Navier-Stokes equations with regularity in one direction

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 partial derivative u/partial derivative x(3) belongs to the space (Lt0Lx0r)-L-s, where 2/s(0)+3/r(0)<= 2 and 9/4 <= r(0)<= 3, then the solution is regular. This extends a result of Beirao 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 partial derivative u/partial derivative x(3) can be substituted with any directional derivative of u. (c) 2007 American Institute of Physics.