Stability analysis for the incompressible Navier-Stokes equations with Navier boundary conditions

TL;DR

This paper investigates the stability of steady states in 2D incompressible Navier-Stokes equations with Navier-slip boundary conditions, revealing how boundary dissipation, viscosity, and slip length influence stability and instability.

Contribution

It provides a detailed stability analysis considering boundary dissipation effects and identifies a critical viscosity threshold for stability transition.

Findings

Nonlinear asymptotic stability when all boundaries are dissipative.

Existence of a sharp critical viscosity separating stability and instability.

Boundary dissipation critically affects the stability of steady states.

Abstract

This paper concerns the instability and stability of the trivial steady states of the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then nonlinear asymptotic stability holds true, otherwise, there is a sharp critical viscosity, which distinguishes the nonlinear stability from instability.