  We consider the initial problem for the Navier-Stokes equations over
${\mathbb R}^3 \times [0,T]$ with a positive time $T$ in the spatially periodic
setting. Identifying periodic vector-valued functions on ${\mathbb R}^3$ with
functions on the $3\,$-dimensional torus ${\mathbb T}^3$, we prove that the
problem induces an open injective mapping ${\mathcal A} _s: B^{s}_1 \to
B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially
constructed function spaces of Bochner-Sobolev type parametrized with the
smoothness index $s \in \mathbb N$. Finally, we prove rather expectable
statement that a map ${\mathcal A} _s$ is surjective if and only if the inverse
image ${\mathcal A} _s ^{-1}(K)$ of any precompact set $K$ from the range of
the map ${\mathcal A} _s $ is bounded in the Bochner space $L^{\mathfrak s}
([0,T], L ^{\mathfrak s} ({\mathbb T}^3))$ with the Ladyzhenskaya-Prodi-Serrin
numbers ${\mathfrak s}$, ${\mathfrak r}$.
