Fast Escape in Incompressible Vector Fields

Abstract

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 $\gamma$ in the plane induced by incompressible vector fields $\text{v}:{\mathbb{R}}^2\to {\mathbb{R}}^2$ satisfying $c_1<\Vert v\Vert <c_2.$ The length $\ell$ a flow curve $\dot{\gamma }(t)=\text{v}(\gamma (t))$ until $\gamma$ 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 ${\text{v}}^{\perp }=(-{\text{v}}_2,{\text{v}}_1)$. We show that a combination does strictly better: there always exists a curve flowing first along ${\text{v}}^{\perp }$ and then along $\text{v}$ which escapes the unit disk before reaching the length $\sqrt{4\pi c_2/c_1}$. Moreover, if the escape length of $\text{v}$ is uniformly $\sim c_2/c_1$, then the escape length of ${\text{v}}^{\perp }$ 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.