Localized Blow-Up Criterion for \( C^{ 1, \alpha } \) Solutions to the 3D Incompressible Euler Equations

IF 1.2 3区 数学 Q2 MATHEMATICS, APPLIED
Dongho Chae, Jörg Wolf
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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.

三维不可压缩欧拉方程\( C^{ 1, \alpha } \)解的局部爆破判据
我们证明了Hölder空间条件下三维不可压缩欧拉方程的一个局域Beale-Kato-Majda型爆破判据。更具体地说,设\(v\in C([0, T); C^{ 1, \alpha } (\Omega ))\cap L^\infty (0, T; L^2(\Omega ))\)为域\(\Omega \subset {\mathbb {R}}^3\)中欧拉方程的解。如果存在一个球\(B\subset \Omega \),使得\( \int \limits \nolimits _{0}^T \Vert \omega (s)\Vert _{ BMO(B )} ds < +\infty , \),其中\( \omega = \nabla \times v\)代表涡度,则\( v\in C([0, T]; C^{ 1, \alpha } (K)) \)对于每个紧化子集\( K \subset B \)。在证明这一结果时,为了处理涡度的局部Hölder范数的时间演化,我们对Hölder空间使用了著名的Campanato空间表示,并且我们的论证依赖于Campanato空间对相应输运方程解的估计。
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来源期刊
CiteScore
2.00
自引率
15.40%
发文量
97
审稿时长
>12 weeks
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.
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