Contour dynamics and global regularity for periodic vortex patches and layers

We study vortex patches for the 2D incompressible Euler equations. Prior works on this problem take the support of the vorticity (i.e., the vortex patch) to be a bounded region. We instead consider the horizontally periodic setting. This includes both the case of a periodic array of bounded vortex patches and the case of vertically bounded vortex layers. We develop the contour dynamics equation for the boundary of the patch in this horizontally periodic setting, and demonstrate global $C^{1,\epsilon }$ regularity of this patch boundary. In the process of formulating the problem, we consider different notions of periodic solutions of the 2D incompressible Euler equations, and demonstrate equivalence of these.