Stable stationary states of non-local interaction equations

Mathematical Models and Methods in Applied Sciences

Volumn 20, Issue 12, 2010, Pages 2267-2291

  Fellner, Klemens ^a   Raoul, Gaël ^b  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

double well potential; Non local interaction equation; numerical simulation; stability analysis

Indexed keywords

A-DENSITY; DOUBLE-WELL POTENTIAL; EXTERNAL POTENTIAL; INTERACTION POTENTIALS; LINEAR STABILITY; NON-LINEAR STABILITIES; NON-LOCAL INTERACTIONS; NUMERICAL EXAMPLE; NUMERICAL SIMULATION; REPULSIVE INTERACTIONS; STABILITY ANALYSIS; STATIONARY STATE;

COMPUTER SIMULATION; STABILITY;

EQUATIONS OF STATE;

EID: 78651233551     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202510004921     Document Type: Article

References (31)

1. P. R. Beesak, Gronwall Inequalities, Carleton Math. Lecture Notes No. 11 (1975).
2. D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, RAIRO Mod́el. Math. Anal. Nuḿer. 31 (1997) 615-641. (Pubitemid 127374202)
3. E. Ben-Naim, P. L. Krapivsky and S. Redner, Bifurcations and patterns in compromise processes, Physica D 183 (2003) 190-204.
4. A. L. Bertozzi and J. Brandman, Finite-time blow-up of L1-weak solutions of an aggregation equation, Comm. Math. Sci., to appear.
5. A. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity 22 (2009) 683-710.
6. A. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys. 274 (2007) 717-735.
7. A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model, SIAM J. Numer. Anal. 46 (2008) 691-721.
8. A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. . Eqns. 44 (2006).
9. S. Boi, V. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal. Real World Appl. 1 (2000) 163-176.
10. 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.
11. M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with non-linear diffusion, Netw. Heterog. Media 3 (2008) 749-785.
12. J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent and D. Slepcev, Global-in-time weak measure solutions, finite-time aggregation and confinement for nonlocal interaction equations, preprint UAB 17 (2009), submitted.
13. 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.
14. J. A. Carrillo and J. Rosado, Uniqueness of bounded solutions to aggregation equations by optimal transport methods, preprint UAB 7 (2009), submitted.
15. J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, in New Trends in Mathematical Physics (World Scientific, 2004), pp. 234-244.
16. Y.-L. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle °ocking: Scalability of cooperative control algorithms using pairwise potentials, IEEE Int. Conf. Robotics and Automation (2007) 2292-2299.
17. K. Fellner and G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, preprint CMLA-ENS Cachan 26 (2009), submitted for the proceeding of the conference Mathematical Methods and Modeling of Biophysical Phenomena.
18. K. Kang, B. Perthame, A. Stevens and J. J. L. Velazquez, An integro-Differential equation model for alignment and orientational aggregation, J. Diff. . Eqns. 246 (2009) 1387-1421.
19. T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Diff. . Eqns. 32 (2007) 1941-1964.
20. H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Rational Mech. Anal. 172 (2004) 407-428. (Pubitemid 38808952)
21. A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol. 38 (1999) 534-570.
22. D. Morale, V. Capasso and K. Oelschlager, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol. 50 (2005) 49-66.
23. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953) 311-338.
24. I. Primi, A. Stevens and J. J. L. Velazquez, Mass-selection in alignment models with nondeterministic e® ects, Comm. Partial Diff. . Eqns. 34 (2009) 416-456.
25. G. Raoul, Local stability of evolutionary attractors for continuous structured populations, preprint CMLA-ENS Cachan 1 (2009), submitted.
26. G. Raoul, Non-local interaction equations: Stationary states and stability analysis, preprint CMLA-ENS Cachan 25 (2009), submitted.
27. G. Raoul, Orbital stability of some steady-states for an aggregation equation, work in progress.
28. 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.
29. C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol. 68 (2006) 1601-1623.
30. F. Theil, A proof of crystallization in two dimensions, Comm. Math. Phys. 262 (2006) 209-236.
31. G. Toscani, One-dimensional kinetic models of granular flows, RAIRO Mod́el. Math. Anal. Nuḿer. 34 (2000) 1277-1291.