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

José A. Carrillo; Yanghong Huang

doi:10.3934/krm.2017007

Article Contents

2017, Volume 10, Issue 1: 171-192. Doi: 10.3934/krm.2017007

This issue Previous Article A logistic equation with nonlocal interactions Next Article Self-organized hydrodynamics with density-dependent velocity

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

José A. Carrillo^1, and
Yanghong Huang^2,

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: January 31, 2016

Revised: May 31, 2016

Published: September 2017

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Abstract

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.

Keywords:

Fredholm integral equations,
stationary solutions.

Mathematics Subject Classification: Primary: 45B05; Secondary: 34K21.

Citation:

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Figure 1. The steady states exist only on the shaded region and their expressions are obtained from the reduced governing equation

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Figure 2. 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

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Figure 3. 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)

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Figure 4. The compactly supported steady states exist only in the shaded region

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Figure 5. 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 $\bar{b}=(2+2d-d^2)/(d+1)$

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Figure 6. The two limits of a sectionally analytical function $\Psi$ along the line segment connecting $-R$ and $R$

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