Inverse image of precompact sets and regular solutions to the Navier–Stokes equations

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