Beltrami Fields with a Nonconstant Proportionality Factor are Rare

By Alberto Enciso, D. Peralta‐Salas. 2014

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We consider the existence of Beltrami fields with a nonconstant proportionality factor f in an open subset U of $${\mathbb{R}^3}$$R3. 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 Ck topology, $${k\geqq 7}$$k≧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).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ac833155706418dc50c6f841b844d09e954966b9