Smooth solutions of the Navier-Stokes equations

We consider smooth solutions of the Cauchy problem for the Navier-Stokes equations on the scale of smooth functions which are periodic with respect to $x\in\mathbb R^3$.
We obtain existence theorems for global (with respect to $t>0$) and local solutions of the Cauchy problem. The statements of these depend on the smoothness and the norm of the initial vector function. Upper bounds for the behaviour of solutions in both classes, which depend on $t$, are also obtained.
Bibliography: 10 titles.