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Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. (English) Zbl 1215.35045
The paper deals with a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The authors develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, the authors show the finite-time total collapse of the solution onto a single point for compactly supported initial measures. The presented approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, the authors exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

MSC:
35D30 Weak solutions to PDEs
35B44 Blow-up in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
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[1] M. Agueh, Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory , Adv. Differential Equations 10 (2005), 309-360. · Zbl 1103.35051
[2] L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures , Lectures Math. ETH Zürich, Birkhäuser, Basel, 2005. · Zbl 1090.35002
[3] L. Ambrosio and G. Savaré, Gradient Flows of Probability Measures , Handb. Differ. Equ. 3 , Elsevier/North-Holland, Amsterdam, 2006.
[4] D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media , RAIRO Modél. Math. Anal. Numér. 31 (1997), 615-641. · Zbl 0888.73006
[5] A. L. Bertozzi and J. Brandman, Finite-time blow-up of \(L^\infty\)-weak solutions of an aggregation equation , Commun. Math. Sci. 8 (2010), 45-65. · Zbl 1197.35061
[6] A. L. Bertozzi, J. A. Carrillo, and T. Laurent, Blowup in multidimensional aggregation equations with mildly singular interaction kernels , Nonlinearity 22 (2009), 683-710. · Zbl 1194.35053
[7] A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in \(\R^n\) , Comm. Math. Phys. 274 (2007), 717-735. · Zbl 1132.35392
[8] A. L. Bertozzi, T. Laurent, and J. Rosado, \(L^p\) Theory for the multidimensional aggregation equation , to appear in Comm. Pure Appl. Math. · Zbl 1218.35075
[9] P. Biler, G. Karch, and P. Laurençot, Blowup of solutions to a diffusive aggregation model , Nonlinearity 22 (2009), 1559-1568. · Zbl 1177.35188
[10] A. Blanchet, V. Calvez, and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model , SIAM J. Numer. Anal. 46 (2008), 691-721. · Zbl 1205.65332
[11] A. Blanchet, J. A. Carrillo, and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in \(\R^2\) , Comm. Pure Appl. Math. 61 (2008), 1449-1481. · Zbl 1155.35100
[12] A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions , Electron. J. Differential Equations (2006), no. 44. MR2226917 · Zbl 1112.35023
[13] M. Bodnar and J. J. L. Velázquez, An integro-differential equation arising as a limit of individual cell-based models , J. Differential Equations 222 (2006), 341-380. · Zbl 1089.45002
[14] S. Boi, V. Capasso, and D. Morale, “Modeling the aggregative behavior of ants of the species Polyergus rufescens ” in Spatial Heterogeneity in Ecological Models (Alcalá de Henares, Spain, 1998) , Nonlinear Anal. Real World Appl. 1 (2000), 163-176. · Zbl 1011.92053
[15] M. Burger, V. Capasso, and D. Morale, On an aggregation model with long and short range interactions , Nonlinear Anal. Real World Appl. 8 (2007), 939-958. · Zbl 1188.92040
[16] M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion , Netw. Heterog. Media 3 (2008), 749-785. · Zbl 1171.35328
[17] J. A. Carrillo, M. R. D’Orsogna, and V. Panferov, Double milling in self-propelled swarms from kinetic theory , Kinet. Relat. Models 2 (2009), 363-378. · Zbl 1195.92069
[18] J. A. Carrillo, R. J. Mccann, and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates , Rev. Mat. Iberoamericana 19 (2003), 971-1018. · Zbl 1073.35127
[19] -, Contractions in the \(2\)-Wasserstein length space and thermalization of granular media , Arch. Ration. Mech. Anal. 179 (2006), 217-263. · Zbl 1082.76105
[20] J. A. Carrillo and J. Rosado, “Uniqueness of bounded solutions to aggregation equations by optimal transport methods” in Proceedings of the 5th European Congress of Mathematicians , Eur. Math. Soc., Zürich, 2010, 3-16. · Zbl 1198.35007
[21] Y.-L. Chuang, Y. R. Huang, M. R. D’Orsogna, and A. L. Bertozzi, “Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials” in 2007 IEEE International Conference on Robotics and Automation , IEEE, Piscataway, N.J., 2007, 2292-2299.
[22] R. Dobrushin, Vlasov equations , Funktsional. Anal. i Prilozhen. 13 (1979), 48-58; English translation in Functional Anal. Appl. 13 (1979), 115-123.
[23] F. Golse, The Mean-Field Limit for the Dynamics of Large Particle Systems , Journées “Équations aux Dérivées Partielles,” exp. no. IX, Univ. Nantes, Nantes, 2003. · Zbl 1211.82037
[24] Y. Huang and A. L. Bertozzi, Self-similar blow-up solutions to an aggregation equation in \(\real^n\) , SIAM. J. Appl. Math. 70 (2010), 2582-2603. · Zbl 1238.35013
[25] R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation , SIAM J. Math. Anal. 29 (1998), 1-17. · Zbl 0915.35120
[26] E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability , J. Theor. Biol. 26 (1970), 399-415. · Zbl 1170.92306
[27] T. Laurent, Local and global existence for an aggregation equation , Comm. Partial Differential Equations 32 (2007), 1941-1964. · Zbl 1132.35088
[28] D. Li and J. Rodrigo, Finite-time singularities of an aggregation equation in \(\R^n\) with fractional dissipation , Comm. Math. Phys. 287 (2009), 687-703. · Zbl 1178.35083
[29] D. Li and J. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation , Adv. Math. 220 (2009), 1717-1738. · Zbl 1168.35037
[30] D. Li and X. Zhang, On a nonlocal aggregation model with nonlinear diffusion , Discrete Contin. Dyn. Syst. 27 (2010), 301-323. · Zbl 1209.35070
[31] H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows , Arch. Ration. Mech. Anal. 172 (2004), 407-428. · Zbl 1116.82025
[32] R. J. Mccann, A convexity principle for interacting gases , Adv. Math. 128 (1997), 153-179. · Zbl 0901.49012
[33] A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm , J. Math. Bio. 38 (1999), 534-570. · Zbl 0940.92032
[34] D. Morale, V. Capasso, and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations , J. Math. Biol. 50 (2005), 49-66. · Zbl 1055.92046
[35] H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles , Trans. Fluid Dynamics 18 (1977), 663-678.
[36] A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives , Interdiscip. Appl. Math. 14 , Springer, Berlin, 2002. · Zbl 1027.92022
[37] F. Otto, The geometry of dissipative evolution equations: The porous medium equation , Comm. Partial Differential Equations 26 (2001), 101-174. · Zbl 0984.35089
[38] C. S. Patlak, Random walk with persistence and external bias , Bull. Math. Biophys. 15 (1953), 311-338. · Zbl 1296.82044
[39] F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation , Methods Appl. Anal. 9 (2002), 533-561. · Zbl 1166.35363
[40] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits , Rev. Modern Phys. 52 (1980), 569-615.
[41] C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups , SIAM J. Appl. Math. 65 (2004), 152-174. JSTOR: · Zbl 1071.92048
[42] C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, A nonlocal continuum model for biological aggregation , Bull. Math. Biol. 68 (2006), 1601-1623. · Zbl 1334.92468
[43] G. Toscani, One-dimensional kinetic models of granular flows , M2AN Math. Model. Nummer. Anal. 34 (2000), 1277-1291. · Zbl 0981.76098
[44] C. Villani, Topics in Optimal Transportation , Grad. Stud. Math., Amer. Math. Soc., Providence, 2003. · Zbl 1106.90001
[45] -, Optimal Transport, Old and New , Grundlehren Math. Wiss. 338 , Springer, Berlin, 2009.
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