Uniform nonlinear stability threshold at large shear amplitude

Determine whether the nonlinear stability threshold for the horizontally dissipative two-dimensional Navier–Stokes equations near the Kolmogorov flow remains uniform in the shear amplitude a when L<2\pi, despite quadratic feedback between decaying modes and the undamped shear kernel.

Background

The linearized dynamics for L<2\pi has decay estimates whose constants do not depend on the shear amplitude a. This suggests that a nonlinear smallness threshold might also be uniform in a.

The nonlinear equation couples horizontal Fourier modes and can transfer quadratic effects into the undamped, horizontally independent shear component. The authors explicitly identify it as unresolved whether this nonlinear interaction destroys amplitude-uniform stability for large a.

References

Whether the nonlinear interaction of the neutral shear modes with the decaying modes destroys this uniformity for large $a$ (through the $a$-dependent transient of the marginal case when $L=2\pi$, or through the quadratic feedback onto the kernel) is, in our view, the most interesting open question in the stable regime.

— Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation  (2609.34610 - Yang et al., 28 Sep 2026) in Section 5, subsection “Nonlinear stability modulo shears for $L<2\pi$”

Establishing such a bifurcation and describing the nonlinear dynamics for $a$ near $a_{\mathrm c}(k,\nu)$ remain open problems.

— Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation  (2609.34610 - Yang et al., 28 Sep 2026) in Section 5, subsection “Nonlinear dynamics near the threshold for $L>2\pi$”