论文标题
对于广义SQG方程的不对称涡流贴片的存在
Existence of asymmetric vortex patch for the generalized SQG equations
论文作者
论文摘要
本文旨在研究整个平面中$α\ in [1,2）$的简单连接贴片的二维广义表面准斑块（GSQG）方程的不对称解，其中$α= 1 $对应于表面Quasi-GeageSi-GeoSi-GeoStophic exostroptrophic方程（SQG）。更确切地说，我们构建非平凡的简单连接的共旋转和行进贴片，并具有不平等的涡度幅度。证明是通过将降低参数与对轮廓动力学方程的线性化的隐式函数定理的结合进行的。我们的结果将Hassainia-Hmidi（DCDS-A，2021年）和Hassainia-Wheeler（Siam J.Math。Anal。，2022）的最新范围扩展到[0,1）$ $α\ in [0,1）$中的范围。
This paper aims to study the existence of asymmetric solutions for the two-dimensional generalized surface quasi-geostrophic (gSQG) equations of simply connected patches for $α\in[1,2)$ in the whole plane, where $α=1$ corresponds to the surface quasi-geostrophic equations (SQG). More precisely, we construct non-trivial simply connected co-rotating and traveling patches with unequal vorticity magnitudes. The proof is carried out by means of a combination of a desingularization argument with the implicit function theorem on the linearization of contour dynamics equation. Our results extend recent ones in the range $α\in[0,1)$ by Hassainia-Hmidi (DCDS-A, 2021) and Hassainia-Wheeler (SIAM J. Math. Anal., 2022) to more singular velocities, filling an open gap in the range of $α$.
