Mathematical approach of boundary layer stability in some fluid mechanics models

In many applications such as acoustics or geophysics, fluid models are written in terms of small dimensionless parameters controlling diffusion or wave propagation. The asymptotic regime of vanishing parameters sometimes generates waves and small scale flow near boundaries called boundary layers. We present recent results based upon partial differential equations techniques on nonlinear stability or instability of boundary layers applied to Ekman layers which arise in the small Rossby number limit of the rotating Navier-Stokes equations relevant to the earth regime. We also study the convergence of the low Mach number limit of the compressible Navier-Stokes equations with boundaries.