Nonlocal cross-interaction systems on graphs: Energy landscape and dynamics

The paper proves existence of weak solutions to the two-species nonlocal upwind system (their equation (20)) in a Finsler-gradient-flow framework, under (K1)–(K2), a symmetry condition K(21)=K(12), and graph-geometry assumptions (W), (MB), (BC), plus a uniform bound (33); see the model, the weak formulation and assumptions in Definition 3.2 and (12), (K1)–(K2), and (W),(MB),(BC), and the existence theorem (Theorem 3.26) with its proof via discrete approximation and stability of curves of maximal slope. These are internally consistent and complete in the presented framework . The candidate solution establishes existence (and uniqueness) by a different route: it rewrites the dynamics as a nonlinear Kolmogorov forward equation on L1(μ)2 with upwind jump rates, obtains global Lipschitz bounds on the vector field using (K2) and the same η-bounds, and then applies Picard–Lindelöf in Banach spaces. It verifies the weak formulation and second-moment bounds, preserving support in supp μ, matching the target notion of solution. This alternative approach is coherent and yields the same existence conclusion under assumptions essentially aligned with the paper (μ finite and uniform η-bounds, as in (33)). The model, however, does not impose K(21)=K(12) (used in the paper to link to the energy gradient-flow structure) and does not identify the solution as a curve of maximal slope; conversely, the paper does not claim uniqueness while the model’s ODE argument delivers it on L1(μ)2. Hence, both are correct, with substantially different proofs and emphases.