Thierry Gallay

Université Grenoble Alpes

Vortex Rings at High Reynolds Number

We consider axisymmetric solutions without swirl of the 3D Navier-Stokes equations which originate from circular vortex filaments at initial time. In the case of a single filament, we construct an asymptotic expansion of the corresponding 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 along its symmetry axis at a speed that agrees with the prediction of the binormal flow. To prove these results we introduce self-similar variables located at the (unknown) position of the vortex center, and we control the evolution of the perturbations using a nonlocal functional which is related to Arnold's variational characterization of steady states for the 2D Euler equations. This talk is based on joint work with Vladimir Sverak (Minneapolis).