论文标题
时间级环境涡流贴片的欧拉方程
Time quasi-periodic vortex patches of Euler equation in the plane
论文作者
论文摘要
我们证明了2 $ d $ - 欧拉尔方程的时间的时间，以$ \ mathbb {r}^2 $中的2 $ d $ - euler方程的存在，接近均匀旋转的kirchhoff椭圆形涡旋，其长宽比属于一组非额外的全勒贝格（Lebesgue）。这个问题被重新构成准线性的哈密顿方程，以从椭圆形径向位移。 KAM证明的主要困难是存在零正常模式频率，这是由于角动量的保护。克服这种退化的主要新颖性是对角动量进行扰动的互合型降低，将其作为符号校友定理精神的符号变量引入符号变量，在有限尺寸中有效。这种方法在无限的尺寸相空间中特别精致：我们的变量的符号变化是对角动量本身产生的传输流的非线性修饰。这是PDE的KAM首次实现这样的想法。其他困难是缺乏方程式的旋转对称性和双曲/椭圆形正常模式的存在。在其他涡流贴片问题中，后者的困难以及正常频率的堕落性不存在，这些问题最近使用了本文中引入的配方进行了研究。
We prove the existence of time quasi-periodic vortex patch solutions of the 2$d$-Euler equations in $\mathbb{R}^2$, close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Other difficulties are the lack of rotational symmetry of the equation and the presence of hyperbolic/elliptic normal modes. The latter difficulties -- as well as the degeneracy of a normal frequency -- are absent in other vortex patches problems which have been recently studied using the formulation introduced in this paper.
