On the vanishing viscosity limit in a disk

TL;DR

This paper characterizes the conditions under which solutions to the Navier-Stokes equations in a disk converge to Euler solutions as viscosity vanishes, focusing on boundary layer energy modes related to eigenfunctions of the Stokes operator.

Contribution

It provides a necessary and sufficient condition for the vanishing viscosity limit in a disk based on the behavior of boundary layer energy modes.

Findings

Vanishing viscosity limit depends on boundary layer energy modes.

Modes with frequencies between L and M determine convergence.

Conditions relate mode frequencies to viscosity scale.

Abstract

We say that the solution u to the Navier-Stokes equations converges to a solution v to the Euler equations in the vanishing viscosity limit if u converges to v in the energy norm uniformly over a finite time interval. Working specifically in the unit disk, 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 the viscosity due to modes (eigenfunctions of the Stokes operator) whose frequencies in the radial or the tangential direction lie between L and M. Here, L must be of order less than 1/(viscosity) and M must be of order greater than 1/(viscosity).