Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition
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| Format: | Preprint |
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2026
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| _version_ | 1866914363237990400 |
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| author | Li, Zhen Shen, Shunlin Zhang, Zhifei |
| author_facet | Li, Zhen Shen, Shunlin Zhang, Zhifei |
| contents | In this paper, we investigate the asymptotic stability threshold problem for the 2-D Navier-Stokes equations in a finite channel with no-slip boundary conditions, around monotone shear flow $(U(t,y),0)$. We establish that the flow is asymptotically stable under perturbations satisfying $\|u^{\mathrm{in}}\|_{H^2}\leq cν^{\frac12}$. To achieve the stability threshold $ν^{\frac{1}{2}}$, the key ingredients of the proof include: sharp resolvent estimates for the vorticity based on weak-type resolvent bounds; weighted space-time estimates for the vorticity; pointwise estimates for the velocity. Furthermore, we handle the nonlinear term through a divergence formulation, which facilitates the sharp application of the aforementioned space-time estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_01797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition Li, Zhen Shen, Shunlin Zhang, Zhifei Analysis of PDEs In this paper, we investigate the asymptotic stability threshold problem for the 2-D Navier-Stokes equations in a finite channel with no-slip boundary conditions, around monotone shear flow $(U(t,y),0)$. We establish that the flow is asymptotically stable under perturbations satisfying $\|u^{\mathrm{in}}\|_{H^2}\leq cν^{\frac12}$. To achieve the stability threshold $ν^{\frac{1}{2}}$, the key ingredients of the proof include: sharp resolvent estimates for the vorticity based on weak-type resolvent bounds; weighted space-time estimates for the vorticity; pointwise estimates for the velocity. Furthermore, we handle the nonlinear term through a divergence formulation, which facilitates the sharp application of the aforementioned space-time estimates. |
| title | Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.01797 |