{"title":"Localized Blow-Up Criterion for \\( C^{ 1, \\alpha } \\) Solutions to the 3D Incompressible Euler Equations","authors":"Dongho Chae, Jörg Wolf","doi":"10.1007/s00021-023-00813-8","DOIUrl":null,"url":null,"abstract":"<div><p>We prove a localized Beale–Kato–Majda type blow-up criterion for the 3D incompressible Euler equations in the Hölder space setting. More specifically, let <span>\\(v\\in C([0, T); C^{ 1, \\alpha } (\\Omega ))\\cap L^\\infty (0, T; L^2(\\Omega ))\\)</span> be a solution to the Euler equations in a domain <span>\\(\\Omega \\subset {\\mathbb {R}}^3\\)</span>. If there exists a ball <span>\\(B\\subset \\Omega \\)</span> such that <span>\\( \\int \\limits \\nolimits _{0}^T \\Vert \\omega (s)\\Vert _{ BMO(B )} ds < +\\infty , \\)</span> where <span>\\( \\omega = \\nabla \\times v\\)</span> stands for the vorticity, then <span>\\( v\\in C([0, T]; C^{ 1, \\alpha } (K)) \\)</span> for every compact subset <span>\\( K \\subset B \\)</span>. In the proof of this result, in order to handle the time evolution of the local Hölder norm of the vorticity we use the well-known Campanato space representation for the the Hölder space, and our argument relies on the Campanato space estimates for the solution to the corresponding transport equation.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00021-023-00813-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We prove a localized Beale–Kato–Majda type blow-up criterion for the 3D incompressible Euler equations in the Hölder space setting. More specifically, let \(v\in C([0, T); C^{ 1, \alpha } (\Omega ))\cap L^\infty (0, T; L^2(\Omega ))\) be a solution to the Euler equations in a domain \(\Omega \subset {\mathbb {R}}^3\). If there exists a ball \(B\subset \Omega \) such that \( \int \limits \nolimits _{0}^T \Vert \omega (s)\Vert _{ BMO(B )} ds < +\infty , \) where \( \omega = \nabla \times v\) stands for the vorticity, then \( v\in C([0, T]; C^{ 1, \alpha } (K)) \) for every compact subset \( K \subset B \). In the proof of this result, in order to handle the time evolution of the local Hölder norm of the vorticity we use the well-known Campanato space representation for the the Hölder space, and our argument relies on the Campanato space estimates for the solution to the corresponding transport equation.
期刊介绍:
The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.