Title: Global regularity and fast small scale formation for Euler patch equation in a disk,
Abstract: It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain
regular if it is regular enough initially. In bounded domains, the regularity
theory for patch solutions is less complete. We study here the Euler vortex
patch in a disk. We prove global in time regularity by providing the upper
bound of the growth of curvature of the patch boundary. For a special symmetric
scenario, we construct an example of double exponential curvature growth,
showing that such upper bound is qualitatively sharp.