Existence of free energy minimizers in the whole space

Establish the existence of minimizers in the entire space ℝ^d for the free energy functionals associated with the local aggregation–diffusion models: namely, for the one-species functional F[ρ] = (1/2)∫(|∇ρ|² − μ²ρ²) dx and for the two-species functional F₂[ρ,η] = ∫(κ/2|∇ρ|² + 1/2|∇η|² + α∇ρ·∇η − μ/2 ρ² − 1/2 η² − ω ρ η) dx, overcoming the lack of control on mass escape at infinity.

Background

The models are formulated as gradient flows of explicit free energy functionals in the 2-Wasserstein metric. While the energies are shown to be nonincreasing along solutions, the existence of global minimizers in unbounded domains is not guaranteed due to potential mass loss at infinity.

Proving existence of minimizers in ℝ^d typically requires tightness or coercivity conditions that prevent escape of mass. The authors state that such control is currently missing for the energies considered here, leaving the existence of minimizers in the whole space unresolved.