Quantitative Propagation of Chaos for 2D Viscous Vortex Model with General Circulations on the Whole Space
Xuanrui Feng, Zhenfu Wang
TL;DR
This work analyzes the mean-field limit of a stochastic N-particle 2D point vortex system with general circulations on ${\mathbb{R}}^2$, proving quantitative propagation of chaos toward the vorticity form of the 2D Navier–Stokes equation. The authors develop a global relative-entropy framework and a local BBGKY-based ODE hierarchy, augmented by extended LLN/LD-type results and sharp logarithmic growth estimates for the limit density, to obtain precise rates in $N$ and $t$. They establish global entropy decay $H_N(G^N|g^N)\le M(1+t)^M(H_N(G_0^N|g_0^N)+1/N)$ and, under large viscosity, sharp local rates $H_k(G^{N,k}|g^k)\le M(1+t)^M(H_k(G_0^{N,k}|g_0^k)+k/N^2)$, leading to $L^1$-type convergence of marginals with rates $\sqrt{k/N}$ globally and $k/N$ locally. Extending prior torus and fixed-circulation results, the paper handles the whole space with mixed-sign circulations, introducing sharp logarithmic-growth estimates and an iterated ODE-hierarchy to control singular Biot–Savart interactions, thus providing refined mean-field approximations for unconfined 2D viscous flows.
Abstract
We derive quantitative propagation of chaos in the sense of relative entropy for the 2D viscous vortex model with general circulations, approximating the vorticity formulation of the 2D Navier-Stokes equation on the whole Euclidean space. Our results work on the general setting that the vortices are positioned on the whole space $\R^2$ and that the circulations are allowed to be in different magnitudes and orientations, which can be adapted to general unconfined realistic fluids with vorticity that may change sign. We provide explicit convergence rates which are optimal in $N$ and optimal in $t$ among existing literature. The key technical tools, which are our major novelty, are the sharp logarithmic growth estimates and a new ODE hierarchy and iterated integral estimates.