Linear inviscid damping and vorticity depletion for shear flows

TL;DR

This paper proves linear damping for 2-D Euler equations around shear flows, revealing a new vorticity depletion mechanism at stationary streamlines that enhances understanding of flow stability.

Contribution

It establishes linear damping under specific spectral conditions and uncovers a novel vorticity depletion phenomenon at stationary streamlines.

Findings

Explicit decay estimates for symmetric shear flows

Confirmation of vorticity depletion at stationary streamlines

New mechanism for flow damping with stationary streamlines

Abstract

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.