Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows

TL;DR

This paper advances the understanding of vanishing viscosity limits for circularly symmetric 2D flows, demonstrating convergence in stronger norms, handling more singular boundary conditions, and analyzing boundary layer behavior including vorticity concentration.

Contribution

It extends previous work by establishing convergence in $L^2$ and $L^p$-Sobolev spaces, allowing for more singular boundary angular velocities, and analyzing boundary layer phenomena and vorticity concentration.

Findings

Convergence in stronger $L^2$ and $L^p$-Sobolev spaces.

Analysis of boundary layer behavior and vorticity concentration.

Extension to flows on an annulus with independent boundary rotation.

Abstract

We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [LMN], on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in $L^2$-norm as long as the prescribed angular velocity $\alpha(t)$ of the boundary has bounded total variation. Here we establish convergence in stronger $L^2$ and $L^p$-Sobolev spaces, allow for more singular angular velocities $\alpha$, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently. [LMN] Lopes Filho, M. C., Mazzucato, A. L. and Nussenzveig Lopes, H. J., Vanishing viscosity limit for incompressible flow inside a rotating circle, preprint 2006.