二维欧拉方程环形涡斑周围的不变 KAM Tori

IF 2.2 1区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL
Zineb Hassainia, Taoufik Hmidi, Emeric Roulley
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引用次数: 0

摘要

我们为平面欧拉方程构建了带一个洞的时间准周期涡斑解。只要这些结构的模量属于大质量伯尔集合,它们就能被捕捉到。该证明基于纳什-莫泽方案和 KAM 理论,并应用于控制补片径向变形的哈密顿系统。与最近在 Hassainia 等人(主动标量方程的 KAM 理论,arXiv:2110.08615)、Hassainia 和 Roulley(欧拉方程准周期解存在的边界效应,arXiv:2202.10053), Hmidi and Roulley (Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations, arXiv:2110.13751) and Roulley (Dyn Partial Differ Equ 20(4):311-366, 2023),由于界面之间的相互作用,出现了一些技术问题。其中一个问题与二阶梅尔尼科夫非共振条件中的新小除数问题有关,该条件来自以不同速度平流的输运方程。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Invariant KAM Tori Around Annular Vortex Patches for 2D Euler Equations

We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash–Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case discussed recently in Hassainia et al. (KAM theory for active scalar equations, arXiv:2110.08615), Hassainia and Roulley (Boundary effects on the existence of quasi-periodic solutions for Euler equations, arXiv:2202.10053), Hmidi and Roulley (Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations, arXiv:2110.13751) and Roulley (Dyn Partial Differ Equ 20(4):311–366, 2023), some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations advected with different velocities.

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来源期刊
Communications in Mathematical Physics
Communications in Mathematical Physics 物理-物理:数学物理
CiteScore
4.70
自引率
8.30%
发文量
226
审稿时长
3-6 weeks
期刊介绍: The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.
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