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It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remainregular if it is regular enough initially. In bounded domains, the regularitytheory for patch solutions is less complete. In this paper, we study Eulervortex patches in a general smooth bounded domain. We prove global in timeregularity by providing an upper bound on the growth of curvature of the patchboundary. For a special symmetric scenario, we construct an example of doubleexponential curvature growth, showing that our upper bound is qualitativelysharp.

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