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 Ω=[0,L1]×[0,L2]×[0, ] with periodic boundary conditions, where L1,L2>0 and (0,1/2). We prove that if where u0 is the initial datum, then there exists a unique global smooth solution with the initial datum u0. Also, if the global regularity of the corresponding solution holds.