Boundary Layers and Initial-Boundary Corner Asymptotics for the 2D Navier-Stokes Equations with Vanishing Vertical Viscosity

Motivated by anisotropic viscosity in geophysical fluid dynamics, where vertical momentum diffusion is often much weaker than horizontal diffusion and may be negligible in the interior yet remains essential near a solid wall, we study the vanishing vertical viscosity limit for the two-dimensional incompressible Navier–Stokes equations in the upper half-plane, with horizontal viscosity fixed at one and vertical viscosity ${\epsilon }^2$. For arbitrary divergence-free $H^4$ no-slip initial data, with no time-differentiated compatibility conditions required, we construct the limiting horizontally viscous flow and the boundary-layer profiles on every prescribed finite interval $[0,T]$. The leading-order corrected approximation has $O(\epsilon )$ error in $L^{\mathrm{\infty }}$, and the full finite-order expansion reduces this error to $O({\epsilon }^{3\mathrm{/}2})$, uniformly down to $t=0$. We further describe the development of the layer from data that vanish initially at the wall. At the initial–boundary corner, we determine the first two self-similar coefficients and characterize matching through exact profile tails, thereby separating the fixed-time scale $\epsilon$ from the short-time scale $\epsilon \sqrt{t}$. When the initial wall acceleration is nonzero, both the leading correction rate and the first normalized corner remainder are sharp. Finally, continuity estimates for fixed data yield joint small-time and small-viscosity limits for the exact solution, as well as finite-$L^p$ asymptotics.