Inverse image of precompact sets and existence theorems for the Navier-Stokes equations in spatially periodic setting

Abstract

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 ${\mathcal{A}}_s:B_1^s\to B_2^{s-1}$ where $B_1^s$, $B_2^{s-1}$ 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 $\mathfrak{s}$, $\mathfrak{r}$.

arXiv admin note: substantial text overlap with arXiv:2007.14911, arXiv:2009.10530