A note on Prandtl boundary layers

This note concerns 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 illposedness results established by Gérard-Varet and Dormy and an analysis by Guo and Tice. We show that the asymptotic boundary layer expansion is not valid for nonmonotonic 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 nonstationary and nonmonotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed.