Small Mass Limit for McKean‐Vlasov Equations With Different Types of Interaction Potentials

ABSTRACT We study the small mass limit for a class of systems described by McKean‐Vlasov equations with different types of interaction potentials. Our aim is to identify the limiting equation and establish bounds on the error between solutions of the associated nonlinear kinetic Fokker‐Planck equation and the limiting equation. We introduce an intermediate system through a coarse‐graining map, enabling us to estimate the error between the spatial densities of the Vlasov‐Fokker‐Planck equation and the intermediate system in terms of the 2‐Wasserstein distance. Subsequently, we derive an inequality for Wasserstein gradient flows by quantifying the error between the intermediate system and the corresponding limiting equation. This approach requires only the weak integrability of the interaction potentials, which quantifies the small mass limit of the nonlinear Fokker‐Planck system with three different types of interaction potentials: bounded and Lipschitz interaction potential, singular interaction potential, and bounded interaction potential. Moreover, our approach includes examples of singular interactions such as the Riesz potential and the Coulomb potential.