On the size of the Navier - Stokes singular set
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Bibliographic record
Abstract
A beautiful and influential subject in the study of thequestion of smoothness of solutions for the Navier - Stokesequations in three dimensions is the theory of partial regularity.A major paper on this topic is Caffarelli, Kohn & Nirenberg [5](1982)which gives an upper bound on the size of the singularset $S(u)$ of a suitable weak solution $u$. In the present paperwe describe a complementary lower bound. More precisely, westudy the situation in which a weak solution fails to becontinuous in the strong $L^2$ topology at some singulartime $t=T$. We identify a closed set in space onwhich the $L^2$ norm concentrates at this time $T$, andwe study microlocal properties of the Fourier transformof the solution in the cotangent bundle T * (R 3)above this set. Our main resultis that $L^2$ concentration can only occur on subsets ofT * (R 3) which are sufficiently large. Anelement of the proof is a new global estimate on weak solutionsof the Navier - Stokes equations which have sufficientlysmooth initial data.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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