We study a geometric flow on curves, immersed in $\mathbb{R}^3$, that have strictly positive torsion. The evolution equation is given by $$X_{t}=\frac{1}{\sqrt{\tau}} \textbf{B}$$ where $\tau$ is the torsion and $\textbf{B}$ is the unit binormal vector. In the case of constant curvature, we find all of the stationary solutions and linearize the PDE for torsion around stationary solutions admitting an explicit formula. Afterwards, we prove the $L^2(\mathbb{R})$ linear stability of the stationary solutions corresponding to helices with constant curvature and constant torsion.

中文翻译：

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空间曲线上保持曲率流动的平稳解

我们研究曲线上的几何流动，沉浸在 $\mathbb{R}^3$ 中，具有严格的正扭转。演化方程由 $$X_{t}=\frac{1}{\sqrt{\tau}} \textbf{B}$$ 给出，其中 $\tau$ 是扭力，$\textbf{B}$ 是单位副法向量。在恒定曲率的情况下，我们找到所有的平稳解并线性化偏微分方程，以在允许明确公式的平稳解周围扭转。之后，我们证明了恒定曲率和恒定扭转螺旋对应的平稳解的$L^2(\mathbb{R})$线性稳定性。