Instability of Boundary Layers with the Navier Boundary Condition

Abstract We study the $$L^{\infty }$$ L ∞ stability of Navier-Stokes equations in half-plane with a viscosity-dependent Navier friction boundary condition around shear profiles which are linearly unstable for Euler equation. The dependence from viscosity is given as $$\partial _y u = \nu ^{-\gamma }u$$ ∂ y u = ν - γ some $$\gamma \in {\mathbb {R}}$$ ∈ R , where tangential velocity. With no-slip condition, corresponds to limit \rightarrow +\infty $$ → + celebrated result E. Grenier (Comm. Pure Appl. Math. 53:1067–1091, 2000) provides an instability order $$\nu ^{1/4}$$ 1 / 4 . M. Paddick (Differ. Integral Equ. 27:893–930, 2014) proved same case =1/2$$ 2 furthermore improving one. In this paper, we extend these two results all obtaining ^{\vartheta ϑ particular $$\vartheta =0$$ 0 \le 1/2$$ ≤ and =1/4$$ \ge 3/4$$ ≥ 3 When denies validity Prandtl layer expansion chosen profile.