  In this paper, we derive sharp bounds on the semigroup of the linearized
incompressible Navier-Stokes equations near a stationary shear layer in the
half plane and in the half space ($\mathbb{R}_+^2$ or $\mathbb{R}_+^3$), with
Dirichlet boundary conditions, assuming that this shear layer in spectrally
unstable for Euler equations. In the inviscid limit, due to the prescribed
no-slip boundary conditions, vorticity becomes unbounded near the boundary. The
novelty of this paper is to introduce boundary layer norms that capture the
unbounded vorticity and to derive sharp estimates on this vorticity that are
uniform in the inviscid limit.
