THE MEAN-FIELD LIMIT FOR HYBRID MODELS OF COLLECTIVE MOTIONS WITH CHEMOTAXIS

The paper proves a mean-field limit for a hybrid chemotaxis–alignment model with explicit dimension-dependent rates and double-exponential-in-time stability. Its main theorem (Theorem 2.1) states W2(((ΦNt)#(ρin)⊗N)N;1, ρt)2 ≤ τ(t)·Cd(N) with Cd(N)=N^{-1/2} (d=1), N^{-1/2}log N (d=2), N^{-1/d} (d>2), and τ(t) ≤ exp(exp(C t)) (see Theorem 2.1 and (32) in the paper ). The candidate solution also derives these rates and time growth, via a direct coupling of particles with mean-field characteristics, a Dobrushin-type estimate, and empirical-measure bounds. The paper proves the result in two steps—Particles → Liouville–Vlasov (Proposition 5.3, using Theorem 5.2/Fournier–Guillin) and Liouville–Vlasov → Vlasov (Proposition 6.1)—while the model uses a single-step direct coupling. Minor differences include using W2 (paper) vs W1 (model) for the empirical-measure control of the chemotactic term; both yield the same Cd(N) rates due to bounded support and Lipschitz kernels. Both arguments are consistent with the paper’s assumptions and conclusions.