Stability of Vortices in Ideal Fluids : the Legacy of Kelvin and Rayleigh

The mathematical theory of hydrodynamic stability started in the middle of the 19th century with the study of model examples, such as parallel flows, vortex rings, and surfaces of discontinuity. We focus here on the equally interesting case of columnar vortices, which are axisymmetric stationary flows where the velocity field only depends on the distance to the symmetry axis and has no component in the axial direction. The stability of such flows was first investigated by Kelvin in 1880 for some particular velocity profiles, and the problem benefited from important contributions by Rayleigh in 1880 and 1917. Despite further progress in the 20th century, notably by Howard and Gupta (1962), the only rigorous results so far are necessary conditions for instability under either two-dimensional or axisymmetric perturbations. This note is a non-technical introduction to a recent work in collaboration with D. Smets, where we prove under mild assumptions that columnar vortices are spectrally stable with respect to general three-dimensional perturbations, and that the linearized evolution group has a subexponential growth as $|t| \to \infty$.