Instability of Boundary Layers with the Navier Boundary Condition

Abstract

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