Instability of Boundary Layers with the Navier Boundary Condition

Abstract

We study the $L^{\infty }$ stability of the 2D Navier-Stokes equations with a viscosity-dependent Navier 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 $\gamma \to +\infty$, a celebrated result from E. Grenier provides an instability of order ${\nu }^{1/4}$. M. Paddick 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 ${\nu }^{\theta }$, where $$\theta :=\{\begin{array}{ll}\frac{1}{4}& \text{if }\gamma \ge \frac{3}{4};\\ \gamma -\frac{1}{2}& \text{if }\frac{1}{2}<\gamma <\frac{3}{4};\\ 0& \text{if }\gamma \le \frac{1}{2}.\end{array}$$ When $\gamma \ge 1/2$, the result denies the validity of the Prandtl boundary layer expansion around the chosen shear profile.

28 pages, 1 figure