Inverse image of precompact sets and existence theorems for the Navier-Stokes equations in spatially periodic setting
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 ${\mathfrak s}$, ${\mathfrak r}$.