Global well-posedness of<i>z</i>-weak solutions to the primitive equations without vertical diffusivity

In this paper, we consider the initial boundary value problem in a cylindrical domain to three dimensional primitive equations with full eddy viscosity momentum but only horizontal diffusivity temperature equation. Global well-posedness of $z$-weak solution is established for any such datum that itself and its vertical derivative belong $L^2$. This not extends results \cite{Cao5} from spatially periodic case general domains also weakens regularity assumptions on data which are required be $H^2$ there.