  In the language of differential geometry, the incompressible inviscid Euler
equations can be written in vorticity-vector potential form as \begin{align*}
\partial_t \omega + {\mathcal L}_u \omega &= 0\\ u &= \delta \tilde \eta^{-1}
\Delta^{-1} \omega \end{align*} where $\omega$ is the vorticity $2$-form,
${\mathcal L}_u$ denotes the Lie derivative with respect to the velocity field
$u$, $\Delta$ is the Hodge Laplacian, $\delta$ is the codifferential (the
negative of the divergence operator), and $\tilde \eta^{-1}$ is the canonical
map from $2$-forms to $2$-vector fields induced by the Euclidean metric $\eta$.
In this paper we consider a generalisation of these Euler equations in three
spatial dimensions, in which the vector potential operator $\tilde \eta^{-1}
\Delta^{-1}$ is replaced by a more general operator $A$ of order $-2$; this
retains the Lagrangian structure of the Euler equations, as well as most of its
conservation laws and local existence theory. Despite this, we give three
different constructions of such an operator $A$ which admits smooth solutions
that blow up in finite time, including an example on ${\bf R}^3$ which is
self-adjoint and positive definite. This indicates a barrier to establishing
global regularity for the three-dimensional Euler equations, in that any such
method must use some property of those equations that is not shared by the
generalised Euler equations considered here.
