Validity of steady Prandtl layer expansions

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