On $H^2$ solutions and $z$-weak solutions of the 3D Primitive Equations

Global in time well-posedness of $H^2$ solutions and $z$-weak solutions of the 3D Primitive equations in a bounded cylindrical domain is proved. More specifically, uniform in time boundedness and bounded absorbing sets are obtained for both $H^2$ solutions and $z$-weak solutions, as well as uniqueness of the $z$-weak solution for the 3D Primitive equations. The result for $H^2$ solutions improves a recent one proved in[6]. The result for $z$-weak solution positively resolves the problem of global existence and uniqueness of $z$-weak solutions of the 3D primitive equations, which has been open since the work of [14].