García Claudia, Haziot Susanna V
Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain.
Research Unit "Modeling Nature" (MNat), Universidad de Granada, 18071 Granada, Spain.
Commun Math Phys. 2023;402(2):1167-1204. doi: 10.1007/s00220-023-04741-6. Epub 2023 Jun 9.
The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu (in Commun Math Phys 350(2):699-747, 2017) via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland to allow for the singularity at the bifurcation point. For both the corotating and the counter-rotating pairs, along the global curve of solutions either the angular fluid velocity vanishes or the two patches self-intersect.
Hmidi和Mateu（于2017年发表在《数学物理通讯》第350卷第2期，第699 - 747页）通过对一对点涡旋进行去奇异化，证明了共旋转和反向旋转涡旋对局部曲线的存在性。在本文中，我们构建了这些局部曲线的全局延拓。也就是说，我们考虑的解不仅仅是平凡解的微小扰动。实际上，虽然局部分析依赖于对平凡解处线性方程的研究，但全局分析需要对非线性问题的拓扑性质有更深入的理解。为了证明，我们采用了Buffoni和Toland提出的强大的解析全局分岔定理，以考虑分岔点处的奇异性。对于共旋转和反向旋转的涡旋对，沿着解的全局曲线，要么流体角速度消失，要么两个斑块自相交。
