Uniqueness of weak solutions for the primitive equations and CAO-system

Establish uniqueness of weak solutions for the primitive equations and for the coupled atmosphere-ocean (CAO) system with nonlinear wind-driven interface conditions.

Background

The paper recalls that Lions, Temam, and Wang established existence of weak solutions for both the primitive equations and the CAO-system. It explicitly states that uniqueness remains unresolved for both systems, independently of the paper’s focus on long-term behavior of globally defined strong solutions.

References

Lions, Temam and Wang proved in their pioneering work the existence of a weak solution to the primitive equations and to the CAO-system ; its uniqueness remains an open problem for both systems until today.

— Long-term behaviour of the CAO-system: Optimal polynomial $H^1$-convergence to degenerate equilibria  (2609.08664 - Binz, 8 Sep 2026) in Section 1, Introduction

Unlike the three-dimensional Navier--Stokes equations, the three-dimensional viscous primitive equations are globally well-posed at the strong level , although uniqueness of Leray--Hopf weak solutions remains open .

— Spectral gap for the stochastic primitive equations under degenerate Brownian forcing  (2610.05814 - Lin et al., 5 Oct 2026) in Section 1, Introduction

However, the uniqueness of weak solutions, in both two and three dimensions, has been a long standing open problem since then.

— Uniqueness of Leray-Hopf weak solutions to the two-dimensional viscous Primitive Equations  (2610.11339 - Li et al., 8 Oct 2026) in Introduction, first paragraph discussing prior mathematical results