Uniqueness of weak solutions for the local aggregation–diffusion models

Prove uniqueness of weak solutions for the local fourth-order aggregation–diffusion equations studied in this paper: (i) the one-species equation ∂tρ = −∇·(ρ∇(Δρ + μ²ρ)) posed with Neumann-type boundary conditions ∂νρ = ∂νΔρ = 0 and parameter regime μ² > 0; and (ii) the two-species system ∂tρ = −∇·(ρ∇(κΔρ + αΔη + μρ + ωη)), ∂tη = −∇·(η∇(αΔρ + Δη + ωρ + η)), under the positivity condition on the diffusion matrix M = [[κ, α],[α, 1]] (i.e., κ > 0 and 0 ≤ α < √κ) and μ > 0. The challenge arises because the associated free energy functionals are nonconvex.

Background

The paper establishes global existence of weak solutions for a class of local fourth-order aggregation–diffusion equations obtained as nonlocal-to-local limits, including both one-species and two-species systems with gradient-flow structure. The energies driving the models are nonconvex, and the construction of solutions uses variational methods (minimising movement schemes) with uniform energy bounds.

Despite existence results, the authors note that uniqueness is not settled. Nonconvexity of the free energies typically prevents standard convexity-based arguments (e.g., displacement convexity in Wasserstein space) from applying, leaving uniqueness as a key unresolved analytic issue.