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Existence of Vortex Rings in Beltrami Flows

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Abstract

Axisymmetric vortex rings are traveling wave solutions to the 3d Euler equations, first constructed by Fraenkel for the case without swirl via the variational principle. In this paper, we consider axisymmetric vortex rings with swirl consisting of Beltrami fields with a non-constant proportionality factor. They provide first examples to \(C^{1}\)-traveling wave solutions, axisymmetric with swirl.

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Acknowledgements

The property (1.4) and the references [16, 18, 31] were informed by Professor Daniel Peralta-Salas. The reference [30] was informed by Professor Yasuhide Fukumoto. The introduction and the proof of Proposition 4.8 were improved by suggestions of the referee. The author is grateful to Professors Daniel Peralta-Salas, Yasuhide Fukumoto and referees for their helpful comments and suggestions. This work is partially supported by JSPS through the Grant-in-aid for Young Scientist 20K14347, Scientific Research (B) 17H02853 and MEXT Promotion of Distinctive Joint Research Center Program Grant Number JPMXP0619217849.

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  1. Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Ken Abe

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Correspondence to Ken Abe.

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Communicated by A. Ionescu.

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Abe, K. Existence of Vortex Rings in Beltrami Flows. Commun. Math. Phys. 391, 873–899 (2022). https://doi.org/10.1007/s00220-022-04331-y

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