发表机构
Imperial College London（帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造了三维环面上非空的光滑自治无散速度场$C^k$-开族以实现快发电机，解决了快发电机猜想，通过拉伸-折叠-剪切机制构造周期快发电机并转化为自治流，证明了谱不稳定性在奇异扰动下保持且估计一致。
AI 中文摘要
我们构造了一个非空的$C^k$-开族（其中$k\in\mathbb{N}$），由三维环面$\mathbb{T}^3$上光滑、自治、无散的速度场组成，这些速度场可产生快发电机。这解决了Arnold问题集中记载的Zeldovich与Sakharov提出的快发电机猜想。我们首先通过适用于一大类速度场的拉伸-折叠-剪切机制，构造了一族光滑的时间周期快发电机。随后，我们利用全局庞加莱截面和适当指定的返回时间函数，在光滑自治流中实现了对应的周期动力学。在两种情形下，证明过程首先在各向异性分布Banach空间上分离出理想感应算子的不稳定分布本征模，随后证明这种谱不稳定性在奇异扰动$\varepsilon \Delta$下依然存在。所得估计在$\varepsilon\to0$时具有一致性，且在速度场的$C^k$扰动下保持稳定，从而得到了一个由光滑快发电机向量场构成的开集。
英文摘要
We construct a nonempty $C^k$-open family of smooth, autonomous, divergence-free velocity fields on $\mathbb{T}^3$ that generate fast dynamos. The proof proceeds by first establishing fast dynamo action for a large class of smooth, time-periodic velocity fields exhibiting the classical stretch--fold--shear mechanism. We then realize the associated period map within a smooth autonomous flow using carefully tuned return dynamics to a global Poincaré section. In both settings, the proof first isolates unstable distributional eigenmodes for the ideal dynamo operator on an anisotropic Banach space of distributions, and then shows that this spectral instability persists under the singular perturbation $\varepsilon Δ$. The resulting estimates are uniform as $\varepsilon\to0$ and stable under $C^k$ perturbations of the velocity field, yielding an open set of smooth fast dynamo vector fields. This result resolves the Fast Dynamo Conjecture of Zeldovich and Sakharov, as recorded in Arnold's book of problems.
Comments67 pages. V2: expositional revision
