Finite-time blow-up of solutions of an aggregation equation in R n

Communications in Mathematical Physics

Volumn 274, Issue 3, 2007, Pages 717-735

  Bertozzi, Andrea L ^a,b   Laurent, Thomas ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b DUKE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34547507325     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-007-0288-1     Document Type: Article

References (29)

1. Beale J.T., Kato T. and Majda A.J. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94: 61-66
2. Bodnar M. and Velazquez J.J.L. (2006). An integro-differential equation arising as a limit of individual cell-based models. J. Differ. Eqs. 222(2): 341-380
3. Brenner M.P., Constantin P., Kadanoff L.P., Schenkel A. and Venkataramani S.C. (1999). Diffusion, attraction, and collapse. Nonlinearity 12: 1071-1098
4. Burger M., Capasso V. and Morale D. (2007). On an aggregation model with long and short range interactions. Nonlinear Analysis: Real World Applications 8(3): 939-958
5. Burger, M., Di Francesco, M.: Large time behavior of nonlocal aggregation models with nonlinear diffusion. Preprint, 2006
6. Chandrasekhar S. (1967). An introduction to the study of stellar structure. Dover, New York
7. Chuang, Y.-L., D'Orsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.: State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Preprint, 2006
8. Constantin P., Lax P.D. and Majda A. (1985). A simple one-dimensional model for the three-dimensional vorticity equation. Commun. Pure. Appl. Math. 38: 715-724
9. Constantin P., Majda A. and Tabak E. (1994). Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7: 1495-1533
10. Constantin P. (1994). Geometric statistics in turbulence. SIAM Review 36(1): 73-98
11. D'Orsogna M.R., Chuang Y.-L., Bertozzi A.L. and Chayes L. (2006). Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96: 104302
12. Ebeling W. and Erdmann U. (2003). Nonequilibrium statistical mechanics of swarms of driven particles. Complexity 8: 23-30
13. Edelstein-Keshet L., Watmough J. and Grunbaum D. (1998). Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts. J. Math. Bio. 36: 515-549
14. Flierl G., Grünbaum D., Levin S. and Olson D. (1999). From individuals to aggregations: the interplay between behavior and physics. J. Theor. Biol. 196: 397-454
15. Holm D.D. and Putkaradze V. (2005). Aggregation of finite size particles with variable mobility. Phys. Rev. Lett. 95: 226106
16. Holm, D.D., Putkaradze, V.: Clumps and patches in self-aggregation of finite size particles. Preprint, 2006
17. Ju N. (2004). Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251: 365-376
18. Keller E.F. and Segel L.A. (1970). J. Theor. Biol. 26: 399-415
19. Laurent, T.: PhD thesis, Duke University, Department of Mathematics, 2006
20. Laurent, T.: Local and global existence for an aggregation equation. Preprint 2006. to appear in Communications in PDE
21. Levine H., Rappel W.J. and Cohen I. (2000). Self-organization in systems of self-propelled particles. Phys. Rev. E 63: 017101
22. Majda A.J. and Bertozzi A.L. (2002). Vorticity and Incompressible Flow. University Press, Cambridge
23. Mogilner A. and Edelstein-Keshet L. (1999). A non-local model for a swarm. J. Math. Bio. 38: 534-570
24. Parrish J. and Edelstein-Keshet L. (1999). Complexity, pattern and evolutionary trade-offs in animal aggregation. Science 294: 99-101
25. Taylor M.E. (1996). Partial Differential Equations, Volume 3. Springer-Verlag, New York
26. Toner J. and Tu Y. (1995). Long-range order in a two-dimensional dynamical xy model: how birds fly together. Phys. Rev. Lett. 75: 4326-4329
27. Topaz, C.M., Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65(1), 152-174 (electronic) (2004)
28. Topaz C.M., Bertozzi A.L. and Lewis M.A. (2006). A nonlocal continuum model for biological aggregation. Bull. Math. Bio. 68(7): 1601-1623
29. Vicsek T., Czirk A., Ben-Jacob E., Cohen I. and Shochet O. (1995). Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75: 1226-1229