AI 中文总结
该研究构造三维环面上的周期分段仿射剪切速度场，证明理想感应方程下大剪切振幅时非零无散度初始场指数增长，时间反转后对应解指数衰减，为相关方程的动力学性质提供了严格分析。
AI 中文摘要
我们在三维环面上构造了一个无散度速度场，该场具有时间周期性，由三个交替出现的分段仿射剪切组成。当剪切振幅足够大时，理想感应方程下，$L^p$中每个非零无散度初始场都会指数增长。证明过程为时间一步映射建立了一致锥条件，并结合聚集不等式排除了几乎处处位于稳定丛中的非平凡无散度场。此外，我们证明对于时间反转后的速度场（其时间一步映射是原映射的逆），存在非平凡、有界、无散度的初始构型，几乎处处取值于其二维稳定丛中，对应的解在每个$L^p$中都指数衰减。
英文摘要
We construct a divergence-free velocity field on the three-dimensional torus that is time-periodic and consists of three alternating piecewise-affine shears. For sufficiently large shear amplitude, every non-zero divergence-free initial field in $L^p$ grows exponentially under the ideal induction equation. The proof establishes a uniform cone condition for the time-one map, and combines it with a bunching inequality to rule out nontrivial divergence-free fields lying almost everywhere in the stable bundle. Additionally, we show that for the time-reversed velocity, whose time-one map is the inverse of the original one, there exist nontrivial, bounded, divergence-free initial configurations taking values almost everywhere in its two-dimensional stable bundle. The corresponding solution decays exponentially in every $L^p$.
Comments30 pages
