Existence of smooth non-Killing admissible infinitesimal generators

Establish the existence of smooth divergence-free vector fields u on bounded spatial domains Q ⊂ ℝ³ whose strain-rate tensor S = (∇u + (∇u)ᵗ)/2 is nowhere zero (non-Killing) and that are admissible, meaning there exists a nowhere-vanishing divergence-free magnetic field B such that ∇×(u×B) = 0, (∇×B)×u + ∇(u·B) = 0, and ∇·B = 0, so that u serves as the infinitesimal generator of a quasisymmetric magnetic field.

Background

The paper defines a quasisymmetric magnetic field B on a bounded domain Q ⊂ ℝ³ as one admitting a divergence-free infinitesimal generator u that satisfies a system of equations ensuring symmetry of guiding-center dynamics. A vector field u is called admissible when there exists such a quasisymmetric B that has u as its infinitesimal generator.

The authors derive necessary PDE constraints on u that do not involve B and show these constraints are typically locally sufficient; they then provide a global sufficiency framework under additional topological conditions on toroidal annuli. However, they only solve the construction half of the problem (building B from a given admissible u) and explicitly leave open the existence of smooth, non-Killing admissible u themselves.

Resolving this existence problem would demonstrate the presence of genuinely three-dimensional quasisymmetric magnetic fields beyond Euclidean rigid motions, addressing a central question in stellarator theory raised by prior no-go results under force balance.