On the vanishing viscosity limit in a disk

We say that a solution of the Navier-Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L (2)) norm uniformly in time as the viscosity nu vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space-time energy density of the solution to the Navier-Stokes equations in a boundary layer of width proportional to nu vanish with nu, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/nu.