不可压缩向量场中的快速逃逸
Fast Escape in Incompressible Vector Fields
摘要：
被离岸流困住的游泳者被建议垂直于水流方向游泳以逃离它。我们以类似的精神描述一个数学原理。更准确地说，我们认为流动$γ$在平面上诱导的不可压缩的向量场$\textbf{v}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$满足$ c_1 < \|v\| < c_2.$的长度$\ell$流动曲线$\dot γ(t) = \textbf{v}(γ(t))$，直到$γ$离开一个圆盘的半径为1中心的初始位置可以是$\ell \sim c_2/c_1$一样长。对于正交流$\textbf{v}^{\perp} = (-\textbf{v}_2, \textbf{v}_1)$也是如此。我们表明，一个组合严格更好：总是存在一个曲线流动的第一次沿$\textbf{v}^{\perp}$，然后沿$\textbf{v}$逃逸单位盘之前达到长度$ \sqrt{4πc_2 / c_1}$。此外，如果$\textbf{v}$的逃逸长度一致地为$\sim c_2/c_1$，则$\textbf{v}^{\perp}$的逃逸长度一致地为$\sim 1$（允许从电流快速逃逸）。我们还证明了一个基本的定量庞加莱-Bendixson定理，似乎是新的。
英文摘要：
Swimmers caught in a rip current flowing away from the shore are advised to swim orthogonally to the current to escape it. We describe a mathematical principle in a similar spirit. More precisely, we consider flows $γ$ in the plane induced by incompressible vector fields $\textbf{v}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ satisfying $ c_1 < \|v\| < c_2.$ The length $\ell$ a flow curve $\dot γ(t) = \textbf{v}(γ(t))$ until $γ$ leaves a disk of radius 1 centered at the initial position can be as long as $\ell \sim c_2/c_1$. The same is true for the orthogonal flow $\textbf{v}^{\perp} = (-\textbf{v}_2, \textbf{v}_1)$. We show that a combination does strictly better: there always exists a curve flowing first along $\textbf{v}^{\perp}$ and then along $\textbf{v}$ which escapes the unit disk before reaching the length $ \sqrt{4πc_2 / c_1}$. Moreover, if the escape length of $\textbf{v}$ is uniformly $\sim c_2/c_1$, then the escape length of $\textbf{v}^{\perp}$ is uniformly $\sim 1$ (allowing for a fast escape from the current). We also prove an elementary quantitative Poincaré-Bendixson theorem that seems to be new.
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