  This note concerns a nonlinear ill-posedness of the Prandtl equation and an
invalidity of asymptotic boundary-layer expansions of incompressible fluid
flows near a solid boundary. Our analysis is built upon recent remarkable
linear ill-posedness results established by G\'erard-Varet and Dormy [2], and
an analysis in Guo and Tice [5]. We show that the asymptotic boundary-layer
expansion is not valid for non-monotonic shear layer flows in Sobolev spaces.
We also introduce a notion of weak well-posedness and prove that the nonlinear
Prandtl equation is not well-posed in this sense near non-stationary and
non-monotonic shear flows. On the other hand, we are able to verify that
Oleinik's monotonic solutions are well-posed.
