  We consider the initial problem for the Navier-Stokes equations over
${\mathbb R}^3 \times [0,T]$ with a positive time $T$ over specially
constructed scale of function spaces of Bochner-Sobolev type. We prove that the
problem induces an open both injective and surjective mapping of each space of
the scale. In particular, intersection of these classes gives a uniqueness and
existence theorem for smooth solutions to the Navier-Stokes equations for
smooth data with a prescribed asymptotic behaviour at the infinity with respect
to the time and the space variables.
