Propagation of chaos for 3D sedimentation with Oseen (Stokes) interaction

Establish propagation of chaos for the three-dimensional sedimentation particle system governed by the first-order inertialess dynamics with non-attractive interaction kernel K(x)=Φ(x) g, where g is the constant gravitational acceleration vector and Φ is the Stokes (Oseen) tensor Φ(x) = (1/(8π|x|))(Id + (x ⊗ x)/|x|^2), corresponding to singularity exponent α=1. Specifically, starting from chaotic (i.i.d.) initial particle positions with law ρ^0, demonstrate that the empirical measure ρ_N(t) converges with probability one, as N→∞, to the solution ρ(t) of the mean-field transport-Stokes equation ∂_t ρ + div((K * ρ) ρ) = 0, without restrictive assumptions on the initial minimal inter-particle distances beyond those typically satisfied by random configurations.

Background

The paper analyzes first-order N-particle mean-field systems with moderately singular, non-attractive kernels K satisfying |K(x)| + |x||∇K(x)| ≲ |x|^{-α}, focusing on quantitative mean-field limits and propagation of chaos. A central example motivating the work is sedimentation, where the leading-order gravity-driven interaction among particles in a Stokes fluid is binary with kernel K(x)=Φ(x) g, g the gravitational acceleration vector, and Φ the Oseen tensor. This kernel is non-attractive and has singularity exponent α=1 in dimension d=3.

Existing derivations of mean-field limits for sedimentation rely on Hauray's method and require strong control of minimal inter-particle distances, which chaotic i.i.d. initial data typically do not satisfy. As a result, earlier results establish convergence only for well-prepared initial configurations and do not achieve propagation of chaos for the sedimentation case. While the present paper relaxes assumptions by exploiting non-attraction and next-to-nearest neighbor control, the sedimentation setting (α=1, d=3) remains critical, and propagation of chaos for this system is explicitly identified as an open problem.