On the local existence and blow-up for generalized SQG patches

We study patch solutions of a family transport equations given by parameter \(\alpha \), \(0< \alpha <2\), with the cases =0\) and =1\) corresponding to Euler surface quasi-geostrophic respectively. In this paper, using several new cancellations, we provide following results. First, prove local well-posedness for \(H^{2}\) patches in half-space setting \(0<\alpha < 1/3\), allowing self-intersection fixed boundary. Furthermore, are able extend range \) which finite time singularities have been shown Kiselev et al. (Commun Pure Appl Math 70(7):1253–1315, 2017) (Ann 3:909–948, 2016). Second, establish that remain regular <2\) as long arc-chord condition regularity order \(C^{1+\delta }\) \(\delta >\alpha /2\) integrable. This finite-time singularity criterion holds lower than numerical simulations Córdoba (Proc Natl Acad Sci USA 102:5949–5952, 2005) Scott Dritschel (Phys Rev Lett 112:144505, 2014) patches, where curvature contour blows up numerically. is first proof or equal numerics. Finally, also improve results Gancedo (Adv 217(6):2569–2598, 2008) Chae 65(8):1037–1066, 2012), giving existence 1\) \(H^3\) \(1<\alpha <2\).