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arXiv · 2603.01797

Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition

Zhen Li·Shunlin Shen·Zhifei Zhang
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Abstract

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.

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Zhen Li, Shunlin Shen, Zhifei Zhang. 2026-03-02. Asymptotic stability threshold of the 2-D monotone shear flow with no-slip boundary condition. https://arxiv.org/abs/2603.01797

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