# Global regularity and fast small scale formation for Euler patch equation in a disk

## TL;DR

This paper proves global regularity for Euler vortex patches in a disk and constructs an example of double exponential curvature growth, highlighting the sharpness of the upper bound.

## Contribution

It establishes global regularity results for vortex patches in bounded domains and demonstrates the sharpness of curvature growth bounds through a symmetric example.

## Key findings

- Global regularity of vortex patches in a disk
- Construction of an example with double exponential curvature growth
- Upper bound on curvature growth is sharp

## 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.