Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909861157011456 |
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| author | Chen, Qionglei Li, Zhen Miao, Changxing |
| author_facet | Chen, Qionglei Li, Zhen Miao, Changxing |
| contents | In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover the key division point at $10ν$ in the frequency interval $(0,1)$ by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval $(0,1)$, and then we establish the space-time estimates on the low-frequency $0\leq |k|\leq 10 ν$ and the intermediate-frequency $ 10 ν\leq |k|<1$, respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be $γ\leq\frac{1}{2}$.Meanwhile, we also show that when the frequencies $|k|\geq ν^{1-}$, the enhanced dissipation effect occurs for the linearized Navier-Stokes equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition Chen, Qionglei Li, Zhen Miao, Changxing Analysis of PDEs In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover the key division point at $10ν$ in the frequency interval $(0,1)$ by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval $(0,1)$, and then we establish the space-time estimates on the low-frequency $0\leq |k|\leq 10 ν$ and the intermediate-frequency $ 10 ν\leq |k|<1$, respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be $γ\leq\frac{1}{2}$.Meanwhile, we also show that when the frequencies $|k|\geq ν^{1-}$, the enhanced dissipation effect occurs for the linearized Navier-Stokes equations. |
| title | Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.18376 |