# Asymptotics for vortex filaments using a modified Biot-Savart kernel

## TL;DR

This paper analyzes the asymptotic behavior of vortex filaments in modified Euler equations with a fractional Biot-Savart kernel, revealing how the velocity on the filament relates to curvature and binormal in the limit.

## Contribution

It introduces a modified Biot-Savart kernel with fractional Laplacian and mollification, deriving asymptotic velocity behavior for vortex filaments as mollification vanishes.

## Key findings

- Velocity on filament scales with curvature and binormal vector
- Asymptotic velocity coefficient is bounded and depends on fractional parameter
- Results hold uniformly over time and parameters

## 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.