Multifractality and intermittency in the limit evolution of polygonal vortex filaments

IF 1.3 2区 数学 Q1 MATHEMATICS
Valeria Banica, Daniel Eceizabarrena, Andrea R. Nahmod, Luis Vega
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引用次数: 0

Abstract

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann’s non-differentiable functions

$$\begin{aligned} R_{x_0}(t) = \sum _{n \ne 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{aligned}$$

These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When \(x_0\) is rational, we show that \(R_{x_0}\) is multifractal and intermittent by completely determining the spectrum of singularities of \(R_{x_0}\) and computing the \(L^p\) norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that \(R_{x_0}\) has a multifractal behavior also when \(x_0\) is irrational. The proofs rely on a careful design of Diophantine sets that depend on \(x_0\), which we study by using the Duffin–Schaeffer theorem and the Mass Transference Principle.

Abstract Image

多边形涡旋丝极限演化中的多分形和间歇性
为了量化涡旋丝的湍流行为,我们研究了广义黎曼无差异函数$$\begin{aligned}族的多重性和间歇性。R_{x_0}(t) = \sum _{n \ne 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}.}{n^2}, \qquad x_0 \in [0,1].\end{aligned}$$这些函数在一定限度内代表了根据二正态流演化的规则多边形涡旋丝的轨迹。当 \(x_0\) 是有理数时,我们通过完全确定 \(R_{x_0}\) 的奇点谱并计算其傅里叶高通滤波器的 \(L^p\) 准则(它们是结构函数的类似物),证明 \(R_{x_0}\) 是多分形和间歇的。我们证明了当\(x_0\)是无理数时,\(R_{x_0}\)也具有多分形行为。证明依赖于对依赖于 \(x_0\) 的 Diophantine 集的精心设计,我们利用 Duffin-Schaeffer 定理和质量转移原理对其进行了研究。
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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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