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

Anuj Kumar, Wojciech Ożański

TL;DR

This paper provides a streamlined Sobolev-space proof of linear instability for 2D parallel shear flows, extending Lin's framework and clarifying the pivotal role of the Plemelj-Sochocki formula in generating unstable spectra without relying on the cone condition. By perturbing a neutral Rayleigh mode $\widetilde{\phi}$ with parameters $(\varepsilon,c)$ and decomposing the perturbation into a projected and an auxiliary part, the authors cast the problem into a reduced equation $G(c,\varepsilon)=0$ where $G(c,\varepsilon)=c\lambda+\varepsilon+o(|c|+|\varepsilon|)$ with $\Im\lambda>0$, derived from the Plemelj limit. A Neumann-series-based solvability of the projected equation combined with Rouché's theorem yields unstable modes with $\Im c>0$ for small $\varepsilon$, providing a shorter, Hölder-free route to instability. The second result localizes the instability in space via an inner-outer gluing construction for $U(y)=U_0(ky)$, showing that as $k\to\infty$ the unstable mode concentrates near the inflection point and that the growth rate scales like $\Im c\sim \varepsilon/k^2$, echoing Kelvin–Helmholtz-type vortex-sheet behavior.

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.

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

TL;DR

This paper provides a streamlined Sobolev-space proof of linear instability for 2D parallel shear flows, extending Lin's framework and clarifying the pivotal role of the Plemelj-Sochocki formula in generating unstable spectra without relying on the cone condition. By perturbing a neutral Rayleigh mode with parameters and decomposing the perturbation into a projected and an auxiliary part, the authors cast the problem into a reduced equation where with , derived from the Plemelj limit. A Neumann-series-based solvability of the projected equation combined with Rouché's theorem yields unstable modes with for small , providing a shorter, Hölder-free route to instability. The second result localizes the instability in space via an inner-outer gluing construction for , showing that as the unstable mode concentrates near the inflection point and that the growth rate scales like , echoing Kelvin–Helmholtz-type vortex-sheet behavior.

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.
Paper Structure (5 sections, 9 theorems, 94 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['T01']}
  3. Proof of Theorem \ref{['T02']}
  4. Plemelj-Sochocki formula
  5. Eigenvalue problems

Key Result

Theorem 1

For all sufficiently small $\varepsilon >0$ there exists $c=c(\varepsilon ) \in \mathbb{C}$ with $\mathrm{Im}\,c>0$ such that the Rayleigh equation rayleigh has a solution $\phi\in H^1_0 (-1,1)$ such that $\phi = \widetilde{\phi} +\psi$ with $\| \psi \|_{H^1} \lesssim |c| + \varepsilon$.

Figures (1)

  • Figure 1: A sketch of the velocity profile $U(y)=U_0 (ky)$ and the cutoffs $\chi_{{\rm in}}$, $\chi_{{\rm out}}$ leading to the inner and outer estimates \ref{['multisc_ests']}.

Theorems & Definitions (17)

  • Theorem 1: Linear instability of shear flow in Sobolev spaces
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4: Approximation lemma
  • proof
  • Lemma 5
  • proof
  • Lemma 6: Remainder norm
  • proof
  • ...and 7 more