Horizontally periodic generalized surface quasigeostrophic patches and layers
David M. Ambrose, Fazel Hadadifard, James P. Kelliher
TL;DR
The paper develops a rigorous framework for horizontally 1-periodic α-SQG patches and layers, establishing short-time well-posedness for the contour-dynamics equation in the periodic setting. It constructs the periodic Green’s function and corresponding kernels, derives a periodic CDE, and proves that CDE solutions yield weak α-SQG-patch solutions, with careful treatment of self-intersection via a chord-arc measure. The main result shows existence and uniqueness of periodic CDE solutions for α in (0,1) with boundary regularity in $H^m$ for $m\ge 3$, using a regularization and compactness (Aubin-Lions) argument to pass to the limit. The work extends the Euler (α=0) theory to the interceding regime between Euler and SQG, handling the extra periodicity and two-sided-front structures, and provides a foundation for future work at α=1 and for more general periodic geometries.
Abstract
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 $H^k$-regularity of the patch boundary, $k \ge 3$, 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.