Validity of Steady Prandtl Layer Expansions

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