Nonlinear asymptotic stability and optimal decay rate around the three-dimensional Oseen vortex filament
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908717880967168 |
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| author | Li, Te Zhang, Ping Zhang, Yibin |
| author_facet | Li, Te Zhang, Ping Zhang, Yibin |
| contents | 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 $Σ(α) \sim |α|^{1/2}$ for the nonlocal Oseen operator $L_\perp - αΛ_\perp$, 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 $Σ(α)$ and $Ψ(α)$. These results provide a rigorous mathematical explanation for the shear-mixing mechanism in the vicinity of the 3D Oseen vortex filament. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15040 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear asymptotic stability and optimal decay rate around the three-dimensional Oseen vortex filament Li, Te Zhang, Ping Zhang, Yibin Analysis of PDEs 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 $Σ(α) \sim |α|^{1/2}$ for the nonlocal Oseen operator $L_\perp - αΛ_\perp$, 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 $Σ(α)$ and $Ψ(α)$. These results provide a rigorous mathematical explanation for the shear-mixing mechanism in the vicinity of the 3D Oseen vortex filament. |
| title | Nonlinear asymptotic stability and optimal decay rate around the three-dimensional Oseen vortex filament |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.15040 |