Quantitative light-particle limit for the Vlasov-Fokker-Planck-Navier-Stokes system

TL;DR

This paper develops the first quantitative convergence theory for the light particle limit of the Vlasov-Fokker-Planck-Navier-Stokes system, providing explicit rates and optimal convergence estimates in various topologies.

Contribution

It introduces a novel relative entropy method tailored to the light particle regime, enabling explicit convergence rates for the kinetic and fluid components.

Findings

Explicit convergence rates for kinetic distribution and fluid velocity

Propagation of quantitative estimates in weak topologies

Optimal convergence rates in bounded Lipschitz distance

Abstract

We investigate the hydrodynamic limit of the Vlasov--Fokker--Planck--Navier--Stokes system in the light particle regime, where the particle relaxation takes place on a singularly fast time scale. Using a relative entropy method adapted to this scaling, we develop the first quantitative convergence theory for the light particle limit. Our analysis yields explicit rates for the convergence of both the kinetic distribution and the fluid velocity, extending the qualitative compactness-based result of Goudon, Jabin, and Vasseur [Indiana Univ. Math. J., 53, (2004), 1495--1515]. Moreover, we show that these quantitative estimates propagate in weak topologies and, in particular, lead to optimal convergence rates in the bounded Lipschitz distance. The results apply on both the torus and the whole space, providing a unified quantitative description of the light particle hydrodynamic limit.