Beltrami Fields with Nonconstant Proportionality Factor

Abstract

We consider the question raised by Enciso and Peralta-Salas in [4]: What nonconstant functions f can occur as the proportionality factor for a Beltrami field u on an open subset U ⊂ R 3 ? We also consider the related question: For any such f , how large is the space of associated Beltrami fields? By applying Cartan's method of moving frames and the theory of exterior differential systems, we are able to improve upon the results given in [4]. In particular, the answer to the second question depends crucially upon the geometry of the level surfaces of f . We conclude by giving a complete classification of Beltrami fields that possess either a translation symmetry or a rotation symmetry. This is clearly an important result; unfortunately, the operator P is extremely cumbersome to compute. Moreover, this necessary condition for f is almost certainly not sufficient; the proof of the theorem shows that there is, in fact, a hierarchy of differential constraints that

Linearisability of divergence-free fields along invariant 2-tori

“…Our results also provide some easy bounds on the size of the space of divergence-free Beltrami fields for a special class of proportionality factors in a similar vein to [17,27].…”

Rigidity of Beltrami fields with a non-constant proportionality factor

“…This rigidity follows by investigating compatibility of a constrained evolution equation equivalent to (1.2), see Remarks 2.1. The constraint in (1.2) is understood as compatibility for the constrained evolution equation [EPS16] and studied via an exterior differential system by using Cartan's method [CK20].…”

“…The proof of Theorem 1.2 is based on the facts that (1.2) can be recasted as a constrained evolution equation on a level set of f [EPS16], [CK20] and that Beltrami fields are solutions to the elliptic equation −∆u = ∇ f × u + f 2 u as explained below. Unfortunately, solutions of (1.1) with π const.…”

“…To prove this, without assuming symmetry of u, we use differential forms and rewrite (1.2) as a constrained evolution equation [EPS16], cf. [CK20],…”

“…Existence of solutions to (1.2) is unknown unless f ≡ const or under symmetry, cf. [CK20]. Axisymmetric solutions with compactly supported vorticity exist [Cha56], [Tur89], [Abe].…”