  The initial problem for the Navier-Stokes type equations over ${\mathbb R}^n
\times [0,T]$, $n\geq 2$, with a positive time $T$ in the spatially periodic
setting is considered. First, we prove that the problem induces an open
injective continuous mapping on scales of specially constructed function spaces
of Bo\-chner-Sobolev type over the $n\,$-dimensional torus ${\mathbb T}^n$.
Next, rejecting the idea of proving a universal a priori estimate for
high-order derivatives, we obtain a surjectivity criterion for the non-linear
mapping under the considerations in terms of boundedness for its inverse images
of precompact sets. Finally, we prove that the mapping is surjective if we
consider the versions of the Navier-Stokes type equations containing no
`pressure'{}. This gives a uniqueness and existence theorem for regular
solutions to this particular ersatz of the Navier-Stokes type equations. The
used techniques consist in proving the closedness of the image by estimating
all possible divergent sequences in the preimage and matching the asymptotics.
The following facts are essential: i) the torus is a compact closed manifold,
ii) the corresponding system is `local'.
