  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.
