In this note, we consider the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disc. By choosing a suitable vector field to deform the patch, we show that each simply-connected rotating vortex patch $D$ with angular velocity $\Omega$, $\Omega\geq \max\{{1}/{2},({2 l^2})/{(1-l^2)^2}\}$ or $\Omega\leq -({2 l^2})/{(1-l^2)^2}$, where $l=\sup_{x\in D}|x|$, must be a disc. The main idea of the proof, which has a variational flavor, comes from a very recent paper of G\'omez-Serrano--Park--Shi--Yao, arXiv:1908.01722, where radial symmetry of rotating vortex patches in the whole plane was studied.

中文翻译：

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关于圆盘中旋转涡块的径向对称性

在本笔记中，我们考虑了单位圆盘中二维不可压缩欧拉方程的旋转涡块的径向对称性。通过选择一个合适的矢量场使补丁变形，我们证明了每个简单连接的旋转涡补丁 $D$ 的角速度为 $\Omega$, $\Omega\geq \max\{{1}/{2},( {2 l^2})/{(1-l^2)^2}\}$ 或 $\Omega\leq -({2 l^2})/{(1-l^2)^2}$ ，其中 $l=\sup_{x\in D}|x|$，必须是光盘。证明的主要思想有变分的味道，来自 G\'omez-Serrano--Park--Shi--Yao, arXiv:1908.01722 的一篇最近的论文，其中旋转涡块的径向对称性在整个飞机进行了研究。