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

Full instability of boundary layers with the Navier boundary condition