Global Well-Posedness and Persistence of H2 Regularity for the Three-Dimensional Primitive Equations with Horizontal and Topography-Adapted Thermal Diffusion

Abstract

We study the three-dimensional primitive equations with full viscosity and horizontal thermal diffusion in bounded domains. We establish global well-posedness for admissible $H^1$ initial data and persistence of $H^2$ regularity on the flat-bottom cylinder. The higher regularity analysis combines fractional hydrostatic Stokes regularity with anisotropic estimates of the vertical transport. We also consider variable-depth domains with positive $h\in C^3(\overline{M})$ and a diffusion operator adapted to the topography compatible with layers $z/h(x,y)=\mathrm{constant}$. This operator reduces to ${\Delta }_H$ at constant depth and generally differs from the Cartesian horizontal Laplacian over a varying bottom. Its weighted formulation yields global strong solutions for admissible $H^1$ data. Persistence of $H^2$ regularity is proved under the additional assumption ${\partial }_{\nu }h=0$ on $\partial M$, where $\nu$ is the outward horizontal unit normal.

arXiv comment66 pages