We show that if velocity and vorticity are orthogonal at each point (and they become orthogonal fast enough) then solutions of the 3D Navier–Stokes equations are smooth. This condition implies that the helicity is identically zero and, in a certain sense, the flow resembles the 2D geometric situation.
Nous démontrons que si la vélocité et le rotationnel sont perpendiculaires partout (avec une borne sur la vitesse avec laquelle ils deviennent perpendiculaires) alors les solutions des équations de Navier–Stokes 3D sont régulières. Cette condition implique que l'élicité est nulle et, dans un certain sens, que le flux ressemble à la situation géométrique bidimensionnelle.
Published online:
@article{CRMATH_2009__347_11-12_613_0,
author = {Berselli, Luigi C. and C\'ordoba, Diego},
title = {On the regularity of the solutions to the {3D} {Navier{\textendash}Stokes} equations: a remark on the role of the helicity},
journal = {Comptes Rendus. Math\'ematique},
pages = {613--618},
publisher = {Elsevier},
volume = {347},
number = {11-12},
year = {2009},
doi = {10.1016/j.crma.2009.03.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.03.003/}
}
TY - JOUR AU - Berselli, Luigi C. AU - Córdoba, Diego TI - On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity JO - Comptes Rendus. Mathématique PY - 2009 SP - 613 EP - 618 VL - 347 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.03.003/ DO - 10.1016/j.crma.2009.03.003 LA - en ID - CRMATH_2009__347_11-12_613_0 ER -
%0 Journal Article %A Berselli, Luigi C. %A Córdoba, Diego %T On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity %J Comptes Rendus. Mathématique %D 2009 %P 613-618 %V 347 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.03.003/ %R 10.1016/j.crma.2009.03.003 %G en %F CRMATH_2009__347_11-12_613_0
Berselli, Luigi C.; Córdoba, Diego. On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 613-618. doi : 10.1016/j.crma.2009.03.003. http://www.numdam.org/articles/10.1016/j.crma.2009.03.003/
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