Collective Behavior Models with Repulsive/Attractive Potentials
Student No.:50
Time:Mon/Wed 15:10-17:00, 2015-07-27 ~ 2015-08-19 (Classes on Aug. 10 and 12 is stopped and will be recovered at 10:10-12:00 on July 31 and Aug. 4)
Instructor:Jose A. Carrillo  [Imperial College London]
Place:Conference Room 4, Floor 2, Jin Chun Yuan West Building
Starting Date:2015-7-27
Ending Date:2015-8-19

Classes on Aug. 10 and 12 is stopped and will be recovered at 10:10-12:00 on July 31 and 13:00-14:50 August 4




This course will have 4 parts corresponding to the different levels of description of collective behavior models. The starting point is particle like models, called Individual Based Models in this field. We will discuss its relation to aggregation-diffusion equations, kinetic models, and hydrodynamic systems. The main emphasis will be in obtaining information about the qualitative properties of their solutions. We are mostly interested in stability properties, rates of decay, pattern shape properties, and the connection and derivation of the different description levels. Numerical simulations and schemes are also integral part of the course leading to motivations and to different questions regarding the approximation of the continuum models.





Partial Differential Equations, Dynamical Systems, Numerical Analysis of Partial Differential Equations





The lectures will include materials and ideas of:

*          J. A. Carrillo, Y.-P. Choi, E. Tadmor, C. Tan, Critical thresholds in 1D Euler equations with nonlocal forces, Preprint.
*          J. A. Carrillo, A. Klar, A. Roth, Single to Double Mill Small Noise Transition via Semi-Lagrangian Finite Volume Methods, Preprint.
*          J. A. Carrillo, M. G. Delgadino, A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Preprint.
*          J. A. Cañizo, J. A. Carrillo, F. S. Patacchini, Existence of Compactly Supported Global Minimisers for the Interaction Energy, to appear in Arch. Rat. Mech. Anal.
*          J. A. Carrillo, A. Chertock, Y. Huang, A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure, Comm. in Comp. Phys. 17, 233-258, 2015.
*          J. A. Carrillo, D. Castorina, B. Volzone, Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction, SIAM J. Math. Anal. 47, 1-25, 2015.
*          G. A. Bonaschi, J. A. Carrillo, M. DiFrancesco, M. A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, Control, Optimization and Calculus of Variations 21, 414--441, 2015.
*          J. A. Carrillo, Y. Huang, S. Martin, Nonlinear stability of flock solutions in second-order swarming models, Nonlinear Analysis: Real World Applications 17, 332–343, 2014.
*          G. Albi, D. Balagué, J. A. Carrillo, J. von Brecht, Stability Analysis of Flock and Mill rings for 2nd Order Models in Swarming, SIAM J. Appl. Math. 74, 794–818, 2014.
*          D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul, Dimensionality of Local Minimizers of the Interaction Energy, Archive for Rational Mechanics and Analysis 209, 1055--1088, 2013.
*          M. Bostan, J. A. Carrillo, Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming, Math. Mod. Meth. in the Appl. Sci. 23, 2353, 2013.
*          D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul, Nonlocal interactions by repulsive-attractive potentials: radial ins/stability, Physica D 260, 5-25, 2013.
*          F. Bolley, J. A. Cañizo, J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Letters 25, 339-343, 2012.
*          F. Bolley, J. A. Cañizo, J. A. Carrillo, Stochastic Mean-Field Limit: Non-Lipschitz Forces & Swarming, Math. Mod. Meth. Appl. Sci. 21, 2179-2210, 2011.
*          A. Blanchet, E. A. Carlen, J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Func. Anal. 262, 2142-2230, 2012.
*          E. A. Carlen, J. A. Carrillo, M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Nat. Acad. USA 107 (46), 19696-19701, 2010.
*          J. A. Carrillo, A. Klar, S. Martin, S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Mod. Meth. Appl. Sci. 20, 1533-1552, 2010.
*          J. A. Carrillo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic Flocking Dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. 42, 218-236, 2010.
*          J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J. 156, 229-271, 2011.
*          J. A. Carrillo, M. R. D'Orsogna, V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinetic and Related Models 2, 363-378, 2009.