## Revista Matemática Iberoamericana

### Wellposedness and regularity of solutions of an aggregation equation

#### Abstract

We consider an aggregation equation in $\mathbb R^d$, $d\ge 2$ with fractional dissipation: $u_t + \nabla\cdot(u \nabla K*u)=-\nu \Lambda^\gamma u$, where $\nu\ge 0$, $0 < \gamma\le 2$ and $K(x)=e^{-|x|}$. In the supercritical case, $0 < \gamma < 1$, we obtain new local wellposedness results and smoothing properties of solutions. In the critical case, $\gamma=1$, we prove the global wellposedness for initial data having a small $L_x^1$ norm. In the subcritical case, $\gamma > 1$, we prove global wellposedness and smoothing of solutions with general $L_x^1$ initial data.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 1 (2010), 261-294.

Dates
First available in Project Euclid: 16 February 2010

https://projecteuclid.org/euclid.rmi/1266330124

Mathematical Reviews number (MathSciNet)
MR2666315

Zentralblatt MATH identifier
1197.35012

#### Citation

Li, Dong; Rodrigo, José L. Wellposedness and regularity of solutions of an aggregation equation. Rev. Mat. Iberoamericana 26 (2010), no. 1, 261--294. https://projecteuclid.org/euclid.rmi/1266330124

#### References

• Bertozzi, A. L. and Brandman, J.: Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation. Commun. Math. Sci. 8 (2010), no. 1, 45-65.
• Bertozzi, A.L. and Laurent, T.: Finite-time blow up of solutions of an aggregation equation in $\mathbbR^n$. Comm. Math. Phys. 274 (2007), no. 3, 717-735.
• 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.
• 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.
• 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.
• 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.
• 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.
• Córdoba, A. and Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249 (2004), no. 3, 511-528.
• 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.
• Constantin, P., Córdoba, D. and Wu, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50 (2001), 97-107.
• 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.
• 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.
• 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.
• Holmes, E., Lewis, M.A., Banks, J. and Veit, R.: PDE in ecology: spatial interactions and population dynamics. Ecology 75 (1994), 17-29.
• 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.
• Ikeda, T.: Stationary solutions of a spatially aggregating population model. Proc. Japan Acad. Ser. A Math. Sci. 60 (1984), no. 2, 46-48.
• Ikeda, T.: Standing pulse-like solutions of a spatially aggregating population model. Japan J. Appl. Math. 2 (1985), no. 1, 111-149.
• Ikeda, T. and Nagai, T.: Stability of localized stationary solutions. Japan J. Appl. Math. 4 (1987), no. 1, 73-97.
• 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.
• Kawasaki, K.: Diffusion and the formation of spatial distributions. Math. Sci. 16 (1978), no. 183, 47-52.
• Laurent, T.: Local and global existence for an aggregation equation. Comm. Partial Differential Equations 32 (2007), no. 10-12, 1941-1964.
• 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.
• Levine, H., Rappel, W. J., and Cohen, I.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63 (2001), paper 017101.
• Li, D. and Rodrigo, J.: Blow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation. Adv. Math. 217 (2008), no. 6, 2563-2568.
• Li, D. and Rodrigo, J.: Finite-time singularities of an aggregation equation in $\mathbb R^n$ with fractional dissipation. Comm. Math. Phys. 287 (2009), no. 2, 687-703.
• Majda, A.J. and Bertozzi, A.L.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics 27. Cambridge University Press, Cambridge, UK, 2002.
• Mogilner, A., Edelstein-Keshet, L., Bent, L., and Spiros, A.: Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol. 47 (2003), no. 4, 353-389.
• Mogilner, A. and Edelstein-Keshet, L.: A non-local model for a swarm. J. Math. Biol. 38 (1999), no. 6, 534-570.
• Murray, J.D.: Mathematical biology I: An introduction. Interdiscip. Appl. Math. 17. Springer, New York, 2002.
• Mimura, M. and Yamaguti, M.: Pattern formation in interacting and diffusing systems in population biology. Adv. Biophys. 15 (1982), 19-65.
• 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.
• Okubo, A.: Diffusion and ecological problems: mathematical models. Biomathematics 10. Springer-Verlag, Berlin-New York, 1980.
• 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.
• dal Passo, R. and Demotoni, P.: Aggregative effects for a reaction-advection equation. J. Math. Biol. 20 (1984), no. 1, 103-112.
• Topaz, C.M., Bertozzi, A.L. and Lewis, M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Bio. 68 (2006), no. 7, 1601-1623.
• Topaz, C.M. and Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl Math. 65 (2004), no. 1, 152-174.
• Toner, J. and Tu, Y.: Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E (3) 58 (1998), no. 4, 4828-4858.
• 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.
• Yudovich, V.I.: Non-stationary flow of an incompressible liquid. Zh. Vychisl. Mat. Mat. Fiz. 3 (1963), 1032-1066.