On the regularity of the Navier-Stokes equation in a thin periodic domain

We address the global regularity Of Solutions of the Navier-Stokes equations in a thin domain Omega = [0, L-1] x [0, L-2] x [0, is an element of] with periodic boundary conditions, where L-1, L-2 > 0 and is an element of is an element of (0, 1/2). We prove that if vertical bar vertical bar del u(0)vertical bar vertical bar(L2(Omega)) <= (nu)/(1/2)(3/2)(C(L1, L2)is an element of)(vertical bar log is an element of vertical bar) where mu(0) is the initial datum, then there exists a unique global smooth solution with the initial datum mu(0). Also, if vertical bar vertical bar u(0)vertical bar vertical bar (H)1/2((Omega)) <= (nu)/(1/4)(C(L1, L2)vertical bar log is an element of vertical bar) the global regularity of the corresponding solution holds. (c) 2006 Elsevier Inc. All rights reserved.