AI 中文总结
本文研究定常欧拉流的叶状结构，证明满足特定条件的C¹定常欧拉流具有切向流表示，通量函数满足法向通量方程，其一般结构包含轴对称特例。
AI 中文摘要
1965年，V. I. 阿诺德建立了一个结构定理，保证了具有非常数压强的一般三维定常欧拉流存在由不变曲面构成的叶状结构。本文研究定常欧拉流中可能出现的叶状结构，考虑一个由通量函数Ψ的水平集叶状化的环面区域，证明满足ι_u dΨ=0和p=p(Ψ)假设的每一个C¹定常欧拉流(u,p)，都存在切向流表示：u=c₁(Ψ)ξ¹ + c₂(Ψ)ξ²，其中ξ¹和ξ²是与环面叶上加权调和1-形式自然基相关的提升无散向量场。此外，通量函数Ψ满足一个称为法向通量方程的标量方程。这些刻画揭示了定常欧拉流的一般叶状结构，其中Clebsch表示和Grad-Shafranov方程可作为轴对称特例被恢复。
英文摘要
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.
Comments20 pages, 3 figures, 1 table
