Error Estimation in the Mean-Field Limit of Kinetic Flocking Models with Local Alignments

The paper proves the quantitative mean-field error bound for the moderately interacting particle system and the intermediate mean-field SDE (Theorem 3.2), namely sup_{0<t<T} sup_i E(|X_{i,ξ,N}-X_{i,ξ}|^2+|V_{i,ξ,N}-V_{i,ξ}|^2) ≤ C T^3 ln(N^α)/N · (1 + T^3 N T^{3α} ln(N^α)), with the logarithmic scaling 1/(δ^2ν^4ε^{4d+2}) ∼ ln(N^α) explicitly stated and used in the analysis . The candidate solution reaches the same bound via a synchronous coupling and concentration-on-a-time-grid approach. While the candidate’s sketch slightly misstates the optimal ε-exponents for ∥∇W_ε∥_∞ and ∥D^2W_ε∥_∞ compared with the paper’s recorded bounds , and it implicitly folds the interaction-Lipschitz contribution into the alignment-Lipschitz constant, these issues are minor and correctable without changing the final rate. The paper’s argument is complete (via decomposition into J1–J3 and Lemma 3.2 for local alignment) , and the model’s proof sketch is methodologically sound though it would benefit from sharpening constants and tracking the ∥D^2W_ε∥_∞ term explicitly.