Local Regularity and Finite-time Singularity for a Class of Generalized SQG Patches on the Half-plane -

Local Regularity and Finite-time Singularity for a Class of Generalized SQG Patches on the Half-plane

Time: 2024-10-30 Published By: He Liu

Speaker(s): Liutang Xue(Beijing Normal University)

Time: 16:00-17:00 December 9, 2024

Venue: Siyuan Lecture Hall, Zhihua Building

In this talk, 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 (Λ)

. When

m (Λ) = Λ^{α}

, where

Λ = (- Δ)^{1 / 2}

and

α \in (0, 2)

, the equation reduces to the well-known

α

-SQG equation. Finite-time singularity formation for patch solutions to the

α

-SQG equation was famously discovered by Kiselev, Ryzhik, Yao, and Zlatos [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}^{\infty} \frac{1}{r (\log r) m (r)} d r < \infty

along with some additional mild conditions. Notably, our result fills the gap between the globally well-posed 2D Euler equation (

α = 0

) and the

α

-SQG equation (

α > 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

α

-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^{α}

near infinity but does not have an explicit formulation. This is based on a joint work with Qianyun Miao, Changhui Tan and Zhilong Xue.