Validity of Steady Prandtl Layer Expansions

TL;DR

This paper rigorously justifies the steady Prandtl boundary layer expansion for small-length 2D Navier-Stokes flows with no-slip boundary conditions, using a fixed-point scheme to establish uniform estimates as viscosity vanishes.

Contribution

It provides the first rigorous proof of steady Prandtl layer validity in the small domain regime, including the Blasius boundary layer, via a novel fixed-point approach.

Findings

Validated the steady Prandtl expansion for small L

Established uniform estimates in viscosity parameter

Included analysis of the Blasius boundary layer

Abstract

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 $x=0.$