Global Minimizers for Free Energies of Subcritical Aggregation Equations with Degenerate Diffusion

Abstract

We prove the existence of non-trivial global minimizers of a class of free energies related to aggregation equations with degenerate diffusion on $\Real^d$. Such equations arise in mathematical biology as models for organism group dynamics which account for competition between the tendency to aggregate into groups and nonlinear diffusion to avoid over-crowding. The existence of non-zero optimal free energy stationary solutions representing coherent groups in $\Real^d$ is therefore of interest. The primary contribution is the investigation of a notion of criticality associated with the global minimizer problem. The notion arises from the scaling of diffusive and aggregative forces as mass spreads and is shown to dictate the existence, and sometimes non-existence, of global minimizers.