2025/12/17 by Li, Te, Zhang, Ping, Zhang, Yibin
Mathematics · Computer Science · Engineering · #Navier-Stokes equation solutions #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2512.15040
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^-κ|α|1/2. At the linear level, this decay mechanism corresponds to a sharp spectral lower bound Σ(α) ∼ |α|1/2 for the nonlocal Oseen operator L_⊥ - αΛ_⊥, and we identify an explicit spectral point attaining this optimal bound. Combined with the spectral estimates obtained in \citeLWZ, our analysis fully resolves the conjecture proposed in \citeGM concerning the asymptotic scaling laws for the spectral and pseudospectral bounds Σ(α) and Ψ(α). These results provide a rigorous mathematical explanation for the shear-mixing mechanism in the vicinity of the 3D Oseen vortex filament.