Showing papers by "Thierry Gallay published in 2019"

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

Abstract: 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 $|t| \to \infty$.

On the Linear Stability of Vortex Columns in the Energy Space

Abstract: We investigate the linear stability of inviscid columnar vortices with respect to finite energy perturbations. For a large class of vortex profiles, we show that the linearized evolution group has a sub-exponential growth in time, which means that the associated growth bound is equal to zero. This implies in particular that the spectrum of the linearized operator is entirely contained in the imaginary axis. This contribution complements the results of our previous work Gallay and Smets (Spectral stability of inviscid columnar vortices, 2018. arXiv:1805.05064), where spectral stability was established for the linearized operator in the enstrophy space.

Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

Abstract: The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].