Remarks on the Steady Prandtl Boundary Layer Expansions

We continue the study of the validity of the Prandtl boundary layer expansions in \cite{GZ}, where by estimating the stream-function of the remainder, we proved if the Euler flow is perturbation of shear flows when the width of domain is small. In this paper, we obtain a new derivatives estimate of stream-function away from the boundary layer and then prove the validity of expansions for any non-shear Euler flow, provided the width of domain is small.