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

TL;DR

This paper studies the Navier-Stokes equations in a periodic setting, establishing the properties of a solution map between specific function spaces and characterizing its surjectivity via inverse image boundedness.

Contribution

It introduces a new framework for analyzing the Navier-Stokes initial value problem using Bochner-Sobolev spaces and characterizes the solution map’s surjectivity through inverse image properties.

Findings

The solution map is an open, injective mapping between specialized function spaces.

Surjectivity of the solution map is equivalent to inverse images of precompact sets being bounded.

Provides new insights into the structure of solutions for Navier-Stokes equations in periodic domains.

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 ${\mathcal A} _s: B^{s}_1 \to B^{s-1}_2$ 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 ${\mathfrak s}$, ${\mathfrak r}$.