# Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

## TL;DR

This paper provides a rigorous quantitative analysis of the large friction limit of a Vlasov equation with nonlocal forces, connecting kinetic models to continuum aggregation equations using Wasserstein distances.

## Contribution

It introduces an intermediate pressureless Euler system to accurately estimate the error between kinetic and continuum models in the large friction regime.

## Key findings

- Quantitative error bounds between kinetic and continuum models.
- Use of Wasserstein distance for error estimation.
- Extension of previous results to nonlocal interaction forces.

## Abstract

We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing $2$-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the $2$-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in $2$-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329--361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.