Vol. 13, No. 6, 2020

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Spectral stability of inviscid columnar vortices

Thierry Gallay and Didier Smets

Vol. 13 (2020), No. 6, 1777–1832
Abstract

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, when analyzing the spectrum of the linearized operator, to concentrate on 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.

Dedicated to the memory of Louis N. Howard

Keywords
hydrodynamic stability, inviscid fluid, columnar vortices
Mathematical Subject Classification 2010
Primary: 35B35, 35Q31, 76B47, 76E07
Milestones
Received: 19 October 2018
Revised: 6 June 2019
Accepted: 13 August 2019
Published: 12 September 2020
Authors
Thierry Gallay
Institut Fourier
Université Grenoble Alpes, CNRS
Gières
France
Didier Smets
Laboratoire Jacques-Louis Lions
Sorbonne Université
Paris
France