Non-invariance of Gaussian Measures under the 2D Euler Flow

TL;DR

This paper investigates conditions under which Gaussian measures supported on certain Sobolev spaces are not invariant under the 2D Euler flow, showing that non-invariance is generic and providing explicit examples.

Contribution

It establishes a sufficient condition for non-invariance of Gaussian measures under 2D Euler equations and demonstrates that this condition is generic in relevant topologies.

Findings

Non-invariance condition holds on an open and dense set

Explicit examples of non-invariant Gaussian measures provided

Non-invariance is generic in a Baire category sense

Abstract

In this article we consider the two-dimensional incompressible Euler equations and give a sufficient condition on Gaussian measures of jointly independent Fourier coefficients supported on $H^{\sigma}(\mathbb{T}^2)$ ($\sigma>3$) such that these measures are not invariant (in vorticity form). We show that this condition holds on an open and dense set in suitable topologies (and so is generic in a Baire category sense) and give some explicit examples of Gaussian measures which are not invariant. We also pose a few related conjectures which we believe to be approachable.