A Quantitative Hydrodynamic Limit for the Vlasov-Fokker-Planck/Navier-Stokes System in the Fine Particle Regime
Abstract
We justify the hydrodynamic limit of the Vlasov-Fokker-Planck equation coupled with the incompressible Navier--Stokes equation in the fine regime with a quantitative convergence rate. In this regime, the limiting dynamics is described by the inhomogeneous incompressible Navier-Stokes system for the fluid-particle mixture. In contrast with the existing relative entropy approaches, the present work extends to a pointwise convergence of the hydrodynamic limits with an explicit rate on any given time interval [0,T]. The proof is based on a Macro-Micro decomposition around the moving local Maxwellian, together with an ε-weighted energy-dissipation structure adapted to the fine particle relaxation scales. Furthermore, we remove the classical small background restriction inherent in high-order energy methods, allowing the limiting profile to be possibly large. This is accomplished by exploiting a hidden Kelvin-Voigt type damped wave equation structure within the macroscopic velocity error system, yielding the critical extra dissipation required to close the uniform estimates.
Related
- Global hydrodynamic limit of the incompressible Navier-Stokes/Vlasov-Fokker-Planck system with large data
- Quantitative propagation of chaos in total variation for unregularized Vlasov-Riesz-Fokker-Planck systems
- Quantitative propagation of chaos on arbitrary finite time intervals for the three-dimensional repulsive Vlasov-Poisson-Fokker-Planck equation
- Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system
- Modulated Gibbs Measure and Mean-Field Limit for 3D Vlasov--Poisson--Fokker--Planck Equations
BibTeX
@misc{fang2026quantitative,
title = {{A Quantitative Hydrodynamic Limit for the Vlasov-Fokker-Planck/Navier-Stokes System in the Fine Particle Regime}},
author = {Zhendong Fang and Jian Li and Kunlun Qi and Huanyao Wen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21119},
eprint = {2610.21119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21119},
}
RIS
TY - UNPB
TI - A Quantitative Hydrodynamic Limit for the Vlasov-Fokker-Planck/Navier-Stokes System in the Fine Particle Regime
AU - Zhendong Fang
AU - Jian Li
AU - Kunlun Qi
AU - Huanyao Wen
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21119
UR - https://arxiv.org/abs/2610.21119
AB - We justify the hydrodynamic limit of the Vlasov-Fokker-Planck equation coupled with the incompressible Navier--Stokes equation in the fine regime with a quantitative convergence rate. In this regime, the limiting dynamics is described by the inhomogeneous incompressible Navier-Stokes system for the fluid-particle mixture. In contrast with the existing relative entropy approaches, the present work extends to a pointwise convergence of the hydrodynamic limits with an explicit rate on any given time interval [0,T]. The proof is based on a Macro-Micro decomposition around the moving local Maxwellian, together with an ε-weighted energy-dissipation structure adapted to the fine particle relaxation scales. Furthermore, we remove the classical small background restriction inherent in high-order energy methods, allowing the limiting profile to be possibly large. This is accomplished by exploiting a hidden Kelvin-Voigt type damped wave equation structure within the macroscopic velocity error system, yielding the critical extra dissipation required to close the uniform estimates.
ER -