Existence of ground states for aggregation-diffusion equations

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.

arXiv comment21 pages

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Cited by in corpus (5)

• Learning Mean-Field Equations from Particle Data Using WSINDy
• Reverse Hardy-Littlewood-Sobolev inequalities
• Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics
• Drift-diffusion equations with saturation
• Infinite-time concentration in Aggregation--Diffusion equations with a given potential