Valeria Banica, Daniel Eceizabarrena, Andrea R. Nahmod, Luis Vega
{"title":"Multifractality and intermittency in the limit evolution of polygonal vortex filaments","authors":"Valeria Banica, Daniel Eceizabarrena, Andrea R. Nahmod, Luis Vega","doi":"10.1007/s00208-024-02971-0","DOIUrl":null,"url":null,"abstract":"<p>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 </p><span>$$\\begin{aligned} R_{x_0}(t) = \\sum _{n \\ne 0} \\frac{e^{2\\pi i ( n^2 t + n x_0 ) } }{n^2}, \\qquad x_0 \\in [0,1]. \\end{aligned}$$</span><p>These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When <span>\\(x_0\\)</span> is rational, we show that <span>\\(R_{x_0}\\)</span> is multifractal and intermittent by completely determining the spectrum of singularities of <span>\\(R_{x_0}\\)</span> and computing the <span>\\(L^p\\)</span> norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that <span>\\(R_{x_0}\\)</span> has a multifractal behavior also when <span>\\(x_0\\)</span> is irrational. The proofs rely on a careful design of Diophantine sets that depend on <span>\\(x_0\\)</span>, which we study by using the Duffin–Schaeffer theorem and the Mass Transference Principle.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Annalen","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00208-024-02971-0","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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.