## Abstract

We establish a quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean–Vlasov equation. Introducing an intermediate system via a coarse-graining map, we quantitatively estimate the error between the spatial densities of the Vlasov–Fokker–Planck equation and the intermediate system in the Wasserstein distance of order 2. We then derive an evolution-variational-like inequality for Wasserstein gradient flows which allows us to quantify the error between the intermediate system and the corresponding limiting equation. Our strategy only requires weak integrability of the interaction potentials, thus in particular it includes the quantified overdamped limit of the kinetic Vlasov–Poisson–Fokker–Planck system to the aggregation-diffusion equation with either repulsive electrostatic or attractive gravitational interactions.

Original language | English |
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Pages (from-to) | 150-207 |

Number of pages | 58 |

Journal | Journal of Differential Equations |

Volume | 330 |

DOIs | |

Publication status | Published - 2022 Sept 5 |

### Bibliographical note

Funding Information:The work of Y.-P.C. is supported by NRF grant (No. 2017R1C1B2012918 and 2022R1A2C100282011) and Yonsei University Research Fund of 2020-22-0505 and 2021-22-0301. O.T. acknowledges support from NWO Vidi grant 016.Vidi.189.102, “Dynamical-Variational Transport Costs and Application to Variational Evolutions” and thanks to Mitia Duerinckx, Mark Peletier, and Upanshu Sharma for insightful discussions in the early stages of this manuscript.

Publisher Copyright:

© 2022 The Authors

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics