In this paper, we construct a family of symmetric vortex patches for the 2D steady incompressible Euler equations in a disk. The result is obtained by studying a variational problem in which the kinetic energy of the fluid is maximized subject to some appropriate constraints for the vorticity. Moreover, we show that these vortex patches “shrink” to a given minimum point of the corresponding Kirchhoff–Routh function as the vorticity strength parameter goes to infinity.

中文翻译：

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磁盘中稳定对称涡旋补丁的存在

在本文中，我们为磁盘中的二维稳定不可压缩Euler方程构造了一个对称的涡旋斑块。通过研究一个变分问题可以得到结果，在变分问题中，流体的动能会受到一些适当的涡度约束而达到最大。此外，我们显示出，随着涡度强度参数达到无穷大，这些涡旋斑会“收缩”至相应的Kirchhoff-Routh函数的给定最小值。