Stability of two-dimensional Navier-Stokes motions in the periodic case

TL;DR

This paper proves the existence of global strong solutions for 2D Navier-Stokes equations with periodic boundary conditions and demonstrates the stability of these solutions when extended to 3D under certain conditions.

Contribution

It establishes the global existence of strong solutions in 2D and shows their stability in 3D when initial data and forces are close, advancing understanding of Navier-Stokes stability.

Findings

Existence of global strong 2D solutions.

Existence of global strong 3D solutions near 2D solutions.

Stability of 2D solutions in 3D motions.

Abstract

We consider the motion described by the Navier-Stokes equations in a box with periodic boundary conditions. First we prove the existence of global strong two-dimensional solutions. Next we show the existence of global strong three-dimensional solutions under the assumption that the initial data and the external force are sufficiently close to the initial data and the external force of the two-dimensional problem in appropriate spaces. The second result can be treated as stability of strong two-dimensional solutions in the set of suitably strong three-dimensional motions.