On the Vanishing Viscosity Limit in a Disk

Let u be a solution to the Navier-Stokes equations in the unit disk with no-slip boundary conditions and viscosity ν > 0, and let u be a smooth solution to the Euler equations. We say that the vanishing viscosity limit holds on [0, T ] if u converges to u in L∞([0, T ];L). We show that a necessary and sufficient condition for the vanishing viscosity limit to hold is the vanishing with the viscosity of the time-space average of the energy of u in a boundary layer of width proportional to ν due to the modes (eigenfunctions of the Stokes operator) whose frequencies in the radial or the tangential direction lie between L(ν) and M(ν). Here, L(ν) must be of order less than 1/ν and M(ν) must be of order greater than 1/ν.