Bibliographic record
Abstract
We consider the question whether starting from a smooth initial condition 3D inviscid Euler flows on a periodic domain $\mathbb{T}^3$ may develop singularities in a finite time. Our point of departure is the well-known result by Kato (1972), which asserts the local existence of classical solutions to the Euler system in the Sobolev space $H^m(\mathbb{T}^3)$ for $m > 5/2$. Thus, potential formation of a singularity must be accompanied by an unbounded growth of the $H^m$ norm of the velocity field as the singularity time is approached. We perform a systematic search for "extreme" Euler flows that may realize such a scenario by formulating and solving a PDE-constrained optimization problem where the $H^3$ norm of the solution at a certain fixed time $T > 0$ is maximized with respect to the initial data subject to suitable normalization constraints. This problem is solved using a state-of-the-art Riemannian conjugate gradient method where the gradient is obtained from solutions of an adjoint system. Computations performed with increasing numerical resolutions demonstrate that, as asserted by the theorem of Kato (1972), when the optimization time window $[0, T]$ is sufficiently short, the $H^3$ norm remains bounded in the extreme flows found by solving the optimization problem, which indicates that the Euler system is well-posed on this "short" time interval. On the other hand, when the window $[0, T]$ is long, possibly longer than the time of the local existence asserted by Kato's theorem, then the $H^3$ norm of the extreme flows diverges upon resolution refinement, which indicates a possible singularity formulation on this "long" time interval. The extreme flow obtained on the long time window has the form of two colliding vortex rings and is characterized by certain symmetries. In particular, the region of the flow in which a singularity might occur is nearly axisymmetric.
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How this classification was reachedexpand
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) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".