Local regularity and finite-time singularity for a class of generalized SQG patches on the half-plane

In this paper, we investigate a class of inviscid generalized surface quasi-geostrophic (SQG) equations on the half-plane with a rigid boundary. Compared to the Biot-Savart law in the vorticity form of the 2D Euler equation, the velocity formula here includes an additional Fourier multiplier operator $m(\mathrm{\Lambda })$. When $m(\mathrm{\Lambda })={\mathrm{\Lambda }}^{\alpha }$, where $\mathrm{\Lambda }=(-\mathrm{\Delta }{)}^{1\mathrm{/}2}$ and $\alpha \in (0,2)$, the equation reduces to the well-known $\alpha$-SQG equation. Finite-time singularity formation for patch solutions to the $\alpha$-SQG equation was famously discovered by Kiselev, Ryzhik, Yao, and Zlatoš [Ann. Math., 184 (2016), pp. 909-948]. We establish finite-time singularity formation for patch solutions to the generalized SQG equations under the Osgood condition $${\int }_2^{\mathrm{\infty }}\frac{1}{r(\log r)m(r)}dr<\mathrm{\infty }$$ along with some additional mild conditions. Notably, our result fills the gap between the globally well-posed 2D Euler equation ($\alpha =0$) and the $\alpha$-SQG equation ($\alpha >0$). Furthermore, in line with Elgindi’s global regularity results for 2D Loglog-Euler type equations [Arch. Rat. Mech. Anal., 211 (2014), pp. 965-990], our findings suggest that the Osgood condition serves as a sharp threshold that distinguishes global regularity and finite-time singularity in these models. In addition, we generalize the local regularity and finite-time singularity results for patch solutions to the $\alpha$-SQG equation, as established by Gancedo and Patel [Ann. PDE, 7 (2021), no. 1, Art. no. 4], extending them to cases where $m(r)$ behaves like $r^{\alpha }$ near infinity but does not have an explicit formulation.