Abstract
We consider the existence of Beltrami fields with a nonconstant proportionality factor f in an open subset U of \({\mathbb{R}^3}\). By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that f must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the C k topology, \({k\geqq 7}\). In particular, there are no nontrivial Beltrami fields whenever f has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis et al. (Commun Pure Appl Math 48:571–582, 1995).
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Communicated by V. Šverák
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Enciso, A., Peralta-Salas, D. Beltrami Fields with a Nonconstant Proportionality Factor are Rare. Arch Rational Mech Anal 220, 243–260 (2016). https://doi.org/10.1007/s00205-015-0931-5
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DOI: https://doi.org/10.1007/s00205-015-0931-5