Sedimentation of particles with very small inertia II: Derivation, Cauchy problem and hydrodynamic limit of the Vlasov–Stokes equation

Abstract

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in ${\mathbb{R}}^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton’s law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling, we show that the evolution of the $N$-particle system is well approximated by the Vlasov–Stokes equation. In contrast to the transport-Stokes equation considered in the first part of this series [Duke Math. J. 174, 615–684 (2025)], the Vlasov–Stokes equation takes into account the (small) inertia. Therefore, we obtain improved error estimates. We also improve previous results on the Cauchy problem for the Vlasov–Stokes equation and on its convergence to the transport-Stokes equation in the limit of vanishing inertia. The proofs are based on relative energy estimates. In particular, we show new stability estimates for the Vlasov–Stokes equation in the 2-Wasserstein distance. By combining a Lagrangian approach with a study of the energy dissipation, we obtain uniform stability estimates for arbitrarily small particle inertia. We show that a corresponding stability estimate continues to hold for the empirical particle density which formally solves the Vlasov–Stokes equation up to an error. To this end, we exploit certain uniform control on the particle configuration thanks to results in the first part.