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

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