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 $\gamma=1$.
Categories: math.AP