Evolution of viscous vortex filaments and soliton-type propagation

Abstract

We study the evolution of a viscous incompressible fluid whose initial vorticity is supported on a smooth open curve. We show that, for $\nu t\ll 1$ and sufficiently small Reynolds number $\Gamma/\nu$, the vorticity is described at leading order by a Lamb--Oseen type vortex concentrated around a curve evolving according to the binormal flow predicted by the localized induction approximation. The solution is written as an explicit leading-order profile plus a lower-order perturbation, which is controlled in a Morrey $\mathcal{M}^{\infty}$ norm. Then, we apply this construction to the Hasimoto soliton. In this case, the estimates are uniform with respect to the torsion parameter, allowing us to consider a large-torsion regime in which the soliton undergoes a macroscopic displacement. We show that the corresponding Navier--Stokes solution contains a localized portion of the kinetic-energy distribution, associated with the binormal velocity, which remains concentrated inside a moving physical region and undergoes an order-one displacement during an admissible time interval.