Multifractality and polygonal vortex filaments
[Multifractalité et des tourbillons filamentaires polygonaux]
Banica, Valeria1 ; Eceizabarrena, Daniel2 ; Nahmod, Andrea R.3 ; Vega, Luis4
1Laboratoire Jacques-Louis Lions Sorbonne Université 4 place Jussieu 75005 Paris France
2BCAM - Basque Center for Applied Mathematics Alameda de Mazarredo 14 48009 Bilbao Spain
3Department of Mathematics and Statistics University of Massachusetts Amherst 710 N Pleasant St, Amherst, MA 01003, USA
4Department of Mathematics University of Basque Country Apdo 644, 48080 Bilbao Spain
Journées équations aux dérivées partielles (2024), Exposé no. 1, 13 p.

In this proceedings article we survey the results in [5] and their motivation, as presented at the 50th Journées Équations aux dérivées partielles 2024. With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality of a family of generalized Riemann’s non-differentiable functions. These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow, the classical model for vortex filaments dynamics. We explain how we determined their spectrum of singularities through a careful design of Diophantine sets, which we study by using the Duffin–Schaeffer theorem and the Mass Transference Principle.

Dans cet acte de conférence nous passons en revue les résultats de [5] et leur motivation, tels qu’ils ont été présentés au 50e Journées Équations aux dérivées partielles 2024. Dans le but de quantifier les comportements turbulents des filaments tourbillonnaires, nous étudions la multifractalité d’une famille de fonctions non différentiables de Riemann généralisées. Ces fonctions représentent, dans une certaine limite, la trajectoire de filaments tourbillonaires polygonaux réguliers qui évoluent selon le flot binormal, le modèle classique pour la dynamique des tourbillons filamentaires. Nous expliquons comment nous avons déterminé pour certaines de ces fonctions le spectre des singularités. La preuve repose sur une construction d’ensembles diophantiens que nous étudions en utilisant le théorème de Duffin–Schaeffer et le principe de transfert de masse.

Publié le :
DOI : 10.5802/jedp.682
Classification : 11J82, 11J83, 26A27, 28A78, 42A16, 76F99
Keywords: Vortex filaments, multifractality, Riemann’s non-differentiable function, Diophantine approximation.

Banica, Valeria  1   ; Eceizabarrena, Daniel  2   ; Nahmod, Andrea R.  3   ; Vega, Luis  4

1 Laboratoire Jacques-Louis Lions Sorbonne Université 4 place Jussieu 75005 Paris France
2 BCAM - Basque Center for Applied Mathematics Alameda de Mazarredo 14 48009 Bilbao Spain
3 Department of Mathematics and Statistics University of Massachusetts Amherst 710 N Pleasant St, Amherst, MA 01003, USA
4 Department of Mathematics University of Basque Country Apdo 644, 48080 Bilbao Spain 
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Banica, Valeria; Eceizabarrena, Daniel; Nahmod, Andrea R.; Vega, Luis. Multifractality and polygonal vortex filaments. Journées équations aux dérivées partielles (2024), Exposé no. 1, 13 p.. doi: 10.5802/jedp.682

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