Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation

Abstract

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.