Remarks on the Steady Prandtl Boundary Layer Expansions

TL;DR

This paper extends the analysis of Prandtl boundary layer expansions, providing new derivative estimates of the stream-function and proving their validity for non-shear Euler flows in small domains.

Contribution

It introduces new derivative estimates and generalizes the validity of boundary layer expansions to non-shear Euler flows in small domains.

Findings

Validates boundary layer expansions for non-shear flows

Provides new derivative estimates of the stream-function

Extends previous results to broader flow conditions

Abstract

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.