Abstract
In this paper, we derive some quantitative estimates for uniformly-rotating vortex patches. We prove that if a non-radial simply-connected patch D is uniformly-rotating with small angular velocity 0<Ω≪1, then the outmost point of the patch must be far from the center of rotation, with distance at least of order Ω−1/2. For m-fold symmetric simply-connected rotating patches, we show that their angular velocity must be close to 12 for m≫1 with the difference at most O(1/m), and also obtain estimates on L∞ norm of the polar graph which parametrizes the boundary.
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