Non-invariance of Gaussian measures under the 2D Euler flow

Abstract

In this article we consider the two-dimensional incompressible Euler equations on ${\mathbb{T}}^2$ and seek to characterize the Gaussian measures of jointly independent Fourier coefficients which are invariant or not. We characterize all such measures with sufficiently rapidly decaying coefficients as exactly those supported only on shear flows or cellular flows. For $H^{\sigma }({\mathbb{T}}^2)$ ($\sigma >3$) regularities, we give a checkable sufficient condition for non-invariance and show that this condition holds on an open and dense set in suitable topologies (and so is generic in a Baire category sense) and check the condition using interval arithmetic for a basic example. We conjecture that the same characterization at high regularities holds in all $H^{\sigma }$ at least down to $\sigma >0$. We also pose a few other related conjectures, which we believe to be approachable.