Horizontally periodic generalised surface quasigeostrophic patches and layers

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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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.