# American Institute of Mathematical Sciences

March  2017, 10(1): 171-192. doi: 10.3934/krm.2017007

## Explicit equilibrium solutions for the aggregation equation with power-law potentials

 1 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom 2 School of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom

Received  February 2016 Revised  June 2016 Published  November 2016

Despite their wide presence in various models in the study of collective behaviors, explicit swarming patterns are difficult to obtain. In this paper, special stationary solutions of the aggregation equation with power-law kernelsare constructed by inverting Fredholm integral operators or byemploying certain integral identities. These solutions are expected tobe the global energy stable equilibria and to characterize the generic behaviorsof stationary solutions for more general interactions.

Citation: José A. Carrillo, Yanghong Huang. Explicit equilibrium solutions for the aggregation equation with power-law potentials. Kinetic & Related Models, 2017, 10 (1) : 171-192. doi: 10.3934/krm.2017007
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##### References:
The steady states exist only on the shaded region and their expressions are obtained from the reduced governing equation
Comparison between the compacted supported solutions (17) and the more singular steady states (22) with unit total mass ($M_0=1$): (a) the energy; (b) the radius of support
The comparison of $K*\rho$ and $K*\rho_\delta$ at $a=5/2$ and $b=2$, for two solutions (17) and (22). The black dots indicate the boundary of the support of (17) and the support of (22)
The radial profiles of the constructed solutions for $a=4$. As $b$ increase from $2-d$ (the Newtonian potential) to $b_{max} = (2+3d-d^2)/(d+1)$, the solution becomes negative starting at $\bar{b}=(2+2d-d^2)/(d+1)$
The two limits of a sectionally analytical function $\Psi$ along the line segment connecting $-R$ and $R$
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