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

TL;DR

This paper proves the existence of non-trivial global minimizers for free energies in aggregation equations with degenerate diffusion, revealing how criticality influences the existence of stable group formations in biological models.

Contribution

It introduces a new notion of criticality based on force scaling that determines the existence of global minimizers in aggregation-diffusion equations.

Findings

Existence of non-zero stationary solutions under certain conditions.

Criticality concept predicts when minimizers exist or do not.

Provides mathematical framework for biological group dynamics.

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.