Abstract
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.
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References
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
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)
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
Balagué, D.; Carrillo, J.; Laurent, T.; Raoul, G.: Nonlocal interactions by repulsive-attractive potentials: radial ins/stability. Phys. D. 260, 5–25 (2013)
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
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
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
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
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
Blanchet, A.: A gradient flow approach to the Keller–Segel systems. to appear in RIMS Kokyuroku's lecture notes, preprint at http://publications.ut-capitole.fr/16518/
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
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
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
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
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
Carrillo, J.A., Hittmeir, S., Volzone, B., Yao, Y.: Nonlinear aggregation–diffusion equations: radial symmetry and long time asymptotics, in preparation
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
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
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, 2007
Craig, K.: Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions, preprint at http://arxiv.org/abs/1512.07255
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)
Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
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
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
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
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, 2009
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
Judovič, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066, 1963
Keller, E., Segel, L.: Initiation of slide mold aggregation viewed as an instability. J. Theor. Biol. 26, 1970
Kim, I., Pozar, N.: Porous medium equation to Hele-Shaw flow with general initial density, preprint at http://arxiv.org/abs/1509.06287
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)
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
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
Lieb, E.H.; Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (1997)
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. http://projecteuclid.org/euclid.cmp/1104159813
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
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. http://www.numdam.org/item?id=AIHPC_1984__1_2_109_0
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
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
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
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
Mellet, A., Perthame, B., Quiros, F.: A Hele-Shaw problem for tumor growth, preprint at http://arxiv.org/abs/1512.069957
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)
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
Rechtsman, M.; Stillinger, F.; Torquato, S.: Optimized interactions for targeted self-assembly: application to a honeycomb lattice. Phys. Rev. Lett. 95(22), 228301 (2005)
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
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)
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
Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(4), 697–718, 1976
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
Vázquez, J.L.: The Porous Medium Equation. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory
Villani, C.: Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003)
Wales, D.: Energy landscapes of clusters bound by short-ranged potentials. Chem. Eur. J. Chem. Phys. 11, 2491–2494 (2010)
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Communicated by D. Kinderlehrer
Katy Craig is supported by a UC President’s Postdoctoral Fellowship and by the NSF Grant DMS-1401867.
Inwon Kim is supported by the NSF Grant DMS-1300445.
Yao Yao is supported by the NSF Grant DMS-1565480.
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Craig, K., Kim, I. & Yao, Y. Congested Aggregation via Newtonian Interaction. Arch Rational Mech Anal 227, 1–67 (2018). https://doi.org/10.1007/s00205-017-1156-6
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DOI: https://doi.org/10.1007/s00205-017-1156-6