Remarks on the inviscid limit for the Navier-Stokes equations for uniformly bounded velocity fields

We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the Navier-Stokes solutions remain bounded in $L^2_t L^\infty_{x}$ independently of the kinematic viscosity, and if they are equicontinuous at $x_2=0$.