数学研究所
数学科学全国重点实验室
学术报告
偏微分方程研讨班
Speaker: Prof. Liutang Xia
（ Beijing Normal university）
Inviter: 阮国兴
Language: English
Title: Regularity and finite-time singularity for a class of generalized SQG
patches on the half-planeTime&Venue: 2025年6月30日（星期一）16:00-17:00
& S813
. When 
, where 
and 
, the equation reduces to the well-known 
-SQG equation. Finite-time singularity formation for patch solutions to the-SQG equation was famously discovered by Kiselev, Ryzhik, Yao, and Zlatos. We establish finite-time singularity formation for patch solutions to the generalized SQG equations under the Osgood condition 
along with some additional mild conditions. Notably, our result fills the gap between the globally well-posed 2D Euler equation (
) and the -SQG equation (
). Furthermore, in line with Elgindi's global regularity results for 2D Loglog-Euler type equations in the whole space, our findings suggest that the Osgood condition serves as a sharp threshold that distinguishes global regularity and finite-time singularity in these models. We also revisit Elgindi's result on 2D Loglog-Euler type equations to give a complete and slightly different proof.