Title: Sublayer of Prandtl boundary layers,
Abstract: 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.