We address global regularity of solutions of the Navier--Stokes equations in a periodic domain $Q=[0,1]^3$. We prove that, in the class of solutions oscillating in the vertical direction, the global solutions are smooth under natural conditions on the derivatives in the horizontal direction and the vertical and horizontal averages of the initial data. There are no restrictions on the size of the data.
"Solutions of the Navier-Stokes equations for large oscillatory data." Adv. Differential Equations 18 (5/6) 549 - 586, May/June 2013.