## Differential and Integral Equations

### Stability and instability of Navier boundary layers

#### Abstract

We study the inviscid limit problem for the incompressible Navier-Stokes equation on a half-plane with a Navier boundary condition depending on the viscosity. On one hand, we prove the $L^{2}$ convergence of Leray solutions to the solution of the Euler equation. On the other hand, we show the nonlinear instability of some WKB expansions in the stronger $L^{\infty}$ and $\dot{H}^{s}$ ($s>1$) norms. These results are not contradictory, and in the periodic setting, we provide an example for which both phenomena occur simultaneously.

#### Article information

Source
Differential Integral Equations, Volume 27, Number 9/10 (2014), 893-930.

Dates
First available in Project Euclid: 1 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.die/1404230050

Mathematical Reviews number (MathSciNet)
MR3229096

Zentralblatt MATH identifier
1340.35249

#### Citation

Paddick, Matthew. Stability and instability of Navier boundary layers. Differential Integral Equations 27 (2014), no. 9/10, 893--930. https://projecteuclid.org/euclid.die/1404230050