论文标题
涡流斑块的周长生长
Growth of perimeter for vortex patches in a bulk
论文作者
论文摘要
我们考虑二维不可压缩的Euler方程。我们在$ t^2 $和$ r^2 $上构建具有光滑边界的涡流补丁，其周长会随着时间而生长。更确切地说，对于任何常数$ m> 0 $，我们在$ t^2 $中构造一个涡旋补丁，其平滑边界在初始时间为1的订单长度为1，以便在有限的时间内成长为给定常数$ m $。结构是通过从几乎方形的贴片中切出一根细的棍子来完成的。与$ r^2 $的薄手柄的几乎圆形贴片相似的结果。
We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on $T^2$ and $R^2$ whose perimeter grows with time. More precisely, for any constant $M > 0$, we construct a vortex patch in $T^2$ whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant $M$ within finite time. The construction is done by cutting a thin stick out of an almost square patch. A similar result holds for an almost round patch with a thin handle in $R^2$.
