Global existence for the primitive equations with small anisotropic viscosity

Abstract

In this paper, we consider the primitive equations with zero vertical viscosity, zero vertical thermal diffusivity, and the horizontal viscosity and horizontal thermal diffusivity of size $\varepsilon^\alpha$ where $0 < \alpha < \alpha_0$. We prove the global existence of a unique strong solution for large data provided that the Rossby number is small enough (the rotation and the vertical stratification are large).