Instability of boundary layers with the Navier boundary condition

We study the L∞- stability of the Navier-Stokes equations in the half-plane with a viscosity-dependent Navier friction boundary condition around shear profiles which are linearly unstable for the Euler equation. The dependence from the viscosity is given in the Navier boundary condition as ∂yu=ν−γu for some γ∈R, where u is the tangential velocity. With the no-slip boundary condition, which corresponds to the limit γ→+∞, a celebrated result from E. Grenier provides an instability of order ν1/4. M. Paddick proved the same result in the case γ= 1/2, furthermore improving the instability to order one. In this paper, we extend these two results to allγ∈R, obtaining an instability of order νθ, where in particular θ= 0 for γ≤1/2 and θ= 1/4 for γ≥3/4. When γ≥1/2, the result denies the validity of the Prandtl boundary layer expansion around the chosen shear profile