A nonlinear instability result to the Navier-Stokes equations with Navier slip boundary conditions

TL;DR

This paper demonstrates nonlinear instability of trivial steady states in incompressible viscous fluids with Navier-slip boundary conditions, using a novel approach based on linear instability and boundary layer analysis.

Contribution

It introduces a new method to prove nonlinear instability for Navier-Stokes equations with Navier-slip conditions, differing from previous approaches.

Findings

Existence of infinitely many linear normal mode solutions.

Nonlinear instability established through boundary layer analysis.

Different approach from prior work by Ding, Li, and Xin.

Abstract

In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of infinitely many normal mode solutions to the linearized equations is shown via the operator method of Lafitte and Nguyen (2022). Hence, we prove the nonlinear instability by adapting the framework of Desjardins and Grenier (2003) studying some classes of viscous boundary layers to obtain two separated solutions at escaping time. Our work performs a different approach from that of Ding, Li and Xin (2018).