Small scales in inviscid limits of steady fluids

In this article, we study the 2D incompressible steady Navier-Stokes equation in a channel $(-L,0)\times (-1,1)$ with the no-slip boundary condition on $\{Y=\pm 1\}$, and consider the inviscid limit $\epsilon \to 0$. In the special case of Euler shear flow $(u_e(Y),0)$, we construct a steady Navier-Stokes solution for $\epsilon \ll 1$, $$\{\begin{array}{rl}{\displaystyle }& {\displaystyle u^{\epsilon }\sim u_e+u_p+O(\sqrt{\epsilon }),}\\ {\displaystyle }& {\displaystyle v^{\epsilon }\sim h(Y)\exp \{X{u}_e(Y)\mathrm{/}\epsilon \}+O(\sqrt{\epsilon }),}\end{array}$$ where $u_p$ represents the classical Prandtl layer profile, and $h(Y)$ is an arbitrary smooth, compactly-supported function with small magnitude. While the classical Prandtl boundary layer $u_p$ exhibits a small scale of order $\sqrt{\epsilon }$ in $Y$ near $Y=\pm 1$, the profile we construct reveals an $\epsilon$ small scale of $X{u}_e(Y)$ in the vertical velocity component.