A class of solutions of the Navier–Stokes equations with large data

We are concerned with global regularity of solutions in a periodic domain Q=[0,1]3Q=[0,1]3. We prove that, in the class of solutions oscillating in the vertical direction, the solutions are smooth under natural conditions on the horizontal derivatives of the horizontal components of the velocity, the derivative in the vertical direction and the vertical average of the initial data. The obtained conditions admit data whose BMO−1BMO−1-norm has algebraic dependence on 1/h1/h where h is the period of oscillation and generate global solutions. Also, the results allow non-zero force and large data which do not decay in time.