Navier-Stokes equations with regularity in one direction

Abstract

We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity u x3 belongs to the space Lt s0 Lx r0, where 2 s0 +3 r0 ≤2 and 94≤ r0 ≤3, then the solution is regular. This extends a result of Beirão da Veiga [Chin. Ann. Math., Ser. B 16, 407-412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405-408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative u x3 can be substituted with any directional derivative of u. © 2007 American Institute of Physics.