Kinetic and Related Models (KRM)

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

Pages: 171 - 192, Volume 10, Issue 1, March 2017      doi:10.3934/krm.2017007

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José A. Carrillo - Department of Mathematics, Imperial College, London, London SW7 2AZ, United Kingdom (email)
Yanghong Huang - School of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom (email)

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 kernels are constructed by inverting Fredholm integral operators or by employing certain integral identities. These solutions are expected to be the global energy stable equilibria and to characterize the generic behaviors of stationary solutions for more general interactions.

Keywords:  Fredholm integral equations, stationary solutions.
Mathematics Subject Classification:  Primary: 45B05; Secondary: 34K21.

Received: February 2016;      Revised: June 2016;      Available Online: November 2016.