Non-uniqueness of the Leray-Hopf solutions in the hyperbolic setting
Bibliographic record
Abstract
The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on R 2 .We show the uniqueness of the Leray-Hopf solutions breaks down on H 2 (-a 2 ), the two dimensional hyperbolic space with constant sectional curvature -a 2 .We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation.Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions.Contents 1. Introduction 43 Acknowledgements 54 2. Preliminaries 54 3. Exponential decay of the gradient of bounded harmonic functions 62 4. The proof that ∇ |∇F | 2 L 1 (H 2 (-a 2 )) is finite 64 5. Finite Dissipation 69 6. Proofs of the main results 71 References 75
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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".