Wellposedness and regularity of solutions of an aggregation equation

Revista Matematica Iberoamericana

Volumn 26, Issue 1, 2010, Pages 261-294

  Li, Dong ^a   Rodrigo, José L ^b  

^a UNIVERSITY OF IOWA   (United States)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Aggregation equations; Higher regularity; Well posedness

Indexed keywords

EID: 77951117235     PISSN: 02132230     EISSN: 22350616     Source Type: Journal    
DOI: 10.4171/RMI/601     Document Type: Article

References (39)

1. ∞-weak solutions of an aggregation equation. Commun. Math. Sci. 8 (2010), no. 1, 45-65.
2. n. Comm. Math. Phys. 274 (2007), no. 3, 717-735.
3. BODNAR, M. AND VELÁZQUEZ, J. J. L.: Derivation of macroscopic equations for individual cell-based model: a formal approach. Math. Methods Appl. Sci. 28 (2005), no. 15, 1757-1779.
4. BODNAR, M. AND VELÁZQUEZ, J. J. L.: An integro-differential equation arising as a limit of individual cell-based models. J. Differential Equations 222 (2006), no. 2, 341-380.
5. BURGER, M. AND DI FRANCESCO, M.: Large time behavior of nonlocal aggregation models with nonlinear diffusion. Netw. Heterog. Media 3 (2008), no. 4, 749-785.
6. BURGER, M., CAPASSO, V. AND MORALE, D.: On an aggregation model with long and short range interactions. Nonlinear Anal. Real World Appl. 8 (2007), no. 3, 939-958.
7. BILER, P. AND WOYCZYŃSKI, W. A.: Global and exploding solutions for nonlocal quadratic evolution problems. SIAM J. Appl. Math. 59 (1998), no. 3, 845-869.
8. CÓRDOBA, A. AND CÓRDOBA, D.: A maximum principle applied to quasigeostrophic equations. Comm. Math. Phys. 249 (2004), no. 3, 511-528.
9. CÓRDOBA, A., CÓRDOBA D. AND FONTELOS, M.: Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math. (2) 162 (2005), no. 3, 1377-1389.
10. CONSTANTIN, P., CÓRDOBA, D. AND WU, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50 (2001), 97-107.
11. EDELSTEIN-KESHET, L.: Mathematical models of swarming and social aggregation. In Proceedings of the 2001 International Symposium on Nonlinear Theory and its Applications, 1-7. Miyagi, Japan, 2001.
12. EDELSTEIN-KESHET, L., WATMOUGH, J., AND GRÜNBAUM, D.: Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts. J. Math. Biol. 36 (1998), no. 6, 515-549. (Pubitemid 128735361)
13. FLIERL, G., GRÜNBAUM, D., LEVIN, S., AND OLSON, D.: From individuals to aggregations: The interplay between behavior and physics. J. Theoret. Biol. 196 (1999), 397-454. (Pubitemid 29082205)
14. HOLMES, E., LEWIS, M. A., BANKS, J. AND VEIT, R.: PDE in ecology: spatial interactions and population dynamics. Ecology 75 (1994), 17-29.
15. HOSONO, Y. AND MIMURA, M.: Localized cluster solutions of nonlinear degenerate diffusion equations arising in population dynamics. SIAM J. Math. Anal. 20 (1989), no. 4, 845-869.
16. IKEDA, T.: Stationary solutions of a spatially aggregating population model. Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), no. 2, 46-48.
17. IKEDA, T.: Standing pulse-like solutions of a spatially aggregating population model. Japan J. Appl. Math. 2 (1985), no. 1, 111-149.
18. IKEDA, T. AND NAGAI, T.: Stability of localized stationary solutions. Japan J. Appl. Math. 4 (1987), no. 1, 73-97.
19. JU, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Comm. Math. Phys. 255 (2005), no. 1, 161-181. (Pubitemid 40900257)
20. KAWASAKI, K.: Diffusion and the formation of spatial distributions. Math. Sci. 16 (1978), no. 183, 47-52.
21. LAURENT, T.: Local and global existence for an aggregation equation. Comm. Partial Differential Equations 32 (2007), no. 10-12, 1941-1964.
22. LEMARIÉ-RIEUSSET, P.: Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics 431. Chapman & Hall/CRC Press, Boca Raton, FL, 2002.
23. LEVINE, H., RAPPEL, W. J., AND COHEN, I.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63 (2001), paper 017101.
24. LI, D. AND RODRIGO, J.: Blow up of solutions for a ID transport equation with nonlocal velocity and supercritical dissipation. Adv. Math. 217 (2008), no. 6, 2563-2568.
25. n with fractional dissipation. Comm. Math. Phys. 287 (2009), no. 2, 687-703.
26. MAJDA, A. J. AND BERTOZZI, A. L.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics 27. Cambridge University Press, Cambridge, UK, 2002.
27. MOGILNER, A., EDELSTEIN-KESHET, L., BENT, L., AND SPIROS, A.: Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. BioJ. 47 (2003), no. 4, 353-389.
28. MOGILNER, A. AND EDELSTEIN-KESHET, L.: A non-local model for a swarm. J. Math. Biol. 38 (1999), no. 6, 534-570.
29. MURRAY, J. D.: Mathematical biology I: An introduction. Interdiscip. Appl. Math. 17. Springer, New York, 2002.
30. MLMURA, M. AND YAMAGUTI, M.: Pattern formation in interacting and diffusing systems in population biology. Adv. Biophys. 15 (1982), 19-65.
31. NAGAI, T. AND MIMURA, M.: Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics. SIAM J. Appl. Math. 43 (1983), no. 3, 449-464.
32. OKUBO, A.: Diffusion and ecological problems: mathematical models. Biomathematics 10. Springer-Verlag, Berlin-New York, 1980.
33. OKUBO, A., GRUNBAUM, D., AND EDELSTEIN-KESHET, L.: The dynamics of animal grouping In Diffusion and ecological problems, 197-237. Interdiscip. Appl. Math. 14. Springer, New York, 1999.
34. DAL PASSO, R. AND DEMOTONI, P.: Aggregative effects for a reactionadvection equation. J. Math. Biol. 20 (1984), no. 1, 103-112.
35. TOPAZ, CM., BERTOZZI, A. L., AND LEWIS, M. A.: A nonlocal continuum model for biological aggregation. Bull. Math. Bio. 68 (2006), no. 7, 1601-1623.
36. TOPAZ, C.M. AND BERTOZZI, A. L.: Swarming patterns in a twodimensional kinematic model for biological groups. SIAM J. Appl Math. 65 (2004), no. 1, 152-174.
37. TONER, J. AND TU, Y.: Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E (3) 58 (1998), no. 4, 4828-4858.
38. VICSEK, T., CZIRÓK, A., FARKAS, I. J., AND HELBING, D.: Application of statistical mechanics to collective motion in biology. Phys. A 274 (1999), 182-189. (Pubitemid 30523123)
39. YUDOVICH, V. I.: Non-stationary flow of an incompressible liquid. Zh. Vychisl. Mat. Mat. Fiz. 3 (1963), 1032-1066.