Local estimates of Hölder exponents in turbulent vector fields.
Local estimates of Hölder exponents in turbulent vector fields.
复制标题
湍流矢量场中 Hölder 指数的局部估计。
DOI：
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发表时间：
2019
影响因子：
2.4
通讯作者：
Bérengère Dubrulle
中科院分区：
文献类型：
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作者：
Florian Nguyen;J. Laval;Pierre Kestener;A. Cheskidov;R. Shvydkoy;Bérengère Dubrulle
目前尚不清楚Navier-Stokes方程的解是否能从规则的初始条件发展为奇点。特别地，一个经典的尚未解决的问题是证明速度场在小尺度上是Hölder连续的，指数为h<；1(即不一定是可微的)。人们已经提出了不同的方法来研究速度场的规律性及其Hölder指数h的估计。第一种方法是通过与粘度无关的惯性耗散D*=Lim_{ℓ→0}D_{ℓ}^{i}的极值来检测势奇性[Duchon and Robert，非线性13249(2000)0951-771510.1088/0951-7715/13/1/312]。另一种可能性是使用多重分形分析的概念，该概念提供指数h的子空间的分维。然而，多重分形分析是一种全局统计方法，它仅通过其出现的概率来提供关于局部Hölder指数的全局信息。为了探索速度场的局部正则性，我们发展了一个局部统计分析，它估计了局部Hölder连续性。我们将我们的分析结果与惯性能量耗散D_{ℓ}^{i}的结果进行了比较。我们观察到，根据我们的估计，对于不那么规则的速度场，耗散项确实变得更大。然而，局域Hölder指数的精确空间分布表现出非平凡的行为，与惯性耗散的分布并不完全匹配。
It is still not known whether solutions to the Navier-Stokes equation can develop singularities from regular initial conditions. In particular, a classical and unsolved problem is to prove that the velocity field is Hölder continuous with some exponent h<1 (i.e., not necessarily differentiable) at small scales. Different methods have already been proposed to explore the regularity properties of the velocity field and the estimate of its Hölder exponent h. A first method is to detect potential singularities via extrema of an "inertial" dissipation D*=lim_{ℓ→0}D_{ℓ}^{I} that is independent of viscosity [Duchon and Robert, Nonlinearity 13, 249 (2000)0951-771510.1088/0951-7715/13/1/312]. Another possibility is to use the concept of multifractal analysis that provides fractal dimensions of the subspace of exponents h. However, the multifractal analysis is a global statistical method that only provides global information about local Hölder exponents, via their probability of occurrence. In order to explore the local regularity properties of a velocity field, we have developed a local statistical analysis that estimates locally the Hölder continuity. We have compared outcomes of our analysis with results using the inertial energy dissipation D_{ℓ}^{I}. We observe that the dissipation term indeed gets bigger for velocity fields that are less regular according to our estimates. The exact spatial distribution of the local Hölder exponents however shows nontrivial behavior and does not exactly match the distribution of the inertial dissipation.