Citation: |
[1] |
L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 217-246.doi: 10.1016/j.anihpc.2010.11.006. |
[2] |
L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math., 61 (2008), 1495-1539.doi: 10.1002/cpa.20223. |
[3] |
L. A. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005. |
[4] |
D. Balagué and J. A. Carrillo, Aggregation equation with growing at infinity attractive-repulsive potentials, In Hyperbolic problems-theory, numerics and applications. Volume 1, volume 17 of Ser. Contemp. Appl. Math. CAM, pages 136-147. World Sci. Publishing, Singapore, 2012. |
[5] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.doi: 10.1007/s00205-013-0644-6. |
[6] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25.doi: 10.1016/j.physd.2012.10.002. |
[7] |
D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numér., 31 (1997), 615-641. |
[8] |
A. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.doi: 10.1088/0951-7715/22/3/009. |
[9] |
A. Bertozzi, T. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.doi: 10.1002/cpa.20334. |
[10] |
A. L. Bertozzi, T. Laurent and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), (suppl. 1) 1140005, 39pp.doi: 10.1142/S0218202511400057. |
[11] |
J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Confinement in nonlocal interaction equations, Nonlinear Anal., 75 (2012), 550-558.doi: 10.1016/j.na.2011.08.057. |
[12] |
J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.doi: 10.1215/00127094-2010-211. |
[13] |
J. A. Carrillo, S. Martin and V. Panferov, A new interaction potential for swarming models, Phys. D, 260 (2013), 112-126.doi: 10.1016/j.physd.2013.02.004. |
[14] |
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.doi: 10.4171/RMI/376. |
[15] |
J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006),217-263.doi: 10.1007/s00205-005-0386-1. |
[16] |
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.doi: 10.3934/krm.2009.2.363. |
[17] |
Y. Chuang, M. R. D'Orsogna, D. Marthaler, L. S. Chayes and A. L. Bertozzi, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.doi: 10.1016/j.physd.2007.05.007. |
[18] |
M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006), 104302.doi: 10.1103/PhysRevLett.96.104302. |
[19] |
Q. Du and P. Zhang, Existence of weak solutions to some vortex density models, SIAM J. Math. Anal., 34 (2003), 1279-1299(electronic).doi: 10.1137/S0036141002408009. |
[20] |
K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.doi: 10.1142/S0218202510004921. |
[21] |
K. Fellner and G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53 (2011), 1436-1450.doi: 10.1016/j.mcm.2010.03.021. |
[22] |
R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.doi: 10.1016/j.physd.2012.11.004. |
[23] |
R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.doi: 10.1088/0951-7715/24/10/002. |
[24] |
E. Geigant, K. Ladizhansky and A. Mogilner, An integrodifferential model for orientational distributions of f-actin in cells, SIAM Journal on Applied Mathematics, 59 (1998), 787-809.doi: 10.1088/0951-7715/24/10/002. |
[25] |
K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, An integro-differential equation model for alignment and orientational aggregation, J. Differential Equations, 264 (2009), 1387-1421.doi: 10.1016/j.jde.2008.11.006. |
[26] |
T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential, Phys. D, 260 (2013), 65-76.doi: 10.1016/j.physd.2012.10.009. |
[27] |
T. Kolokonikov, H. Sun, D. Uminsky, and A. Bertozzi., Stability of ring patterns arising from 2d particle interactions, Phys. Rev. E, 84 (2011), 015203.doi: 10.1103/PhysRevE.84.015203. |
[28] |
H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), 407-428.doi: 10.1007/s00205-004-0307-8. |
[29] |
F. Lin and P. Zhang, On the hydrodynamic limit of ginzburg-landau vortices, Discrete Contin. Dynam. Systems, 6 (2000), 121-142.doi: 10.3934/dcds.2000.6.121. |
[30] |
E. Mainini, A global uniqueness result for an evolution problem arising in superconductivity, Boll. Unione Mat. Ital. (9), 2 (2009), 509-528. |
[31] |
N. Masmoudi and P. Zhang, Global solutions to vortex density equations arising from sup-conductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 441-458.doi: 10.1016/j.anihpc.2004.07.002. |
[32] |
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.doi: 10.1007/s002850050158. |
[33] |
I. Primi, A. Stevens and J. J. L. Velázquez, Mass-selection in alignment models with non-deterministic effects, Comm. Partial Differential Equations, 34 (2009), 419-456.doi: 10.1080/03605300902797171. |
[34] |
G. Raoul, Nonlocal interaction equations: Stationary states and stability analysis, Differential Integral Equations, 25 (2012), 417-440.doi: 10.1016/S0012-9593(00)00122-1. |
[35] |
D. Ruelle, Statistical Mechanics: RIgorous Results, W. A. Benjamin, Inc., New York-Amsterdam, 1969. |
[36] |
E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup., 33 (2000), 561-592.doi: 10.1016/S0012-9593(00)00122-1. |
[37] |
E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg-Landau Model, Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser Boston, Inc., Boston, MA, 2007. |
[38] |
H. Sun, D. Uminsky and A. L. Bertozzi, Stability and clustering of self-similar solutions of aggregation equations, J. Math. Phys., 53 (2012), 115610, 18pp.doi: 10.1063/1.4745180. |
[39] |
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.doi: 10.1137/S0036139903437424. |
[40] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.doi: 10.1007/s11538-006-9088-6. |
[41] |
G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.doi: 10.1051/m2an:2000127. |
[42] |
J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics, J. Nonlinear Sci., 22 (2012), 935-959.doi: 10.1007/s00332-012-9132-7. |
[43] |
J. von Brecht, D. Uminsky, T. Kolokolnikov and A. Bertozzi, Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22 (2012), (suppl. 1), 1140002, 31pp.doi: 10.1142/S0218202511400021. |
[44] |
E. Weinan, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77 (1994), 383-404.doi: 10.1016/0167-2789(94)90298-4. |