Regularized vortex approximation for 2D Euler equations with transport noise

TL;DR

This paper introduces a mean field approximation for the 2D Euler vorticity equation with transport noise, using interacting point vortices with a regularized Biot-Savart kernel, converging as the number of particles increases and regularization decreases.

Contribution

It provides a novel approximation method for stochastic 2D Euler equations via interacting point vortices with a regularized kernel, incorporating transport noise.

Findings

Convergence of vortex approximation as particle number increases

Effective regularization of Biot-Savart kernel in stochastic setting

Validation of mean field limit for Euler equations with noise

Abstract

We study a mean field approximation for the 2D Euler vorticity equation driven by a transport noise. We prove that the Euler equations can be approximated by interacting point vortices driven by a regularized Biot-Savart kernel and the same common noise. The approximation happens by sending the number of particles $N$ to infinity and the regularization $\epsilon$ in the Biot-Savart kernel to $0$, as a suitable function of $N$.