Admissible Invariant-Torus Foliations for Steady Euler Flows

Abstract

In 1965, V. I. Arnold established a structure theorem guaranteeing the existence of a foliation by invariant surfaces for general three-dimensional steady Euler flows with non-constant pressure. In this paper, we investigate what foliation structures can arise in steady Euler flows. We consider a toroidal domain foliated by the level sets of a flux function $\Psi$, and prove that every $C^1$ steady Euler flow $(\mathbf{u},p)$ satisfying the assumptions ${\iota }_{\mathbf{u}}d\Psi =0$ and $p=p(\Psi )$ admits the tangential flow representation \[\boldsymbol{u}=c_1(Ψ)\boldsymbolξ^{1}+c_2(Ψ)\boldsymbolξ^{2},\] for some lifted solenoidal vector fields ${\mathbf{\xi }}^1$ and ${\mathbf{\xi }}^2$ associated with a natural basis of weighted harmonic one-forms on the toroidal leaves. Moreover, the flux function $\Psi$ satisfies a single scalar equation, referred to as the normal flux equation. These characterizations reveal the general foliation structure of steady Euler flows, with the Clebsch representation and the Grad--Shafranov equation recovered as the axisymmetric special case.

20 pages, 3 figures, 1 table