Grad猜想的反例
Counterexamples to Grad's conjecture
- Brown University（布朗大学）
- Bar-Ilan University（巴伊兰大学）
- University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了磁流体静力学方程在实心环面上的光滑解，其对称群仅为循环群，从而给出Grad猜想的反例，并提供了Lean 4验证。
AI中文摘要：
对于每个足够大的$N$，我们在嵌入的实心环面上构造磁流体静力学方程的光滑解，其正则压力水平面为嵌套环面，磁场恰好在圆形磁轴上消失，且其欧几里得对称群（即使允许反转场符号的等距）恰好是循环群$C_N$，即旋转$2\pi/N$整数倍的群。对于每个这样的$N$，这些平衡态构成一个光滑的单参数族，在嵌入构型的模空间中局部非平凡，并且不具有Grad所 conjectured 的平面反射、轴向或螺旋对称性。上述结果还提供了Lean 4形式化验证。
英文摘要：
For each sufficiently large $N$, we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magnetic field vanishes exactly on a round magnetic axis, and whose group of Euclidean symmetries, even when isometries reversing the sign of the field are admitted, is exactly the cyclic group $C_N$ of rotations through multiples of $2π/N$. For each such $N$, these equilibria form a smooth one-parameter family which is locally nontrivial in the moduli space of embedded configurations and have none of the plane-reflection, axial, or helical symmetries conjectured by Grad. A Lean 4 certification of the above results is also provided.
