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 $Ψ$, and prove that every $C^{1}$ steady Euler flow $(\boldsymbol{u},p)$ satisfying the assumptions $ι_{\boldsymbol{u}}dΨ=0$ and $p=p(Ψ)$ admits the tangential flow representation \[\boldsymbol{u}=c_1(Ψ)\boldsymbolξ^{1}+c_2(Ψ)\boldsymbolξ^{2},\] for some lifted solenoidal vector fields $\boldsymbolξ^{1}$ and $\boldsymbolξ^{2}$ associated with a natural basis of weighted harmonic one-forms on the toroidal leaves. Moreover, the flux function $Ψ$ 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.