A Rigorous Derivation of the Vlasov-Navier-Stokes Model from Multicomponent Boltzmann Equations
Abstract
The rigorous justification of the Vlasov-Navier-Stokes system remains an outstanding open problem, both as a mean-field limit of an N-particle fluid-interacting system and as a hydrodynamic limit derived from multiphase Boltzmann equations. Inspired by the work of Bernard, Desvillettes, Golse, and Ricci [{\it Commun. Math. Sci.}, {\bf 15}(6), 1703-1741, 2017], we present a rigorous derivation of the incompressible Vlasov-Navier-Stokes system from the two-component Boltzmann system for elastic hard-sphere collisions. To this end, we establish the well-posedness of the rescaled multicomponent Boltzmann system for small initial data. Specifically, we obtain estimates for the solution that hold uniformly with respect to both the thermal speed ratio ε and the mass ratio η. Furthermore, under the Vlasov-Navier-Stokes scaling assumption O(1)ε3 leqslant ηleqslant o(1)ε2 as ε → 0, we establish the weak convergence of the solution to the fluid-kinetic coupled limit system.
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BibTeX
@misc{li2026rigorous,
title = {{A Rigorous Derivation of the Vlasov-Navier-Stokes Model from Multicomponent Boltzmann Equations}},
author = {Shuping Li and Linjie Xiong},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27864},
eprint = {2608.27864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27864},
}
RIS
TY - UNPB
TI - A Rigorous Derivation of the Vlasov-Navier-Stokes Model from Multicomponent Boltzmann Equations
AU - Shuping Li
AU - Linjie Xiong
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.27864
UR - https://arxiv.org/abs/2608.27864
AB - The rigorous justification of the Vlasov-Navier-Stokes system remains an outstanding open problem, both as a mean-field limit of an N-particle fluid-interacting system and as a hydrodynamic limit derived from multiphase Boltzmann equations. Inspired by the work of Bernard, Desvillettes, Golse, and Ricci [{\it Commun. Math. Sci.}, {\bf 15}(6), 1703-1741, 2017], we present a rigorous derivation of the incompressible Vlasov-Navier-Stokes system from the two-component Boltzmann system for elastic hard-sphere collisions. To this end, we establish the well-posedness of the rescaled multicomponent Boltzmann system for small initial data. Specifically, we obtain estimates for the solution that hold uniformly with respect to both the thermal speed ratio ε and the mass ratio η. Furthermore, under the Vlasov-Navier-Stokes scaling assumption O(1)ε^3 leqslant ηleqslant o(1)ε^2 as ε → 0, we establish the weak convergence of the solution to the fluid-kinetic coupled limit system.
ER -