Sublayer of Prandtl boundary layers

The aim of this paper is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: $\nu \to 0$. In \cite{Grenier}, one of the authors proved that there exists no asymptotic expansion involving one Prandtl's boundary layer with thickness of order $\sqrt\nu$, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order $\nu^{3/4}$. In this paper, we point out how the stability of the classical Prandtl's layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in $L^\infty$. That is, either the Prandtl's layer or the boundary sublayer is nonlinearly unstable in the sup norm.