Travelling helices and the vortex filament conjecture in the incompressible Euler equations
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Bibliographic record
Abstract
Abstract We consider the Euler equations in $$\mathbb R^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math> expressed in vorticity form $$\begin{aligned} \left\{ \begin{array}{l} \vec \omega _t + (\mathbf{u}\cdot {\nabla } ){\vec \omega } =( \vec \omega \cdot {\nabla } ) \mathbf{u} \\ \mathbf{u} = \mathrm{curl}\vec \psi ,\ -\Delta \vec \psi = \vec \omega . \end{array}\right. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfenced><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>·</mml:mo><mml:mi>∇</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:mi>∇</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>curl</mml:mi><mml:mover><mml:mi>ψ</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mspace/><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mover><mml:mi>ψ</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math> A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of helical filaments , associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple polygonal helical filaments travelling and rotating together.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it