A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area.
A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area.
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DOI：
10.1098/rspa.2015.0629
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发表时间：
2016-01
期刊：
影响因子：
--
通讯作者：
McCue SW
中科院分区：
文献类型：
--
作者：
Dallaston MC;McCue SW
受流体力学问题的启发，我们考虑了标准曲线缩短流动问题的推广，对于一个封闭的嵌入平面曲线，曲线所包围的面积被迫以规定的速率减少。利用形式渐近和数值技术，我们推导出曲线收缩到一个点时可能的消光形状，这取决于面积减少的速率；我们发现存在比标准曲线缩短更广泛的消光形状，对于标准曲线缩短，最初的简单闭合曲线总是渐近圆形。我们还提供了数值证据，证明在非凸初始条件下自交是可能的，区分了曲线内部的掐断和合并。
Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.