A simple proof of linear instability of shear flows with application to vortex sheets

TL;DR

This paper presents a simplified proof of linear instability in shear flows, highlighting the role of the Plemelj-Sochocki formula and extending the approach to vortex sheet instability.

Contribution

It offers an alternative, straightforward Sobolev space proof of shear flow instability that avoids the cone condition and applies to vortex sheet Kelvin-Helmholtz instability.

Findings

Simplified proof of shear flow instability

Mathematical role of Plemelj-Sochocki formula clarified

Approximation of Kelvin-Helmholtz instability achieved

Abstract

We consider the construction of linear instability of parallel shear flows, which was developed by Zhiwu Lin (SIAM J. Math. Anal. 35(2), 2003). We give an alternative simple proof in Sobolev setting of the problem, which exposes the mathematical role of the Plemelj-Sochocki formula in the emergence of the instability, as well as does not require the cone condition. Moreover, we localize this approach to obtain an approximation of the Kelvin-Helmholtz instability of a flat vortex sheet.