  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.
