Vortex filament solutions of the Navier-Stokes equations

TL;DR

This paper establishes well-posedness results for 3D Navier-Stokes solutions with vortex filament initial data, including perturbations of the Oseen vortex and smooth closed curves, advancing understanding of self-similar solutions.

Contribution

It proves global and local well-posedness for vortex filament initial data in critical spaces, including large self-similar solutions, which was previously unaddressed.

Findings

Global well-posedness for perturbations of the Oseen vortex

Local well-posedness for smooth, closed vortex filaments

First results on well-posedness near large self-similar solutions

Abstract

We consider solutions of the Navier-Stokes equations in $3d$ with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of $3d$ Navier-Stokes, as well as solutions which are locally approximately self-similar.