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

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.