Quantitative estimates for uniformly-rotating vortex patches

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 < Omega << 1, then the outmost point of the patch must be far from the center of rotation, with distance at least of order Omega-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 infinity norm of the polar graph which parametrizes the boundary.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

This paper provides quantitative estimates for uniformly-rotating vortex patches. It proves that if a non-radial simply-connected patch D has a small angular velocity 0 < Omega << 1, the distance from the center of rotation to the outmost point of the patch must be at least of order Omega-1/2. For m-fold symmetric simply-connected rotating patches, the paper shows that their angular velocity must be close to 12 for m >> 1, with a difference at most O(1/m), and also obtains estimates on the L infinity norm of the polar graph which parametrizes the boundary.