Piecewise smooth stationary Euler flows with support in a neighborhood of a helix

We construct stationary solutions of the three-dimensional incompressible Euler equations with helical symmetry and support in a neighborhood of a helix.

The solutions are piecewise smooth and arise from a nonlinear overdetermined elliptic boundary value problem associated with a stream-function formulation.

A distinguishing feature is that the vortex cross-sections are intrinsically anisotropic: after rescaling, the leading-order shape is elliptic rather than radial, and the boundary exhibits a nontrivial third Fourier mode reflecting helical effects absent in previous axisymmetric constructions.

A key step in the proof is the analysis of a genuinely anisotropic overdetermined elliptic problem with prescribed Dirichlet and nonconstant Neumann conditions.