J.A., Chipot M., Huang Y.: On global minimizers of repulsive-attractive power-law interaction energies. Philos. Trans. R. Soc. A 372, 20130399 (2014)
Carrillo J.A., Di Francesco M., Figalli A., Laurent T., Slepčev D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156, 229–271 (2011)
Carrillo J.A., Di Francesco M., Figalli A., Laurent T., Slepčev D.: Confinement in nonlocal interaction equations. Nonlinear Anal. 75(2), 550–558 (2012)
Carrillo J.A., Ferreira L.C.F., Precioso J.C.: A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity. Adv. Math. 231(1), 306–327 (2012)
Carrillo J.A., Huang Y., Martin S.: Nonlinear stability of flock solutions in second-order swarming models. Nonlinear Anal. Real World Appl. 17, 332–343 (2014)
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)
Carrillo J.A., McCann R.J., Villani C.: Contractions in the 2-wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179, 217–263 (2006)
Chafaï D., Gozlan N., Zitt P.-A.: First order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24, 2371–2413 (2014)
Choksi R., Fetecau R., Topaloglu I.: On minimizers of interaction functionals with competing attractive and repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 1283–1305 (2015)
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)
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)
Fellner K., Raoul G.: Stable stationary states of non-local interaction equations. Math. Models Methods Appl. Sci. 20(12), 2267–2291 (2010)
Fellner K., Raoul G.: Stability of stationary states of non-local equations with singular interaction potentials. Math. Comput. Model. 53(7–8), 1436–1450 (2011)
Fetecau R.C., Huang Y., Kolokolnikov T.: Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24(10), 2681–2716 (2011)
Fetecau R.C., Huang Y.: Equilibria of biological aggregations with nonlocal repulsive–attractive interactions. Physica D 260, 49–64 (2013)
Frostman, O.: Potentiel d’Equilibre et Capacité des Ensembles. Ph.D. thesis, Faculté des Sciences de Lund (1935)
Givens C.R., Shortt R.M.: A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31(2), 231–240 (1984)
Gustafsson B.: A simple proof of the regularity theorem for the variational inequality of the obstacle problem. Nonlinear Anal. 12(10), 1487–1490 (1986)
Hagan M.F., Chandler D.: Dynamic pathways for viral capsid assembly. Biophys. J. 91, 42–54 (2006)
Kolokolnikov T., Carrillo J.A., Bertozzi A., Fetecau R., Lewis M.: Emergent behaviour in multi-particle systems with non-local interactions. Physica D: Nonlinear Phenomena 260, 1–4 (2013)
Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications, vol. 88 of Pure and Applied Mathematics. Academic Press, New York-London (1980)
Li H., Toscani G.: Long-time asymptotics of kinetic models of granular flows. Arch. Rat. Mech. Anal. 172(3), 407–428 (2004)
López-García, A.: Greedy energy points with external fields. Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., vol. 507, pp. 189–207. Amer. Math. Soc., Providence, RI (2010)
Mattila, P.: Geometry of sets and measures in Euclidean spaces, vol. 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). Fractals and rectifiability.
Mogilner A., Edelstein-Keshet L.: A non-local model for a swarm. J. Math. Bio. 38, 534–570 (1999)
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)
Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)
Petz, D., Hiai, F.: Logarithmic energy as an entropy functional. Advances in differential equations and mathematical physics (Atlanta, GA, 1997). Contemp. Math., vol. 217, pp. 205–221. Amer. Math. Soc., Providence, RI (1998)
Raoul G.: Non-local interaction equations: Stationary states and stability analysis. Differential Integral Equations 25(5-6), 417–440 (2012)
Rechtsman, M.C., Stillinger, F.H., Torquato, S.: Optimized interactions for targeted self-assembly: application to a honeycomb lattice. Phys. Rev. Lett. 95, 22 (2005)
Ruelle D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin Inc, New York-Amsterdam (1969)
Serfaty, S., Vázquez, J.L.: Hydrodynamic limit of nonlinear diffusions with fractional Laplacian operators. Calc. Var. PDE (2013)
Silvestre L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)
Simione R., Slepčev D., Topaloglu I.: Existence of minimizers of nonlocal interaction energies. J. Stat. Phys. 159, 972–986 (2015)
Stein E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Theil F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262, 209–236 (2006)
Villani C.: Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2003)
von Brecht J., Uminsky D.: On soccer balls and linearized inverse statistical mechanics. J. Nonlinear Sci. 22(6), 935–959 (2012)
Wales D.J.: Energy landscapes of clusters bound by short-ranged potentials. Chem. Eur. J. Chem. Phys. 11, 2491–2494 (2010)
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Communicated by C. Mouhot
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Carrillo, J.A., Delgadino, M.G. & Mellet, A. Regularity of Local Minimizers of the Interaction Energy Via Obstacle Problems. Commun. Math. Phys. 343, 747–781 (2016). https://doi.org/10.1007/s00220-016-2598-7
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DOI: https://doi.org/10.1007/s00220-016-2598-7