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Proof of Taylor’s Conjecture on Magnetic Helicity Conservation

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Abstract

We prove the mathematical version of Taylor’s conjecture which says that in 3D MHD, magnetic helicity is conserved in the ideal limit in bounded, simply connected, perfectly conducting domains. When the domain is multiply connected, magnetic helicity depends on the vector potential of the magnetic field. In that setting we show that magnetic helicity is conserved for a large and natural class of vector potentials but not in general for all vector potentials. As an analogue of Taylor’s conjecture in 2D, we show that mean square magnetic potential is conserved in the ideal limit, even in multiply connected domains.

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Acknowledgements

We thank Hussein Aluie for making for making us aware of his very interesting related work on magnetohydrodynamic turbulence.We also thank the anonymous referee for making us aware of the related work done in [13,16].

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Daniel Faraco

  2. ICMAT CSIC-UAM-UC3M-UCM, 28049, Madrid, Spain

    Daniel Faraco

  3. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsingin Yliopisto, Finland

    Sauli Lindberg

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  1. Daniel Faraco
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Correspondence to Daniel Faraco.

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Communicated by C. De Lellis.

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Daniel Faraco was partially supported by ICMAT Severo Ochoa Projects SEV-2011-0087 and SEV-2015-556, the Grants MTM2014-57769-P-1 and MTM2017-85934-P-1 (Spain) and the ERC Grant 307179-GFTIPFD. Sauli Lindberg was supported by the ERC Grant 307179-GFTIPFD and by the AtMath Collaboration at the University of Helsinki.

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Faraco, D., Lindberg, S. Proof of Taylor’s Conjecture on Magnetic Helicity Conservation. Commun. Math. Phys. 373, 707–738 (2020). https://doi.org/10.1007/s00220-019-03422-7

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