Existence Theorems for Regular Spatially Periodic Solutions to the Navier-Stokes Equations

TL;DR

This paper proves existence and uniqueness of regular spatially periodic solutions to the Navier-Stokes equations on the three-dimensional torus using advanced functional analysis techniques.

Contribution

It establishes a new existence and uniqueness theorem for regular solutions in the periodic setting by analyzing the induced mapping on Bochner-Sobolev spaces.

Findings

Proves the induced mapping is open, injective, and surjective.

Establishes a regular solution existence theorem for Navier-Stokes.

Uses techniques involving the closedness of the image and asymptotic matching.

Abstract

We consider the initial value problem for the Navier-Stokes equations over $R^{3} \times [0,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-valued functions on $R^{3}$ with functions on the three-dimensional torus $T^{3}$, we prove that the problem induces an open both injective and surjective mapping of specially constructed function spaces of Bochner-Sobolev type. This gives a uniqueness and existence theorem for regular solutions to the Navier-Stokes equations. Our techniques consist in proving the closedness of the image by estimating all possible divergent sequences in the preimage and matching the asymptotics.