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 $\varepsilon$ and the mass ratio $η$. Furthermore, under the Vlasov-Navier-Stokes scaling assumption $O(1)\varepsilon^3 \leqslant η\leqslant o(1)\varepsilon^2$ as $\varepsilon \rightarrow 0$, we establish the weak convergence of the solution to the fluid-kinetic coupled limit system.