Travelling helices and the vortex filament conjecture in the incompressible Euler equations
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".