Global Regularity and Fast Small Scale Formation for Euler Patch Equation in a Smooth Domain

TL;DR

This paper establishes global regularity results for Euler vortex patches in smooth bounded domains, providing bounds on boundary curvature growth and demonstrating sharpness through a symmetric example.

Contribution

It extends regularity theory for Euler patches to bounded domains and constructs an example showing the bounds are optimal.

Findings

Global regularity of vortex patches in smooth bounded domains

Upper bounds on curvature growth of patch boundaries

Existence of symmetric example with double exponential curvature growth

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. In this paper, we study Euler vortex patches in a general smooth bounded domain. We prove global in time regularity by providing an upper bound on the growth of curvature of the patch boundary. For a special symmetric scenario, we construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.