{"title":"Quasi-periodic Solutions for the Navier–Stokes Equation on Irrational Tori","authors":"Shangling Song","doi":"10.1007/s00009-024-02727-9","DOIUrl":null,"url":null,"abstract":"<p>This paper is concerned with the existence and stability of quasi-periodic solutions for the Navier–Stokes equation on irrational tori. We prove the persistence of quasi-periodic invariant tori under the small quasi-periodic external force. Furthermore, we obtain the orbital and asymptotic stability in the appropriate Sobolev space. Using the norm estimate of the heat propagator <span>\\(\\textrm{e}^{t\\Delta _{\\alpha }}\\)</span>, we get the solutions convergence exponentially to the invariant tori as <span>\\(t\\rightarrow \\infty \\)</span>, with the rate of convergence <span>\\(O(\\textrm{e}^{-\\rho t})\\)</span> for any <span>\\(\\rho \\in (0,1)\\)</span>.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02727-9","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
This paper is concerned with the existence and stability of quasi-periodic solutions for the Navier–Stokes equation on irrational tori. We prove the persistence of quasi-periodic invariant tori under the small quasi-periodic external force. Furthermore, we obtain the orbital and asymptotic stability in the appropriate Sobolev space. Using the norm estimate of the heat propagator \(\textrm{e}^{t\Delta _{\alpha }}\), we get the solutions convergence exponentially to the invariant tori as \(t\rightarrow \infty \), with the rate of convergence \(O(\textrm{e}^{-\rho t})\) for any \(\rho \in (0,1)\).
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.