Singular patterns for an aggregation model with a confining potential

We consider the aggregation equation with an attractive–repulsive force law. Recent studies (Kolokolnikov et al. (2011) [22]; von Brecht et al. (2012)  [23]; Balague et al. (2013) [15]) have demonstrated that this system exhibits a very rich solution structure, including steady states consisting of rings, spots, annuli, N -fold symmetries, soccer-ball patterns etc. We show that many of these patterns can be understood as singular perturbations off lower-dimensional equilibrium states. For example, an annulus is a bifurcation from a ring; soccer-ball patterns bifurcate off solutions that consist of delta-point concentrations. We apply asymptotic methods to classify the form and stability of many of these patterns. To characterize spot solutions, a class of “semi-linear” aggregation problems is derived, where the repulsion is described by a nonlinear term and the attraction is linear but non-symmetric. For a special class of perturbations that consists of a Newtonian repulsion, the spot shape is shown to be an ellipse whose precise dimensions are determined via a complex variable method. For annular shapes, their width and radial density profile are described using perturbation techniques.