We rigorously construct continuous curves of rotating vortex patch solutions to the two-dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [WOZ84,Ove86] of singular limiting patches with 90$^\circ$ corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are "Cat's eyes"-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.

中文翻译：

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旋转涡块的全局分叉

我们严格地构造了二维欧拉方程的旋转涡块解的连续曲线。曲线较大，当参数趋于无穷大时，旋转坐标系中流体角速度沿界面的最小值任意变小。这与推测存在 [WOZ84,Ove86] 的具有 90$^\circ$ 角的奇异限制斑块一致，在该角点处相对流体速度消失。对于靠近圆盘的解，我们证明了流动中存在“猫眼”型结构，并提供了数值证据表明这些结构沿着整个解曲线持续存在并且与角的形成有关。我们还表明，对于任何旋转涡块，只要边界足够规则，边界就是解析的。