{"title":"Long Time Evolution of Concentrated Vortex Rings with Large Radius","authors":"Paolo Buttà, Guido Cavallaro, Carlo Marchioro","doi":"10.1007/s10955-024-03381-x","DOIUrl":null,"url":null,"abstract":"<div><p>We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on <i>N</i> annuli of radii of the order of <span>\\(r_0\\)</span> and thickness <span>\\(\\varepsilon \\)</span>. We prove that when <span>\\(r_0= |\\log \\varepsilon |^\\alpha \\)</span>, <span>\\(\\alpha >1\\)</span>, the vorticity field of the fluid converges for <span>\\(\\varepsilon \\rightarrow 0\\)</span> to the point vortex model, in an interval of time which diverges as <span>\\(\\log |\\log \\varepsilon |\\)</span>. This generalizes previous result by Cavallaro and Marchioro in (J Math Phys 62:053102, 2021), that assumed <span>\\(\\alpha >2\\)</span> and in which the convergence was proved for short times only.</p></div>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":"191 12","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10955-024-03381-x","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on N annuli of radii of the order of \(r_0\) and thickness \(\varepsilon \). We prove that when \(r_0= |\log \varepsilon |^\alpha \), \(\alpha >1\), the vorticity field of the fluid converges for \(\varepsilon \rightarrow 0\) to the point vortex model, in an interval of time which diverges as \(\log |\log \varepsilon |\). This generalizes previous result by Cavallaro and Marchioro in (J Math Phys 62:053102, 2021), that assumed \(\alpha >2\) and in which the convergence was proved for short times only.
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.