Evolution of viscous vortex filaments and desingularization of the Biot-Savart integral

Abstract

We consider a viscous fluid with kinematic viscosity $\nu $ and initial data consisting of a smooth closed vortex filament with circulation $\Gamma $. We show that, for short enough time, the solution consists of a deformed Lamb-Oseen vortex whose center (a filament) follows the binormal flow dynamics plus leading order corrections that depend locally on the filament curvature and the nonlocal interactions with distant parts of the filament. In order to achieve this scale separation we require $\Gamma /\nu $ to be sufficiently small.