SIAM Journal on Applied Mathematics, Volume 80, Issue 2, Page 1034-1056, January 2020.
In this paper, we consider the evolution of the vortex filament equation (VFE): $X_t = X_s \wedge X_{ss},$ taking $M$-sided regular polygons with nonzero torsion as initial data. Using algebraic techniques backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point $X(0,t)$ is not planar and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as those already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann's nondifferentiable function that are as close to smooth curves as desired when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the self-similar solutions of VFE have finite renormalized energy.

中文翻译：

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具有非零扭转的正则$ M $-多边形涡旋长丝方程的演化

SIAM应用数学杂志，第80卷，第2期，第1034-1056页，2020年1月。
在本文中，我们考虑了涡流丝方程（VFE）的演化：$ X_t = X_s \ wedge X_ {ss}，$将具有非零扭转的$ M $边的规则多边形作为初始数据。使用数值模拟支持的代数技术，我们证明了解决方案在有理时间是多边形，如零扭转情况。但是，与那种情况不同的是，演变不是时间上的周期性；此外，点$ X（0，t）$的多重分形轨迹不是平面的，并且在很长时间内似乎是一个螺旋。VFE的这些新解可以用数字来说明，由螺旋和直线给出的VFE的平滑解与已经为圆建立的VFE的不稳定性相同。这通过显示所谓的黎曼变体的存在来实现 s的不可微函数，在正确的拓扑结构中进行测量时，它们尽可能接近平滑曲线。最近关于VFE的适定性的一些结果推动了这种拓扑的发展，这些结果证明了VFE的自相似解具有有限的重新归一化能量。