Instability of Boundary Layers with the Navier Boundary Condition

Abstract

We study the L^{infty } 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 partial _y u = nu ^{-gamma }u for some gamma in {mathbb {R}}, where u is the tangential velocity. With the no-slip boundary condition, which corresponds to the limit gamma rightarrow +infty , a celebrated result from E. Grenier (Comm. Pure Appl. Math. 53:1067–1091, 2000) provides an instability of order nu ^{1/4}. M. Paddick (Differ. Integral Equ. 27:893–930, 2014) proved the same result in the case gamma =1/2, furthermore improving the instability to order one. In this paper, we extend these two results to all gamma in {mathbb {R}}, obtaining an instability of order nu ^{vartheta }, where in particular vartheta =0 for gamma le 1/2 and vartheta =1/4 for gamma ge 3/4. When gamma ge 1/2, the result denies the validity of the Prandtl boundary layer expansion around the chosen shear profile.