Geodesic
Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity
Applications of Mathematics, Tome 54 (2009) no. 1, pp. 47-52

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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.
DOI : 10.1007/s10492-009-0003-y
Classification : 35B65, 35Q30, 76D03, 76D05
Keywords: Navier-Stokes; fluid mechanics; regularity; PRodi-Serrin criteria 
@article{10_1007_s10492_009_0003_y,
     author = {Vasseur, Alexis},
     title = {Regularity criterion for {3D} {Navier-Stokes} equations in terms of the direction of the velocity},
     journal = {Applications of Mathematics},
     pages = {47--52},
     publisher = {mathdoc},
     volume = {54},
     number = {1},
     year = {2009},
     doi = {10.1007/s10492-009-0003-y},
     mrnumber = {2476020},
     zbl = {1212.35354},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0003-y/}
}
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
SP  - 47
EP  - 52
VL  - 54
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0003-y/
DO  - 10.1007/s10492-009-0003-y
LA  - en
ID  - 10_1007_s10492_009_0003_y
ER  - 
%0 Journal Article
%A Vasseur, Alexis
%T Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity
%J Applications of Mathematics
%D 2009
%P 47-52
%V 54
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0003-y/
%R 10.1007/s10492-009-0003-y
%G en
%F 10_1007_s10492_009_0003_y
Vasseur, Alexis. Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity. Applications of Mathematics, Tome 54 (2009) no. 1, pp. 47-52. doi: 10.1007/s10492-009-0003-y

Cité par Sources :