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 $$\mathbf v :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$$ satisfying $$ c_1< \Vert v\Vert < c_2.$$ The length $$\ell $$ a flow curve $$\dot{\gamma }(t) = \mathbf v (\gamma (t))$$ 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 $$\mathbf v ^{\perp } = (-\mathbf v _2, \mathbf v _1)$$ . We show that a combination does strictly better: there always exists a curve flowing first along $$\mathbf v ^{\perp }$$ and then along $$\mathbf v $$ which escapes the unit disk before reaching the length $$ \sqrt{4\pi c_2 / c_1}$$ . Moreover, if the escape length of $$\mathbf v $$ is uniformly $$\sim c_2/c_1$$ , then the escape length of $$\mathbf v ^{\perp }$$ is uniformly $$\sim 1$$ (allowing for a fast escape from the current). We also prove an elementary quantitative Poincare–Bendixson theorem that seems to be new.