Validity of Steady Prandtl Layer Expansions

Let the viscosity $\varepsilon \rightarrow 0$ for the 2D steady Navier-Stokes equations in the region $0\leq x\leq L$ and $0\leq y<\infty$ 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 $\varepsilon$ 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 $\partial _{x}$ of the vorticity equation for the normal velocity $v$, and $\mathcal{L}$ the compatibility ODE for $[u^{0}, v^0]$ at $x=0.$