Global well-posedness of z-weak solutions to the primitive equations without vertical diffusivity

Abstract

In this paper, we consider the initial boundary value problem in a cylindrical domain to the three-dimensional primitive equations with full eddy viscosity in momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of a z-weak solution is established for any such initial datum such that itself and its vertical derivative belong to L2. This not only extends the results in the work of Cao, Li, and Titi [Physica D 412, 132606 (2020)] from the spatially periodic case to general cylindrical domains but also weakens regularity assumptions on the initial data, which are required to be H2 there.