Convergence of transport noise to Ornstein-Uhlenbeck for 2D Euler equations under the enstrophy measure

TL;DR

This paper studies how the stochastic 2D Euler equations with transport noise and white noise initial conditions converge to the stochastic 2D Navier-Stokes equations driven by space-time white noise, highlighting a noise-induced regularization effect.

Contribution

It demonstrates the convergence of the 2D Euler equations with transport noise to the Navier-Stokes equations under specific conditions, revealing a new connection between these models.

Findings

Convergence of Euler to Navier-Stokes equations under transport noise

Identification of conditions for convergence

Insight into noise-induced regularization effects

Abstract

We consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition. It is shown that, under certain conditions, this equation converges to the 2D Navier-Stokes equation driven by the space-time white noise.