Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity

Vasseur, Alexis. "Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity." Applications of Mathematics 54.1 (2009): 47-52. <http://eudml.org/doc/37806>.

@article{Vasseur2009,
abstract = {In this short note we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation and the behavior of the direction of the velocity $u/|u|$. It is shown that the control of $\{\rm Div\}(u/|u|)$ in a suitable $L_t^p(L_x^q)$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.},
author = {Vasseur, Alexis},
journal = {Applications of Mathematics},
keywords = {Navier-Stokes; fluid mechanics; regularity; PRodi-Serrin criteria; Navier-Stokes; fluid mechanics; regularity},
language = {eng},
number = {1},
pages = {47-52},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity},
url = {http://eudml.org/doc/37806},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Vasseur, Alexis
TI - Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 47
EP - 52
AB - In this short note we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation and the behavior of the direction of the velocity $u/|u|$. It is shown that the control of ${\rm Div}(u/|u|)$ in a suitable $L_t^p(L_x^q)$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.
LA - eng
KW - Navier-Stokes; fluid mechanics; regularity; PRodi-Serrin criteria; Navier-Stokes; fluid mechanics; regularity
UR - http://eudml.org/doc/37806
ER -