We prove the existence of time-periodic solutions of the two-dimensional incompressible Euler equations bifurcating from a translating vortex pair. The reference configuration consists of two symmetric vortex patches of equal strength and opposite sign traveling at constant speed. In a regime of large separation between the vortices, the dynamics may be viewed as a small perturbation of an integrable system. Working in a co-moving frame and using the contour dynamics formulation, we reduce the problem to a nonlinear transport equation for the vortex boundaries. The linearized operator exhibits degeneracies associated with symmetries and transport effects. By combining a Lyapunov-Schmidt reduction, Nash-Moser scheme, and spectral analysis with sharp asymptotic expansions of the eigenvalues in order to overcome the degeneracy, we construct families of non-rigid, time-periodic vortex patch solutions for a large Cantor set of parameters. The analysis reveals that translating dipoles possess a surprisingly rich nearby dynamics, far beyond the classical rigid paradigm usually associated with vortex patch motion. More generally, the approach developed in this work is flexible and robust, and is expected to extend to a broader class of nonlocal PDEs from Fluid Mechanics with degenerate behaviours.
Existence and stability of Sadovskii vortices: from patch to smooth vortices
Osaka Metropolitan University（大阪公立大学）；Ulsan National Institute of Science and Technology（蔚山科学技术院）；Korea Institute for Advanced Study（韩国高等科学研究院）；Universität Basel（巴塞尔大学）
Interacting vortex pairs in inviscid and viscous planar flows
这一贡献的目的是使最近的两个结果之间的联系有关的不可压缩平面流动中的旋涡动力学。第一种是以点涡为初值的二维Navier-斯托克斯方程解在粘性极限为零时的渐近展开。在这种情况下，已知（Gallay，2011）解表现为前导阶，就像Oseen涡的线性叠加，其中心根据点涡系统演变，但也可以计算高阶修正，其描述了由于相互作用引起的涡核变形。第二个结果是D. Smets和J. van Schaftingen的二维欧拉方程的“去奇异化”解。这些解在匀速旋转或平移坐标系中是静止的，并且当尺寸参数$ε$趋于零时收敛到单个涡或涡对。我们认为在这里的一对相同的旋涡的特殊情况下，我们表明，弱粘性的Navier-斯托克斯方程的解决方案是准确地描述在时间$t$的近似稳定状态的旋转欧拉方程，这是一个desingularized解决方案的意义上的Smets和范Schaftingen，高斯的轮廓和大小$ε= \sqrt{νt}$。
