Asymptotics for vortex filaments using a modified Biot-Savart kernel

Abstract

We consider a family of approximations to the Euler equations obtained by adding $(-\Delta)^{-\alpha/2}$ to the non-locality in the Biot-Savart kernel together with a mollification (with parameter $\varepsilon$). We consider the evolution of a thin vortex tube. We show that the velocity on the filament (core of the tube) in the limit as $\varepsilon\to 0$ is given $\frac{C(\alpha,t)}{\alpha} \kappa B + \mathcal O(1)$ where $\kappa$ and $B$ are the curvature and binormal of the curve, and $C$, $C^{-1}$ are uniformly bounded.