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 $\omega_{ in}$, showing that stability holds for perturbations of order $\nu^{ 1/2}$ 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 $\nu^{ 1/2}(1+\ln(1/\nu))^{ -1/2}$. Our result removes the logarithmic loss and identifies the natural scaling $\nu^{ 1/2}$ as the critical size of perturbations for nonlinear stability in this setting.
Categories: math.AP