Some stability problems to the Navier-Stokes equations in the periodic case

TL;DR

This paper investigates the stability of two-dimensional solutions to the Navier-Stokes equations in a periodic domain, proving existence and stability of three-dimensional solutions close to the 2D solutions under certain conditions.

Contribution

It establishes the global existence of regular 2D solutions and demonstrates their stability, leading to the existence of stable 3D solutions near the 2D ones.

Findings

Existence of global regular 2D solutions in periodic domains.

Stability of 2D solutions under small perturbations.

Existence of global 3D solutions close to 2D solutions.

Abstract

The Navier-Stokes motions in a box with periodic boundary conditions are considered. First the existence of global regular two-dimensional solutions is proved. The solutions are such that continuous with respect to time norms are controlled by the same constant for all $t\in\mathbb{R}_+$. Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of the two-dimensional solutions we prove existence of global three-dimensional regular solutions which remain close to the two-dimensional solutions for all time. In this way we mean stability of two-dimensional solutions.