Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in $\mathbb R^2$

TL;DR

This paper establishes the existence of invariant measures for 2D Euler equations on R^2 by analyzing the inviscid limit of hyperviscous stochastic Navier-Stokes equations, providing new insights into the stochastic dynamics of inviscid flows.

Contribution

It proves the existence of invariant measures for 2D Euler equations via the inviscid limit of hyperviscous stochastic Navier-Stokes equations, a novel approach in unbounded domains.

Findings

Existence of invariant measure  for 2D Euler equations.

Convergence of hyperviscous stochastic Navier-Stokes solutions to Euler solutions as viscosity tends to zero.

Construction of stationary solutions for the Euler equations in R^2.

Abstract

We prove the existence and some moment estimates for an invariant measure $\mu$ for the two-dimensional ($2$D) deterministic Euler equations on the unbounded domain $\mathbb R^2$ and with highly regular initial data. The result is achieved by first showing the existence of Markov stationary processes which solve the hyperviscous $2$D Navier-Stokes equations with kinematic viscosity $\nu>0$ and an additive stochastic noise scaling as $\sqrt \nu$. We then study the inviscid limit and prove that, as $\nu$ tends to $0$, these processes converge, in an appropriate trajectory space, to a pathwise stationary solution to the Euler equations. Its law is the sought invariant measure $\mu$.