Validity of steady Prandtl layer expansions

Let the viscosity ε→0$\varepsilon \rightarrow 0$ for the 2D steady Navier‐Stokes equations in the region 0≤x≤L$0\le x\le L$ and 0≤y<∞$0\le y<\infty$ with no slip boundary conditions at y=0$y=0$ . For L<<1$L<<1$ , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are achieved through a fixed‐point scheme: u0,v0⟶DNS−1v⟶L−1u0,v0$$\begin{equation*} \hspace*{5pc}{\left[u^{0}, v^0\right]} \overset{\text{DNS}^{-1}}{\longrightarrow }v\overset{\mathcal {L}^{-1}}{ \longrightarrow }{\left[u^{0}, v^0\right]} \end{equation*}$$for solving the Navier‐Stokes equations, where [u0,v0]$[u^{0}, v^0]$ are the tangential and normal velocities at x=0$x=0$ , DNS stands for ∂x$\partial _{x}$ of the vorticity equation for the normal velocity v, and L$\mathcal {L}$ the compatibility ODE for [u0,v0]$[u^{0}, v^0]$ at x=0$x=0$ .