Horizontally periodic generalized surface quasigeostrophic patches and layers

We study solutions to the α-SQG equations, which interpolate between the incompressible Euler and surface quasi-geostrophic equations. We extend prior results on existence of bounded patches, proving propagation of Hᵏ-regularity of the patch boundary, k ≥ 3, for finite time for patches that are periodic in one spatial dimension. Such periodic patches also encompass layers, or two-sided fronts. As the authors have treated the Euler case in prior work, we now primarily focus on the range of α for which α-SQG lies strictly between the Euler and SQG equations.