The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations

We consider the periodic 3D Navier-Stokes equations and we take the initial data of the form u(0) = v(0) + w(0), where v(0) does not depend on the third variable. We prove that, in order to obtain global existence and uniqueness, it suffices to assume that \\w(0)\\x exp(\\v(0)\\(2)(L2(pi2))/Cv(2)) less than or equal to Cv, where X is a space with a regularity H-delta in the first two directions and H (1/2-delta) in the third direction or, if delta = 0, a space which is L-2 in the first two directions and B-2,1(1/2) in the third direction. We also consider the same equations on the torus with the thickness in the third direction equal to epsilon and we study the dependence on epsilon of the constant C above. We show that if v(0) is the projection of the initial data on the space of functions independent of the third variable, then the constant C can be chosen independent of epsilon.