带有多个涡旋空洞的均匀旋转欧拉构型
Uniformly Rotating Euler Configurations with Multiple Vorticity Holes
AI总结：
本文构造了含m≥2个集中内部分量的均匀旋转欧拉方程涡斑解族，通过对称约化等方法完成分析，是首次对多分量同时坍缩的去奇异性 regime 的解析构造。
AI中文摘要：
我们构造了二维不可压缩欧拉方程的新型均匀旋转涡斑解族，该解族由一个单连通的外部涡斑和多个可几何解释为空洞的内部界面组成。更确切地说，每个解包含一个单一的外部涡斑，其包围着m≥2个相同的高度集中的内部分量，这些分量排列在正m边形的顶点处；整个构型沿顺时针方向做刚性旋转。当集中参数趋于零时，内部分量收缩并同时向原点坍缩，而外部边界收敛到单位圆。对应的涡度在测度意义下收敛到一个Rankine涡，其中心补充了一个环量为−m的点涡。该证明基于轮廓动力学公式、对称约化得到的两个非线性边界方程，以及合适的奇异重标度。随后，我们在适配对称性的Hölder空间中应用带连续参数的隐函数定理。据我们所知，这是首次对去奇异性 regime 进行分析构造，该 regime 中多个集中的内部分量包含在一个共同的外部涡斑中并同时向其中心坍缩。
英文摘要：
We construct new families of uniformly rotating vortex-patch solutions of the two-dimensional incompressible Euler equations consisting of a simply connected outer patch and multiple interior interfaces, which can be interpreted geometrically as holes. More precisely, each solution consists of a single outer vortex patch enclosing $\mathbf m\geq2$ identical, highly concentrated inner components arranged at the vertices of a regular $\mathbf m$-gon; the entire configuration rotates rigidly in the clockwise direction. As the concentration parameter tends to zero, the inner components shrink and collapse simultaneously toward the origin, while the outer boundary converges to the unit circle. The corresponding vorticities converge, in the sense of measures, to a Rankine vortex supplemented by a point vortex of circulation $-\mathbf m$ at its center. The proof is based on a contour-dynamics formulation, a symmetry reduction to two nonlinear boundary equations, and a suitable singular rescaling. We then apply an implicit function theorem with a continuous parameter in symmetry-adapted Hölder spaces. To the best of our knowledge, this is the first analytical construction of a desingularization regime in which several concentrated inner components are contained in a common outer patch and collapse simultaneously toward its center.
