Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion

Abstract

We prove the existence of global minimizers of a class of free energies related to aggregation equations with degenerate diffusion on $R^d$. 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 overcrowding. The existence of non-trivial stationary solutions with minimal energy representing coherent groups in $R^d$ is therefore of interest. A scaling criticality that measures the balance between the diffusive and aggregative forces as mass spreads is shown to govern the existence and non-existence of global minimizers. The primary difficulty confronted here is the inability to verify strict subadditivity conditions for biologically relevant problems which violate homogeneity-type assumptions known to be sufficient. To recover, we show that sufficiently degenerate diffusion provides a weaker condition from which tightness of symmetrized infimizing sequences can be recovered, even when the nonlocal attractive force is extremely weak.