Transition threshold for the 2-D Couette flow in whole space via Green's function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913320048525312 |
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| author | Wang, Gaofeng Wang, Weike |
| author_facet | Wang, Gaofeng Wang, Weike |
| contents | In this paper, we investigate the transition threshold problem concerning the 2-D Navier-Stokes equations in the context of Couette flow $(y,0)$ at high Reynolds number $Re$ in whole space. By utilizing Green's function estimates for the linearized equations around Couette flow, we initially establish refined dissipation estimates for the linearized Navier-Stokes equations with a precise decay rate $(1+t)^{-1}.$ As an application, we prove that if the initial perturbation of vorticity satisfies$$\|ω_{0}\|_{H^{1}\cap L^1}\leq c_0ν^{\frac{3}{4}}$$ for some small constant $c_0$ independent of the viscosity $ν$, then we can reach the conclusion that the solution remains within $O\left( ν^{\frac{3}{4}}\right) $ of the Couette flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transition threshold for the 2-D Couette flow in whole space via Green's function Wang, Gaofeng Wang, Weike Analysis of PDEs 35Q30 In this paper, we investigate the transition threshold problem concerning the 2-D Navier-Stokes equations in the context of Couette flow $(y,0)$ at high Reynolds number $Re$ in whole space. By utilizing Green's function estimates for the linearized equations around Couette flow, we initially establish refined dissipation estimates for the linearized Navier-Stokes equations with a precise decay rate $(1+t)^{-1}.$ As an application, we prove that if the initial perturbation of vorticity satisfies$$\|ω_{0}\|_{H^{1}\cap L^1}\leq c_0ν^{\frac{3}{4}}$$ for some small constant $c_0$ independent of the viscosity $ν$, then we can reach the conclusion that the solution remains within $O\left( ν^{\frac{3}{4}}\right) $ of the Couette flow. |
| title | Transition threshold for the 2-D Couette flow in whole space via Green's function |
| topic | Analysis of PDEs 35Q30 |
| url | https://arxiv.org/abs/2404.11878 |