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

TL;DR

This paper proves the global existence and uniqueness of $z$-weak solutions to the three-dimensional primitive equations in cylindrical domains, even with limited vertical diffusivity, extending previous results to more general settings.

Contribution

It extends the well-posedness results for primitive equations from periodic to cylindrical domains and relaxes initial data regularity requirements.

Findings

Establishes global well-posedness of $z$-weak solutions in cylindrical domains.

Extends previous results from periodic to cylindrical geometries.

Reduces regularity assumptions on initial data.

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