In this work, we study the well-posedness of the primitive equations for the ocean and the atmosphere on two specific domains: a bounded domain $\Omega_1\mathrel{ \mathop:}=(-1,1)^3$ with periodic boundary conditions, and the strip $\Omega_2\mathrel{ \mathop:}=\mathbb{ R}^2\times(-1,1)$ with periodic boundary conditions in the vertical direction. In a first time, we establish a global existence and uniqueness theorem for small initial data in a suitable anisotropic Besov space. Then, we also justify, in a similar functional framework, the singular limit from the anisotropic Navier-Stokes equations to this system.
Categories: math.AP