Quantitative Propagation of Chaos for 2D Viscous Vortex Model with General Circulations on the Whole Space

TL;DR

This paper establishes quantitative propagation of chaos for the 2D viscous vortex model with general circulations, providing explicit optimal convergence rates in a setting that models realistic unconfined fluids with sign-changing vorticity.

Contribution

It introduces a novel approach with sharp growth estimates and a new ODE hierarchy to achieve optimal convergence rates for the vortex model with general circulations.

Findings

Explicit convergence rates in N and t are derived.

Results apply to unconfined fluids with sign-changing vorticity.

Method introduces new technical tools for vortex dynamics analysis.

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.