Global regularity for solutions of the three dimensional Navier–Stokes equation with almost two dimensional initial data
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Bibliographic record
Abstract
Abstract In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier–Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates between the global existence of smooth solutions for the two dimensional Navier–Stokes equation with arbitrarily large initial data, and the global existence of smooth solutions for the Navier–Stokes equation in three dimensions with small initial data in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mo>̇</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> </mml:math> . This result states that the closer the initial data is to being two dimensional, the larger the initial data can be in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mo>̇</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> </mml:math> while still guaranteeing the global existence of smooth solutions. In the whole space, this set of almost two dimensional initial data is unbounded in the critical space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mo>̇</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> </mml:math> , but is bounded in the critical Besov spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mo>̇</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msubsup> </mml:math> for all 2 < p ⩽ +∞. On the torus, however, this approach does give examples of arbitrarily large initial data in the endpoint Besov space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mo>̇</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> that generate global smooth solutions to the Navier–Stokes equation. In addition to these new results, we will also sharpen the constants in a number of previously known estimates for the growth of solutions to the Navier–Stokes equation and clarify the relationship between certain component reduction type regularity criteria.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| 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 it