二维拓扑流体动力学中的指数增长性
Exponential growth in two-dimensional topological fluid dynamics
摘要：
本文描述了二维流体的杆搅拌的拓扑运动学。主要的工具是瑟斯顿-尼尔森（TN）理论，这意味着取决于搅拌协议的基本拓扑长度的材料线增长指数或线性。我们给出了一个应用程序的被动平流标量的梯度的增长，亥姆霍兹-开尔文定理，然后产生的应用程序欧拉流。的主要定理表明，有周期性搅拌协议，通用的初始涡量产生的解决方案欧拉方程，这是不是周期性的，进一步的$L^\infty$和$L^1$ -范数的梯度，其涡量指数增长的时间。
英文摘要：
This paper describes topological kinematics associated with the stirring by rods of a two-dimensional fluid. The main tool is the Thurston-Nielsen (TN) theory which implies that depending on the stirring protocol the essential topological length of material lines grows either exponentially or linearly. We give an application to the growth of the gradient of a passively advected scalar, the Helmholtz-Kelvin Theorem then yields applications to Euler flows. The main theorem shows that there are periodic stirring protocols for which generic initial vorticity yields a solution to Euler's equations which is not periodic and further, the $L^\infty$ and $L^1$-norms of the gradient of its vorticity grow exponentially in time.
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