Admissible Invariant-Torus Foliations for Steady Euler Flows

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 $\mathrm{\Psi }$, and prove that every $C^1$ steady Euler flow $(\mathit{u},p)$ satisfying the assumptions ${\iota }_{\mathit{u}}d\mathrm{\Psi }=0$ and $p=p(\mathrm{\Psi })$ admits the tangential flow representation $$\mathit{u}=c_1(\mathrm{\Psi }){\mathit{\xi }}^1+c_2(\mathrm{\Psi }){\mathit{\xi }}^2,$$ for some lifted solenoidal vector fields ${\mathit{\xi }}^1$ and ${\mathit{\xi }}^2$ associated with a natural basis of weighted harmonic one-forms on the toroidal leaves. Moreover, the flux function $\mathrm{\Psi }$ 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.