Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particular symmetry. Our result applies to a large family of velocity profiles, including the most common models in atmospheric flows and engineering applications. The proof is based on a homotopy argument, which allows us to concentrate in the spectral analysis of the linearized operator to a small neighborhood of the imaginary axis, where unstable eigenvalues can be excluded using integral identities and a careful study of the so-called critical layers.

中文翻译：

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无粘柱状涡的光谱稳定性

柱状涡是轴对称的三维欧拉方程的平稳解，其中速度场只取决于到轴的距离，在轴向上没有分量。开尔文勋爵于 1880 年首次研究了此类流动的稳定性，但尽管历史悠久，但迄今为止唯一可用的分析结果为平面或轴对称扰动下的不稳定性提供了必要条件。本文的目的是表明柱状涡旋对于没有特定对称性的三维扰动是光谱稳定的。我们的结果适用于一大类速度剖面，包括大气流动和工程应用中最常见的模型。证明基于同伦论证，