Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

TL;DR

This paper proves the uniqueness of axisymmetric viscous flows from circular vortex filaments with large circulation, extending understanding of vortex ring solutions in Navier-Stokes equations beyond existence results.

Contribution

It establishes the uniqueness of such flows and introduces a method for deriving the leading term in short-time asymptotic expansions for these vortex solutions.

Findings

Proved uniqueness of axisymmetric vortex filament solutions

Developed a method for asymptotic expansion of solutions

Extended understanding of viscous vortex rings in Navier-Stokes

Abstract

The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].