Stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains under Navier-boundary conditions

TL;DR

This paper proves the global existence and stability of two-dimensional solutions to the Navier-Stokes equations in cylindrical domains with Navier boundary conditions, and shows three-dimensional solutions remain close to these under certain conditions.

Contribution

It establishes the stability of 2D solutions to Navier-Stokes in cylindrical domains with Navier boundary conditions, including the existence of 3D solutions near these 2D solutions.

Findings

Existence of global regular 2D solutions in cylindrical domains.

Boundedness of solutions uniformly over time.

Stability of 2D solutions under small perturbations leading to 3D solutions.

Abstract

The Navier-Stokes motions in cylindrical domain with Navier boundary conditions are considered. First the existence of global regular two-dimensional solutions are proved. The solutions are bounded by the same constant for all time. 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 global existence of three-dimensional solutions which remain close to the two-dimensional solutions for all time. In this way we mean stability of two-dimensional solutions.