On the uniqueness of weak solutions of the two-dimensional primitive equations

Abstract

The uniqueness of weak solutions of the primitive equations with Dirichlet boundary conditions at the bottom is an open problem even in the two dimensional case. The aim of this paper is to prove the uniqueness of weak solutions when we replace the Dirichlet boundary condition at the bottom by a friction condition. With this boundary condition at the bottom, we establish an additional regularity result for the vertical derivative of the horizontal velocity which allows us to conclude the uniqueness of weak solutions.