Fast Escape in Incompressible Vector Fields

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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