Inviscid limit of stochastic damped 2D Navier-Stokes equations

TL;DR

This paper proves that as viscosity approaches zero, the stationary solutions of stochastic damped 2D Navier-Stokes equations converge to those of the stochastic damped Euler equations, with enstrophy dissipation vanishing.

Contribution

It establishes the convergence of stationary solutions and enstrophy dissipation rates in the inviscid limit for stochastic damped 2D Navier-Stokes equations.

Findings

Stationary solutions converge to Euler solutions as viscosity vanishes.

Enstrophy dissipation rate approaches zero in the inviscid limit.

Limit measure of invariant measures describes the convergence.

Abstract

We consider the inviscid limit of the stochastic damped 2D Navier- Stokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier-Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The rates are computed with respect to a limit measure of the unique invariant measure of the stochastic damped Navier-Stokes equations.