On a nonlocal aggregation model with nonlinear diffusion

Discrete and Continuous Dynamical Systems

Volumn 27, Issue 1, 2010, Pages 301-323

  Li, Dong ^a   Zhang, Xiaoyi ^a  

^a UNIVERSITY OF IOWA   (United States)

Author keywords

Aggregation; Nonlinear diffusion; Nonlocal

Indexed keywords

EID: 77955386860     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2010.27.301     Document Type: Article

References (35)

1. D. G. Aronson and L. A. Caffarelli, Optimal regularity for one-dimensional porous medium flow, Rev. Mat. Iberoamericana, 2 (1986), 357-366.
2. D. G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461-467.
3. D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), 1-46, Lecture Notes in Math., 1224, Springer, Berlin, (1986).
4. ∞ -regularity and concavity for flows in one- dimensional porous media, Arch. Rational Mech. Anal., 99 (1987), 329-412.
5. p for solutions of the porous media equation, in "Contributions to Nonlinear Partial Differential Equations" (eds. C. Bardos, A. Damlamian, J.I. Diaz and J. Hernandez J.), Research Notes in Math., 89, Pitman London, (1983), 39-58.
6. n, Comm in Math. Phys., 274 (2007), 717-735.
7. M. Bodnar and J. J. L. Velázquez, Derivation of macroscopic equations for individual cell-based model: A formal approach, Math. Methods Appl. Sci., 28 (2005), 1757-1779.
8. M. Bodnar and J. J. L. Velázquez, An integrodifferential equation arising as a limit of individual cell-based models, J. Differential Equations., 222 (2006), 341-380.
9. M. Burger and M. Di Francesco, Large time behaviour of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.
10. M. Burger, V. Capasso and D. Morale, On an aggregation equation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), 939-958.
11. J. A. Carrillo, R. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal., 179 (2006), 217-264.
12. L. A. Caffarelli and A. Friedman, Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc., 252 (1979), 99-113.
13. L. A. Caffarelli, J. L. Vázquez and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J., 36 (1987), 373-101.
14. E. Dibenedetto, Regularity results for the porous media equation, Ann. Mat. Pura Appl. (4), 121 (1979), 249-262.
15. L. Edelstein-Keshet, J. Watmough and D. Grünbaum, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts, J. Math. Biol., 36 (1998), 515-549.
16. B. H. Gilding, Hölder continuity of solutions of parabolic equations, J. London Math. Soc.(2), 13 (1976), 103-106.
17. Darryl D. Holm and Vakhtang Putkaradze, Aggregation of finite size particles with variable mobility, Phys. Rev. Lett., 95 (2005), 226106.
18. K. Kawasaki, Diffusion and the formation of spatial distributions, Math. Sci., 46 (1978), 47-52.
19. S. N. Kruzhkov, Results on the character of the regularity of solutions of parabolic equations and some of their applications, Math. Notes, 6 (1969), 517-523.
20. T. Laurent, Local and global existence for an aggregation equation, Comm. Partial Differential Equations, 32 (2007), 1941-1964.
21. G. L. Lieberman, "Second Order Parabolic Differential Equations," World Scientific publishing Co. Pte. Ltd., 1996.
22. n with fractional dissipation, Comm. Math. Phys., 287 (2009), 687-703.
23. D. Li and J. Rodrigo, Wellposedness and regularity of solutions of an aggregation equation, to appear in Revista Matematica Iberoamericana.
24. D. Li and J. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. Math., 220 (2009), 1717-1738.
25. D. Li, J. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, to appear in Revista Matematica Iberoamericana.
26. D. Li and X. Zhang, Global iueuposedness an d blowup of solutions to a nonlocal evolution problem with singular kernels, submitted.
27. A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Bio)., 38 (1999), 534-570.
28. Oleinik, Kalashnikov and Chzou Yui-Lin', The Cauchy problem and boundary-value problems for equations of unsteady filtration type, Izv. Akad. Nauk SSR Ser. Mat., 22 (1958), 667-704.
29. M. Otani and Y. Sugiyama, Gradient estimates of solutions of some non-Newtonian filtration problems, Pitman Research Notes in Mathematics Series, 363 (1996), 247-253.
30. ∞ and its application to porous medium equations, J. Math. Soc. Japan, 53 (2001), 745-789.
31. ∞-solutions of genearlized porous medium equations with perturbation, Adv. Math. Sci. Appl., 12 (2002), 21-87.
32. T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, I, generation of the weak solution, Adv. Differential Equations, 14 (2009), 433-476.
33. C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68 (2006), 1601-1623.
34. C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensionai kinematic model for biological groups, SIAM J. Appl Math., 65 (2004), 152-174.
35. J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory," Oxford Math. Monogr. Oxford University Press, Oxford, 2007.