Regularity of Desingularised Models for Vortex Filaments in Incompressible Viscous Flows: A Geometrical Approach

TL;DR

This paper proves the regularity of weak solutions for a family of desingularised vortex filament models in 3D viscous flows, extending classical models through a geometric and analytical Sobolev space approach.

Contribution

It introduces a generalized geometric framework for vortex filament models and establishes regularity results for their weak solutions in viscous flows.

Findings

Proved regularity of weak solutions for desingularised vortex models

Extended classical vortex filament models to more general geometries

Used Sobolev space inequalities to analyze vorticity geometry

Abstract

We establish the regularity of weak solutions for the vorticity equation associated to a family of desingularised models for vortex filament dynamics in 3D incompressible viscous flows. These include and generalise the classical model "of an allowance for the thickness of the vortices" due to Louis Rosenhead in 1930. Our approach is based on an interplay between the geometry of vorticity and analytic inequalities in Sobolev spaces.