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arXiv · 2607.16141

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

Daniel Peralta-Salas·Jie Wan
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Abstract

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.

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Daniel Peralta-Salas, Jie Wan. 2026-07-17. Piecewise smooth stationary Euler flows with support in a neighborhood of a helix. https://arxiv.org/abs/2607.16141

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