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<p>We consider the construction of linear instability of parallel shear flows, which was developed by Lin (SIAM J Math Anal 35(2):318–356, 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.</p>

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A Simple Proof of Linear Instability of Shear Flows with Application to Vortex Sheets

  • 期刊论文

  • 发表日期：

    2025年5月5日

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  • Anuj Kumar,
  • Wojciech Ożański

摘要

We consider the construction of linear instability of parallel shear flows, which was developed by Lin (SIAM J Math Anal 35(2):318–356, 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.