Regularized and approximate equations for sharp fronts in the surface quasi-geostrophic equation and its generalizations**Supported by the NSF under grant number DMS-1616988.

Abstract

We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (GSQG) equations. We derive a cubic approximation of the contour dynamics equation, and prove the short-time well-posedness of the approximate equations for generalized surface quasi-geostrophic fronts and weak well-posedness for surface quasi-geostrophic fronts.