Full instability of boundary layers with the Navier boundary condition

Abstract

We consider the problem of the stability of the Navier-Stokes equations in $\mathbb{T}\times \mathbb{R}_+$ near shear flows which are linearly unstable for the Euler equation. In \cite{greniernguyen}, the authors prove an $L^{\infty}$ instability result for the no-slip boundary condition which also denies the validity of the Prandtl boundary layer expansion. In this paper, we generalise this result to a Navier slip boundary condition with viscosity dependent slip length: $\partial_y u =\nu^{-\gamma}u$ at $y=0$, where $\gamma >1/2$. This range includes the physical slip rate $\gamma=1$.