Refined asymptotic decay beyond the established polynomial rate

Prove that, for solutions of the coupled atmosphere-ocean system with nonlinear wind-driven interface conditions, the differences between the atmospheric and oceanic velocities and their respective means decay faster than the polynomial rate established in Theorem 3.1, thereby rigorously establishing the refined asymptotic behavior suggested by the center-manifold analysis.

Background

The paper proves optimal polynomial convergence rates for the full solution in the relevant phase-space norm. It then notes that the velocity fluctuations relative to their means are expected to decay more rapidly than this general rate. A rigorous proof is not obtained because the relevant equations involve nonlinear and nonlocal dynamics, although the derived equations and center-manifold description suggest a possible route.

References

First, a rigorous proof of the refined asymptotic behaviour described at the end of last subsection remains open. In particular, showing that the difference between the velocities and their means decays faster than the polynomial rate from \autoref{thm:long-term}. Although leveraging the derived equations and the center manifold description offers a plausible approach, the non-linear and non-local dynamics make this a highly non-trivial task. Because the central aim of the present work is to establish polynomial convergence rates, we defer a rigorous treatment of this faster decay to future research.

— Long-term behaviour of the CAO-system: Optimal polynomial $H^1$-convergence to degenerate equilibria  (2609.08664 - Binz, 8 Sep 2026) in Section 1.1, “Main challenges, new contributions and open problems”