Thierry Gallay: Axisymmetric Vortex Rings at High Reynolds Number

Abstract:

We consider axisymmetric solutions without swirl of the 3D Navier–Stokes equations originating from circular vortex filaments at initial time. In the case of a single filament, we construct an asymptotic expansion of the viscous vortex ring in the high Reynolds number regime, where the kinematic viscosity is small compared to the circulation of the vortex.

We then show that the unique solution of the axisymmetric Navier–Stokes equations remains close to our approximation over a long time interval, during which the vortex ring moves long its symmetry axis at a speed that was predicted by Kelvin in 1867. To prove that, we introduce self-similar variables located at the (unknown) position of the ring, and we control the evolution of the perturbations using an energy functional related to Arnold’s variational characterization of steady states for the 2D Euler equations.

This talk is based on joint work with Vladimir Sverak.