Well-posedness for SQG sharp fronts with unbounded curvature

Abstract

Patch solutions for the surface quasigeostrophic (SQG) equation model sharp temperature fronts in atmospheric and oceanic flows. Boundedness of curvature plays an important role in the theoretical [F. Gancedo and R. M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014) 635–639] and numerical [D. Córdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005) 5949–5952; R. K. Scott and D. G. Dritschel, Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system, Phys. Rev. Lett. 112 (2014) 144505] study of singularity formation. In this paper, we establish local well-posedness for SQG sharp fronts of low Sobolev regularity, [Formula: see text] for arbitrarily small [Formula: see text]. This is the first construction for SQG sharp front solutions with unbounded curvature.