Nonlinear asymptotic stability and optimal decay rate around the three-dimensional Oseen vortex filament

TL;DR

This paper proves the nonlinear asymptotic stability of the 3D Oseen vortex filament in high-Reynolds-number flows, establishing optimal decay rates for perturbations and resolving conjectures on spectral bounds related to shear-mixing mechanisms.

Contribution

It introduces an anisotropic self-similar coordinate system to analyze stability and decay, and fully resolves conjectures on spectral and pseudospectral bounds for the vortex filament.

Findings

Nonlinear asymptotic stability of the Oseen vortex filament established.

Optimal decay rates for non-axisymmetric perturbations proven.

Spectral bounds for the Oseen operator identified and confirmed.

Abstract

In the high-Reynolds-number regime, this work investigates the long-time dynamics of the three-dimensional incompressible Navier-Stokes equations near the Oseen vortex filament. The flow exhibits a strong interplay between vortex stretching, shearing, and mixing, which generates ever-smaller spatial scales and thereby significantly amplifies viscous effects. By adopting an anisotropic self-similar coordinate system adapted to the filament geometry, we establish the nonlinear asymptotic stability of the Oseen vortex filament. All non-axisymmetric perturbations are shown to decay at the optimal rate $t^{-\kappa |\alpha|^{1/2}}$. At the linear level, this decay mechanism corresponds to a sharp spectral lower bound $\Sigma(\alpha) \sim |\alpha|^{1/2}$ for the nonlocal Oseen operator $L_\perp - \alpha \Lambda_\perp$, and we identify an explicit spectral point attaining this optimal bound. Combined with the spectral estimates obtained in \cite{LWZ}, our analysis fully resolves the conjecture proposed in \cite{GM} concerning the asymptotic scaling laws for the spectral and pseudospectral bounds $\Sigma(\alpha)$ and $\Psi(\alpha)$. These results provide a rigorous mathematical explanation for the shear-mixing mechanism in the vicinity of the 3D Oseen vortex filament.