Explicit Equilibrium Solutions For the Aggregation Equation with Power-Law Potentials

TL;DR

This paper constructs explicit stationary solutions for the aggregation equation with power-law kernels, providing insights into stable equilibria and collective behavior in models of swarming and aggregation.

Contribution

It introduces methods to explicitly solve for equilibrium solutions of the aggregation equation with power-law potentials, advancing understanding of stable collective states.

Findings

Explicit solutions constructed for aggregation equations with power-law kernels.

Solutions are expected to be globally energy stable equilibria.

Characterizes generic behaviors of stationary solutions in collective models.

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.