Nonlinear stability of 3D Couette flow on $\mathbb{T}\times {\mathbb{R}}^2$ under critical transverse low-frequency control

We establish nonlinear stability of three-dimensional Couette flow for the incompressible Navier–Stokes equations on $\mathbb{T}\times {\mathbb{R}}^2$. Divergence-free perturbations satisfying $\mathrm{\parallel }{u}_{\mathrm{i}\mathrm{n}}{\mathrm{\parallel }}_{H^2}+\mathrm{\parallel }{u}_{0,\mathrm{i}\mathrm{n}}^{\perp }{\mathrm{\parallel }}_{{\dot{B}}_{2,\mathrm{\infty }}^{-2}}\le c_{\ast }\nu$ generate unique global $H^2$ solutions that remain close to Couette flow, where $u_{0,\mathrm{i}\mathrm{n}}^{\perp }$ denotes the streamwise average of the transverse velocity. This result retains the $O(\nu )$ stability threshold despite the absence of a spanwise spectral gap. The nonzero streamwise modes exhibit enhanced dissipation, their wall-normal component satisfies a viscosity-uniform time-integrated inviscid-damping estimate, and the transverse zero mode decays in $L^2$ at rate $(1+\nu t{)}^{-1}$. The endpoint space ${\dot{B}}_{2,\mathrm{\infty }}^{-2}$ controls the transverse low frequencies, accommodates divergence-form nonlinear forcing, and is critical for linear lift-up from the wall-normal zero mode. The proof combines propagation of this endpoint norm, Fourier splitting, and a two-level roll-to-streak representation with an adaptation of the nonzero-mode framework of Wei–Zhang (CPAM, 2021).