Quantitative Propagation of Chaos for the Mixed-Sign Viscous Vortex Model on the Torus

TL;DR

This paper establishes a quantitative propagation of chaos result for a mixed-sign vortex system on the torus with Brownian noise, introducing a novel pairing method and tensorized vorticity PDE to leverage existing chaos theory.

Contribution

It introduces a new pairing between vortices of opposite sign and a tensorized vorticity PDE to achieve optimal propagation of chaos results for mixed-sign vortex systems.

Findings

Optimal rate of chaos propagation derived

Tensorized vorticity PDE well-posedness established

Method extends chaos results to mixed-sign vortex systems

Abstract

We derive a quantiative propagation of chaos result for a mixed-sign point vortex system on $\mathbb{T}^2$ with independent Brownian noise, at an optimal rate. We introduce a pairing between vortices of opposite sign, and using the vorticity formulation of 2D Navier-Stokes, we define an associated tensorized vorticity equation on $\mathbb{T}^2\times\mathbb{T}^2$ with the same well-posedness theory as the original equation. Solutions of the new PDE can be projected onto solutions of Navier-Stokes, and the tensorized equation allows us to exploit existing propagation of chaos theory for identical particles.