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- W2963820009 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 $$ c_1< Vert vVert < c_2.$$ 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 $$ sqrt{4pi c_2 / c_1}$$ . 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." @default.
- W2963820009 created "2019-07-30" @default.
- W2963820009 creator A5054778975 @default.
- W2963820009 date "2017-10-24" @default.
- W2963820009 modified "2023-09-26" @default.
- W2963820009 title "Fast escape in incompressible vector fields" @default.
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- W2963820009 doi "https://doi.org/10.1007/s00605-017-1118-y" @default.
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