{"title":"二维欧拉方程环形涡斑周围的不变 KAM Tori","authors":"Zineb Hassainia, Taoufik Hmidi, Emeric Roulley","doi":"10.1007/s00220-024-05141-0","DOIUrl":null,"url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"405 11","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Invariant KAM Tori Around Annular Vortex Patches for 2D Euler Equations\",\"authors\":\"Zineb Hassainia, Taoufik Hmidi, Emeric Roulley\",\"doi\":\"10.1007/s00220-024-05141-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>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.</p></div>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":\"405 11\",\"pages\":\"\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-10-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00220-024-05141-0\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-024-05141-0","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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。
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.
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.