AI 中文总结
该研究构造了三维环面上的一个自主Lipschitz速度场，证明其为运动学感应方程的快速发电机，满足指数增长等关键性质，且具有特定的Lipschitz、熵等相关特征。
AI 中文摘要
我们构造了一个实值、无散度、与时间无关的速度场 $u\in\mathrm{W}^{1,\infty}(\mathbb{T}^3;\mathbb{R}^3)$，它是平坦三维环面上运动学感应方程的快速发电机。对于每个足够小的正磁扩散率，对应的感应算子存在一个实部由与扩散率无关的正常数下界约束的特征值；对于每个此类扩散率，存在一个非零实值、无散度的感应方程解，其 $\mathrm{L}^2$ 范数遵循精确的指数增长规律，增长率一致为正。对于同一速度场，粒子流及其逆的Lipschitz常数随时间最多线性增长，每次映射的拓扑熵为零，理想感应群在算子范数下的指数增长率为零。该速度场处处可微，除一个单圆外光滑，但非 $\mathrm{C}^1$。
英文摘要
We construct an autonomous Lipschitz fast dynamo on the flat three-torus whose particle flow has zero topological entropy and whose ideal induction equation has no exponential growth. For every sufficiently small positive magnetic diffusivity, the induction operator has an eigenvalue with real part bounded below by a positive constant independent of the diffusivity. A corresponding non-zero real-valued magnetic field satisfies an exact exponential growth law in $\mathrm{L}^2$. For the same velocity field, every ideal solution grows at most linearly in $\mathrm{L}^2$. Moreover, the particle flow and its inverse have Lipschitz constants growing at most linearly in time. The velocity field is real-valued, divergence-free, differentiable everywhere and smooth away from one circle, but is not $\mathrm{C}^1$.
CommentsTitle changed; proof substantially simplified; results unchanged. Added a reference to arXiv:2609.04153, which resolves Arnold's fast dynamo conjecture
