A locally anisotropic regularity criterion for the Navier–Stokes equation in terms of vorticity
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Bibliographic record
Abstract
In this paper, we will prove a regularity criterion that guarantees solutions of the Navier–Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript t Superscript 4 Baseline upper L Subscript x Superscript 2"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">L^4_t L^2_x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier–Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe’s regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli’s regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier–Stokes equation must be fully three dimensional at all length scales.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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