Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2309.08114 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915384257413120 |
|---|---|
| author | Banica, Valeria Eceizabarrena, Daniel Nahmod, Andrea R. Vega, Luis |
| author_facet | Banica, Valeria Eceizabarrena, Daniel Nahmod, Andrea R. Vega, Luis |
| contents | 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{equation} R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2πi ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{equation} 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 crucially using the Duffin-Schaeffer theorem and the Mass Transference Principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08114 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multifractality and intermittency in the limit evolution of polygonal vortex filaments Banica, Valeria Eceizabarrena, Daniel Nahmod, Andrea R. Vega, Luis Analysis of PDEs Mathematical Physics Classical Analysis and ODEs 11J82, 11J83, 26A27, 28A78, 42A16, 76F99 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{equation} R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2πi ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{equation} 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 crucially using the Duffin-Schaeffer theorem and the Mass Transference Principle. |
| title | Multifractality and intermittency in the limit evolution of polygonal vortex filaments |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs 11J82, 11J83, 26A27, 28A78, 42A16, 76F99 |
| url | https://arxiv.org/abs/2309.08114 |