Stability threshold of close-to-Couette shear flows with no-slip boundary conditions in 2D
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909854385307648 |
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| author | Bedrossian, Jacob He, Siming Iyer, Sameer Li, Linfeng Wang, Fei |
| author_facet | Bedrossian, Jacob He, Siming Iyer, Sameer Li, Linfeng Wang, Fei |
| contents | In this paper, we develop a stability threshold theorem for the 2D incompressible Navier-Stokes equations on the channel, supplemented with the no-slip boundary condition. The initial datum is close to the Couette flow in the following sense: the shear component of the perturbation is small, but independent of the viscosity $ν$. On the other hand, the $x$-dependent fluctuation is assumed small in a viscosity-dependent sense, namely, $O(ν^{\frac12}|\log ν|^{-2})$. Under this setup, we prove nonlinear enhanced dissipation of the vorticity and a time-integrated inviscid damping for the velocity. These stabilizing phenomena guarantee that the Navier-Stokes solution stays close to an evolving shear flow for all time. The analytical challenge stems from a time-dependent nonlocal term that appears in the associated linearized Navier-Stokes equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability threshold of close-to-Couette shear flows with no-slip boundary conditions in 2D Bedrossian, Jacob He, Siming Iyer, Sameer Li, Linfeng Wang, Fei Analysis of PDEs In this paper, we develop a stability threshold theorem for the 2D incompressible Navier-Stokes equations on the channel, supplemented with the no-slip boundary condition. The initial datum is close to the Couette flow in the following sense: the shear component of the perturbation is small, but independent of the viscosity $ν$. On the other hand, the $x$-dependent fluctuation is assumed small in a viscosity-dependent sense, namely, $O(ν^{\frac12}|\log ν|^{-2})$. Under this setup, we prove nonlinear enhanced dissipation of the vorticity and a time-integrated inviscid damping for the velocity. These stabilizing phenomena guarantee that the Navier-Stokes solution stays close to an evolving shear flow for all time. The analytical challenge stems from a time-dependent nonlocal term that appears in the associated linearized Navier-Stokes equations. |
| title | Stability threshold of close-to-Couette shear flows with no-slip boundary conditions in 2D |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.16378 |