Well-posedness for SQG sharp fronts with unbounded curvature

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