Title: Well-Posedness and Finite Time Singularity for Touching g-SQG Patches on the Plane Show Chinese title

Title: 适定性及平面上接触g-SQG片的有限时间奇异性

Abstract: We prove local well-posedness as well as singularity formation for the g-SQG patch model on the plane (so on a domain without a boundary), with $\alpha\in(0,\frac 16]$ and patches being allowed to touch each other. We do this by bypassing any auxiliary contour equations and tracking patch boundary curves directly instead of their parametrizations. In our results, which are sharp in terms of $\alpha$, the patch boundaries have $L^2$ curvatures and a singularity occurs when at least one of these $L^2$-norms blows up in finite time. Show Chinese abstract

Abstract: 我们证明了在平面上（即在没有边界的区域上）g-SQG 薄片模型的局部适定性以及奇异性形成，其中$\alpha\in(0,\frac 16]$和薄片允许彼此接触。 我们通过绕过任何辅助轮廓方程，直接跟踪薄片边界曲线而不是它们的参数化方式来实现这一点。 在我们的结果中，相对于$\alpha$而言是精确的，薄片边界具有$L^2$曲率，当这些$L^2$-范数中的至少一个在有限时间内爆破时，就会发生奇异性。