Bounds on Kolmogorov spectra for the Navier - Stokes equations
Bibliographic record
Abstract
Let $u(x,t)$ be a (possibly weak) solution of the Navier - Stokes equations on all of ${\mathbb R}^3$, or on the torus ${\mathbb R}^3/ {\mathbb Z}^3$. The {\it energy spectrum} of $u(\cdot,t)$ is the spherical integral \[ E(κ,t) = \int_{|k| = κ} |\hat{u}(k,t)|^2 dS(k), \qquad 0 \leq κ< \infty, \] or alternatively, a suitable approximate sum. An argument involking scale invariance and dimensional analysis given by Kolmogorov (1941) and Obukhov (1941) predicts that large Reynolds number solutions of the Navier - Stokes equations in three dimensions should obey \[ E(κ, t) \sim C_0\varepsilon^{2/3}κ^{-5/3} \] over an inertial range $κ_1 \leq κ\leq κ_2$, at least in an average sense. We give a global estimate on weak solutions in the norm $\|{\mathcal F}\partial_x u(\cdot, t)\|_\infty$ which gives bounds on a solution's ability to satisfy the Kolmogorov law. A subsequent result is for rigorous upper and lower bounds on the inertial range, and an upper bound on the time of validity of the Kolmogorov spectral regime.
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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 | 0.004 | 0.026 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.004 |
| 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".