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

TL;DR

This paper proves a new conditional uniqueness result for weak solutions to the primitive equations in anisotropic function spaces, using homogeneous toroidal Besov spaces, and establishes energy equality within this class.

Contribution

It introduces a novel anisotropic approach to prove uniqueness of weak solutions in scaling invariant spaces for the primitive equations.

Findings

Uniqueness of weak solutions under specific anisotropic conditions

Energy equality holds for solutions in the uniqueness class

Different from existing $z$-weak solutions framework

Abstract

We consider the 3D or 2D primitive equations for oceans and atmosphere in the isothermal setting. 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.