Extension of polynomial convergence to the full (p,q)-scale

Extend the polynomial convergence result for the coupled atmosphere-ocean system with nonlinear wind-driven interface conditions from the Hilbert-space case p=q=2 to the full (p,q)-scale, including the appropriate Besov-space phase spaces and the cases in which the coupling conditions produce a curved Banach manifold.

Background

The paper’s main long-term convergence theorem is established in the Hilbert-space setting p=q=2. Earlier global well-posedness results are available in an Lp_t Lq_x framework, yielding a continuous semiflow on suitable Besov spaces. However, for sufficiently large p and q, the nonlinear coupling conditions alter the phase space into a curved Banach manifold, while the present proof relies on precise energy estimates that do not directly transfer to this setting. The authors therefore identify extension of the main theorem to the full scale as unresolved and requiring new analytic tools.

References

A second compelling question is whether our main result, \autoref{thm:long-term}, can be extended to the full $(p,q)$-scale. The global well-posedness result in holds in the $Lp_t Lq_x$-setting, giving rise to a continuous semi-flow on appropriate Besov spaces. However, for sufficiently large $p$ and $q$, the coupling conditions inherently modify the phase space, transforming it from a flat Banach space into a curved Banach manifold. Since our current methodology relies heavily on precise energy estimates, it cannot be readily adapted to this broader $(p,q)$-framework. Therefore, the study of the CAO-system on the full $(p,q)$-scale requires new analytic tools and we leave it for future studies.

— 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”