论文标题
3D流体动力学中的螺旋唯一性猜想
The helicity uniqueness conjecture in 3D hydrodynamics
论文作者
论文摘要
我们证明，螺旋度是唯一的常规CASIMIR功能，用于在封闭的三维歧管$ M $上，在平滑精确的无偏见的矢量字段上，在平滑精确的无偏见的矢量字段上，音量扩张差异组$ \ text {sdiff}（m）$。更准确地说，在$ c^\ infty $的空间上定义的任何常规$ c^1 $（更一般而言，$ c^k $，$ k \ ge 4 $）精确的无散射矢量字段和在任意数量增强量的范围内的差异性差异性差异可以表示为$ c^1 $ c^1 $ C^1 $ C^1 $ c^1 $ c^1 $ c^1 $ c^1 $。这给出了3D中$ \ text {sdiff}（m）$的伴奏的完整描述，并完成了3D中的$ \ text {sdiff}（m）$，并完成了Arnold-khesin的1998年猜想的证明，该猜想是琐碎的$ m $，带有琐碎的第一同源性组。我们的证明利用了动态系统理论的不同工具，包括无差异矢量场的正常形式，Poincaré-Birkhoff定理以及具有双曲线零的矢量场的划分引理。
We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergence-free vector fields on a closed three-dimensional manifold $M$. More precisely, any regular $C^1$ functional defined on the space of $C^\infty$ (more generally, $C^k$, $k\ge 4$) exact divergence-free vector fields and invariant under arbitrary volume-preserving diffeomorphisms can be expressed as a $C^1$ function of the helicity. This gives a complete description of Casimirs for adjoint and coadjoint actions of $\text{SDiff}(M)$ in 3D and completes the proof of Arnold-Khesin's 1998 conjecture for a manifold $M$ with trivial first homology group. Our proofs make use of different tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincaré-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.
