We continue the study on the validity of the Prandtl boundary layer expansions in [Gao et al., Sci. China Math. 66, 679–722 (2023)], whereby estimating the stream-function of the remainder, we proved the case when the Euler flow is the perturbation of shear flow in a narrow domain. 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 that the width of the domain is small.

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