Stability of the Couette flow for 3D Navier-Stokes equations with rotation
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917868889702400 |
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| author | Huang, Wenting Sun, Ying Xu, Xiaojing |
| author_facet | Huang, Wenting Sun, Ying Xu, Xiaojing |
| contents | Rotation significantly influences the stability characteristics of both laminar and turbulent shear flows. This study examines the stability threshold of the three-dimensional Navier-Stokes equations with rotation, in the vicinity of the Couette flow at high Reynolds numbers ($\mathbf{Re}$) in the periodical domain $\mathbb{T} \times \mathbb{R} \times \mathbb{T}$, where the rotational strength is equivalent to the Couette flow. Compared to the classical Navier-Stokes equations, rotation term brings us more two primary difficulties: the linear coupling term involving in the equation of $u^2$ and the lift-up effect in two directions. To address these difficulties, we introduce two new good unknowns that effectively capture the phenomena of enhanced dissipation and inviscid damping to suppress the lift-up effect. Moreover, we establish the stability threshold for initial perturbation $\left\|u_{\mathrm{in}}\right\|_{H^σ} < δ\mathbf{Re}^{-2}$ for any $σ> \frac{9}{2}$ and some $δ=δ(σ)>0$ depending only on $σ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11005 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of the Couette flow for 3D Navier-Stokes equations with rotation Huang, Wenting Sun, Ying Xu, Xiaojing Analysis of PDEs Rotation significantly influences the stability characteristics of both laminar and turbulent shear flows. This study examines the stability threshold of the three-dimensional Navier-Stokes equations with rotation, in the vicinity of the Couette flow at high Reynolds numbers ($\mathbf{Re}$) in the periodical domain $\mathbb{T} \times \mathbb{R} \times \mathbb{T}$, where the rotational strength is equivalent to the Couette flow. Compared to the classical Navier-Stokes equations, rotation term brings us more two primary difficulties: the linear coupling term involving in the equation of $u^2$ and the lift-up effect in two directions. To address these difficulties, we introduce two new good unknowns that effectively capture the phenomena of enhanced dissipation and inviscid damping to suppress the lift-up effect. Moreover, we establish the stability threshold for initial perturbation $\left\|u_{\mathrm{in}}\right\|_{H^σ} < δ\mathbf{Re}^{-2}$ for any $σ> \frac{9}{2}$ and some $δ=δ(σ)>0$ depending only on $σ$. |
| title | Stability of the Couette flow for 3D Navier-Stokes equations with rotation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.11005 |