2019/01/09 by Thierry Gallay, Gallay, Thierry · 2 citations
Engineering · Mathematics · #35B35 #35Q31 #76B47 #76E07 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1901.02815
openalex publication_date 2019/01/09 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28
The mathematical theory of hydrodynamic stability started in the middle of\nthe 19th century with the study of model examples, such as parallel flows,\nvortex rings, and surfaces of discontinuity. We focus here on the equally\ninteresting case of columnar vortices, which are axisymmetric stationary flows\nwhere the velocity field only depends on the distance to the symmetry axis and\nhas no component in the axial direction. The stability of such flows was first\ninvestigated by Kelvin in 1880 for some particular velocity profiles, and the\nproblem benefited from important contributions by Rayleigh in 1880 and 1917.\nDespite further progress in the 20th century, notably by Howard and Gupta\n(1962), the only rigorous results so far are necessary conditions for\ninstability under either two-dimensional or axisymmetric perturbations. This\nnote is a non-technical introduction to a recent work in collaboration with D.\nSmets, where we prove under mild assumptions that columnar vortices are\nspectrally stable with respect to general three-dimensional perturbations, and\nthat the linearized evolution group has a subexponential growth as |t| \→\n\∞.\n