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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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 | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".