Blow-up for the 3-dimensional axially symmetric harmonic map flow into <inline-formula><tex-math id="M1">$ S^2 $</tex-math></inline-formula>
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Bibliographic record
Abstract
We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $ S^2 $, \begin{document}$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $\end{document} with $ u(x,t): \bar \Omega\times [0,T) \to S^2 $. Here $ \Omega $ is a bounded, smooth axially symmetric domain in $ \mathbb{R}^3 $. We prove that for any circle $ \Gamma \subset \Omega $ with the same axial symmetry, and any sufficiently small $ T>0 $ there exist initial and boundary conditions such that $ u(x,t) $ blows-up exactly at time $ T $ and precisely on the curve $ \Gamma $, in fact \begin{document}$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $\end{document} for a regular function $ u_*(x) $, where $ \delta_\Gamma $ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5,6].
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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