Uniqueness of weak solutions to the primitive equations in some anisotropic spaces

We consider the both 3D and 2D viscous primitive equations for ocean in the isothermal setting. While strong global well-posedness of the viscous primitive equations for large data in $H^1$ has already proved, the uniqueness of the weak solutions of Leray–Hope type for given initial data in $L^2$ remains an outstanding open problem. In this paper, we establish a new conditional uniqueness result for weak solutions to the primitive equations, that is, if a weak solution belongs some scaling invariant function spaces, and satisfies some additional assumptions, then the weak solution is unique. In particular, our result can be obtained as different one from z-weak solutions framework by adopting some anisotropic approaches with the homogeneous toroidal Besov spaces. As an application of the proof, we establish the energy equality for weak solutions in the uniqueness class given in the main theorem.