Improved stability threshold for 2D Navier-Stokes Couette flow in an infinite channel

Abstract

We study the nonlinear stability of the two-dimensional Navier-Stokes equations around the Couette shear flow in the channel domain $\mathbb{R}\times [-1,1]$ subject to Navier slip boundary conditions. We establish a quantitative stability threshold for perturbations of the initial vorticity $\ddot{I}{‰}_{in}$, showing that stability holds for perturbations of order $\hat{I}{½}^{1/2}$ measured in an anisotropic Sobolev space. This sharpens the recent work of Arbon and Bedrossian [Comm. Math. Phys., 406 (2025), Paper No. 129] who proved stability under the threshold $\hat{I}{½}^{1/2}(1+\ln (1/\hat{I}½){)}^{-1/2}$. Our result removes the logarithmic loss and identifies the natural scaling $\hat{I}{½}^{1/2}$ as the critical size of perturbations for nonlinear stability in this setting.

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