Existence of ground states for aggregation-diffusion equations

TL;DR

This paper investigates the conditions under which ground states exist for aggregation-diffusion models, revealing sharp criteria for certain diffusive behaviors and interaction potentials in multi-agent systems.

Contribution

It provides new criteria for the existence of ground states in aggregation-diffusion equations, especially clarifying the case of linear diffusion.

Findings

No ground states or local minimizers exist under certain conditions.

Sharp conditions identified for homogeneous functionals with degenerate diffusions.

Complete characterization of ground state existence for linear diffusion cases.

Abstract

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.