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arXiv · 2510.16378

Stability threshold of close-to-Couette shear flows with no-slip boundary conditions in 2D

Jacob Bedrossian·Siming He·Sameer Iyer·Linfeng Li·Fei Wang
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Abstract

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 $\nu$. On the other hand, the $x$-dependent fluctuation is assumed small in a viscosity-dependent sense, namely, $O(\nu^{\frac12}|\log \nu|^{-2})$. 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.

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Jacob Bedrossian, Siming He, Sameer Iyer, Linfeng Li, Fei Wang. 2025-10-18. Stability threshold of close-to-Couette shear flows with no-slip boundary conditions in 2D. https://arxiv.org/abs/2510.16378

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